FORMALIZED MATHEMATICS Volume 12, Number 3, 2004 University of Białystok

Complex Valued Functions Space

Noboru Endou Gifu National College of Technology

Summary. This article is an extension of [9] to complex valued functions.

MML Identifier: CFUNCDOM.

The articles [14], [5], [16], [10], [17], [3], [4], [1], [12], [11], [15], [2], [8], [13], [9], [7], and [6] provide the notation and terminology for this paper.

1. Operation of Complex Functions

We adopt the following convention: x_{1}, x_{2}, z are sets, A is a non empty set, and f, g, h are elements of ℂ^{A}.

Let us consider A. The functor +_{ℂ^{A}} yielding a binary operation on ℂ^{A} is defined by:

(Def. 1) For all elements f, g of ℂ^{A} holds +_{ℂ^{A}}(f, g) = (+_{ℂ})^{∘}(f, g).

Let us consider A. The functor ·_{ℂ^{A}} yielding a binary operation on ℂ^{A} is defined as follows:

(Def. 2) For all elements f, g of ℂ^{A} holds ·_{ℂ^{A}}(f, g) = (·_{ℂ})^{∘}(f, g).

Let us consider A. The functor ·_{ℂ^{A}}^{ℂ} yielding a function from [: ℂ, ℂ^{A}:] into ℂ^{A} is defined by:

(Def. 3) For every complex number z and for every element f of ℂ^{A} and for every

element x of A holds ·_{ℂ^{A}}^{ℂ}(⟨⟨z, f⟩⟩)(x) = z · f(x). Let us consider A. The functor 0_{ℂ^{A}} yielding an element of ℂ^{A} is defined by:

(Def. 4) 0_{ℂ^{A}} = A ⟼→ 0_{ℂ}.

Let us consider A. The functor 1_{ℂ^{A}} yields an element of ℂ^{A} and is defined by:

(Def. 5) 1_{ℂ^{A}} = A ⟼→ 1_{ℂ}.

One can prove the following propositions:

(1) h = +_{ℂ^{A}}(f, g) iff for every element x of A holds h(x) = f(x) + g(x).

(2) h = ·_{ℂ^{A}}(f, g) iff for every element x of A holds h(x) = f(x) · g(x).

(3) For every element x of A holds 1_{ℂ^{A}}(x) = 1_{ℂ}.

(4) For every element x of A holds 0_{ℂ^{A}}(x) = 0_{ℂ}.

(5) 0_{ℂ^{A}} ≠ 1_{ℂ^{A}}.

In the sequel a, b denote complex numbers.

The following proposition is true

(6) h = ·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩) iff for every element x of A holds h(x) = a · f(x).

In the sequel u, v, w are vectors of ⟨ℂ^{A}, 0_{ℂ^{A}}, +_{ℂ^{A}}, ·_{ℂ^{A}}^{ℂ}⟩.

One can prove the following propositions:

(7) +_{ℂ^{A}}(f, g) = +_{ℂ^{A}}(g, f).

(8) +_{ℂ^{A}}(f, +_{ℂ^{A}}(g, h)) = +_{ℂ^{A}}(+_{ℂ^{A}}(f, g), h).

(9) ·_{ℂ^{A}}(f, g) = ·_{ℂ^{A}}(g, f).

(10) ·_{ℂ^{A}}(f, ·_{ℂ^{A}}(g, h)) = ·_{ℂ^{A}}(·_{ℂ^{A}}(f, g), h).

(11) ·_{ℂ^{A}}(1_{ℂ^{A}}, f) = f.

(12) +_{ℂ^{A}}(0_{ℂ^{A}}, f) = f.

(13) +_{ℂ^{A}}(f, ·_{ℂ^{A}}^{ℂ}(⟨⟨− 1_{ℂ}, f⟩⟩)) = 0_{ℂ^{A}}.

(14) ·_{ℂ^{A}}^{ℂ}(⟨⟨1_{ℂ}, f⟩⟩) = f.

(15) ·_{ℂ^{A}}^{ℂ}(⟨⟨a, ·_{ℂ^{A}}^{ℂ}(⟨⟨b, f⟩⟩)⟩⟩) = ·_{ℂ^{A}}^{ℂ}(⟨⟨a · b, f⟩⟩).

(16) +_{ℂ^{A}}(·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩), ·_{ℂ^{A}}^{ℂ}(⟨⟨b, f⟩⟩)) = ·_{ℂ^{A}}^{ℂ}(⟨⟨a + b, f⟩⟩).

(17) ·_{ℂ^{A}}(f, +_{ℂ^{A}}(g, h)) = +_{ℂ^{A}}(·_{ℂ^{A}}(f, g), ·_{ℂ^{A}}(f, h)).

(18) ·_{ℂ^{A}}(·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩), g) = ·_{ℂ^{A}}^{ℂ}(⟨⟨a, ·_{ℂ^{A}}(f, g)⟩⟩).

2. Complex Linear Space of Complex Valued Functions

One can prove the following propositions:

(19) There exist f, g such that

(i) for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1_{ℂ} and if z ≠ x_{1}, then f(z) = 0_{ℂ}, and

(ii) for every z such that z ∈ A holds if z = x_{1}, then g(z) = 0_{ℂ} and if

z ≠ x_{1}, then g(z) = 1_{ℂ}.

(20) Suppose that (i) x_{1} ∈ A, (ii) x_{2} ∈ A, (iii) x_{1} ≠ x_{2}, (iv) for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1_{ℂ} and if

z ≠ x_{1}, then f(z) = 0_{ℂ}, and (v) for every z such that z ∈ A holds if z = x_{1}, then g(z) = 0_{ℂ} and if

z ≠ x_{1}, then g(z) = 1_{ℂ}. Let given a, b. If +_{ℂ^{A}}(·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩), ·_{ℂ^{A}}^{ℂ}(⟨⟨b, g⟩⟩)) = 0_{ℂ^{A}}, then a = 0_{ℂ} and b = 0_{ℂ}.

(21) If x_{1} ∈ A and x_{2} ∈ A and x_{1} ≠ x_{2}, then there exist f, g such that for all a, b such that +_{ℂ^{A}}(·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩), ·_{ℂ^{A}}^{ℂ}(⟨⟨b, g⟩⟩)) = 0_{ℂ^{A}} holds a = 0_{ℂ} and b = 0_{ℂ}.

(22) Suppose that (i) A = {x_{1}, x_{2}}, (ii) x_{1} ≠ x_{2}, (iii) for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1_{ℂ} and if

z ≠ x_{1}, then f(z) = 0_{ℂ}, and (iv) for every z such that z ∈ A holds if z = x_{1}, then g(z) = 0_{ℂ} and if

z ≠ x_{1}, then g(z) = 1_{ℂ}. Let given h. Then there exist a, b such that h = +_{ℂ^{A}}(·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩), ·_{ℂ^{A}}^{ℂ}(⟨⟨b, g⟩⟩)).

(23) If A = {x_{1}, x_{2}} and x_{1} ≠ x_{2}, then there exist f, g such that for every h there exist a, b such that h = +_{ℂ^{A}}(·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩), ·_{ℂ^{A}}^{ℂ}(⟨⟨b, g⟩⟩)).

(24) Suppose A = {x_{1}, x_{2}} and x_{1} ≠ x_{2}. Then there exist f, g such that for all a, b such that +_{ℂ^{A}}(·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩), ·_{ℂ^{A}}^{ℂ}(⟨⟨b, g⟩⟩)) = 0_{ℂ^{A}} holds a = 0_{ℂ} and b = 0_{ℂ} and for every h there exist a, b such that h = +_{ℂ^{A}}(·_{ℂ^{A}}^{ℂ}(⟨⟨a, f⟩⟩), ·_{ℂ^{A}}^{ℂ}(⟨⟨b, g⟩⟩)).

(25) ⟨ℂ^{A}, 0_{ℂ^{A}}, +_{ℂ^{A}}, ·_{ℂ^{A}}^{ℂ}⟩ is a complex linear space.

Let us consider A. The functor ComplexVectSpace(A) yields a strict complex linear space and is defined by:

(Def. 6) ComplexVectSpace(A) = ⟨ℂ^{A}, 0_{ℂ^{A}}, +_{ℂ^{A}}, ·_{ℂ^{A}}^{ℂ}⟩.

We now state the proposition

(26) There exists a strict complex linear space V and there exist vectors u, v of V such that for all a, b such that a · u + b · v = 0_{V} holds a = 0_{ℂ} and b = 0_{ℂ} and for every vector w of V there exist a, b such that w = a·u+b·v.

Let us consider A. The functor CRing(A) yielding a strict double loop structure is defined by:

(Def. 7) CRing(A) = ⟨ℂ^{A}, +_{ℂ^{A}}, ·_{ℂ^{A}}, 1_{ℂ^{A}}, 0_{ℂ^{A}}⟩. Let us consider A. Observe that CRing(A) is non empty. We now state two propositions:

(27) Let x, y, z be elements of CRing(A). Then x+y = y+x and (x+y)+z = x+(y+z) and x+0_{CRing(A)} = x and there exists an element t of CRing(A) such that x + t = 0_{CRing(A)} and x · y = y · x and (x · y) · z = x · (y · z) and x · 1_{CRing(A)} = x and 1_{CRing(A)} · x = x and x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x.

(28) CRing(A) is a commutative ring.

We introduce complex algebra structures which are extensions of double loop structure and CLS structure and are systems

⟨ a carrier, a multiplication, an addition, an external multiplication, a unity, a zero ⟩, where the carrier is a set, the multiplication and the addition are binary operations on the carrier, the external multiplication is a function from [: ℂ, the carrier :] into the carrier, and the unity and the zero are elements of the carrier.

Let us mention that there exists a complex algebra structure which is non empty.

Let us consider A. The functor CAlgebra(A) yielding a strict complex algebra structure is defined as follows:

(Def. 8) CAlgebra(A) = ⟨ℂ^{A}, ·_{ℂ^{A}}, +_{ℂ^{A}}, ·_{ℂ^{A}}^{ℂ}, 1_{ℂ^{A}}, 0_{ℂ^{A}}⟩.

Let us consider A. Observe that CAlgebra(A) is non empty.

Next we state the proposition

(29) Let x, y, z be elements of CAlgebra(A) and given a, b. Then x+y = y+x and (x + y) + z = x + (y + z) and x + 0_{CAlgebra (A)} = x and there exists an element t of CAlgebra(A) such that x + t = 0_{CAlgebra (A)} and x · y = y · x and (x·y)·z = x·(y ·z) and x·1_{CAlgebra (A)} = x and x·(y +z) = x·y +x·z and a·(x·y) = (a·x)·y and a·(x+y) = a·x+a·y and (a+b)·x = a·x+b·x and (a · b) · x = a · (b · x).

Let I_{1} be a non empty complex algebra structure. We say that I_{1} is complex algebra-like if and only if the condition (Def. 9) is satisfied.

(Def. 9) Let x, y, z be elements of I_{1} and given a, b. Then x · 1_{(I_{1})} = x and x · (y + z) = x · y + x · z and a · (x · y) = (a · x) · y and a · (x + y) = a · x + a · y and (a + b) · x = a · x + b · x and (a · b) · x = a · (b · x).

Let us note that there exists a non empty complex algebra structure which is strict, Abelian, add-associative, right zeroed, right complementable, commutative, associative, and complex algebra-like.

A complex algebra is an Abelian add-associative right zeroed right complementable commutative associative complex algebra-like non empty complex algebra structure.

One can prove the following proposition

(30) CAlgebra(A) is a complex algebra.

References

[1] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[2] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164,

complex valued functions space 235

[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[6] Czesław Byliński and Andrzej Trybulec. Complex spaces. Formalized Mathematics, 2(1):151–158, 1991.

[7] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[8] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[9] Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555–561, 1990.

[10] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[11] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[12] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[13] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

[14] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[15] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[16] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[17] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Banach Algebra of Bounded Complex Linear Operators

Noboru Endou Gifu National College of Technology

Summary. This article is an extension of [16].

MML Identifier: CLOPBAN2.

The terminology and notation used here are introduced in the following articles: [18], [8], [20], [5], [7], [6], [3], [1], [17], [13], [19], [14], [2], [4], [15], [10], [11], [9], and [12].

One can prove the following propositions:

(1) Let X, Y, Z be complex linear spaces, f be a linear operator from X into Y, and g be a linear operator from Y into Z. Then g · f is a linear operator from X into Z.

(2) Let X, Y, Z be complex normed spaces, f be a bounded linear operator from X into Y, and g be a bounded linear operator from Y into Z. Then

(i) g · f is a bounded linear operator from X into Z, and (ii) for every vector x of X holds ∥(g ·f)(x)∥ (BdLinOpsNorm(Y, Z))(g)·

(BdLinOpsNorm(X, Y))(f) · ∥x∥ and (BdLinOpsNorm(X, Z))(g · f) (BdLinOpsNorm(Y, Z))(g) · (BdLinOpsNorm(X, Y))(f).

Let X be a complex normed space and let f, g be bounded linear operators from X into X. Then g · f is a bounded linear operator from X into X.

Let X be a complex normed space and let f, g be elements of BdLinOps(X, X). The functor f + g yields an element of BdLinOps(X, X) and is defined by:

(Def. 1) f + g = (Add (BdLinOps(X, X), CVSpLinOps(X, X)))(f, g).

Let X be a complex normed space and let f, g be elements of BdLinOps(X, X). The functor g · f yields an element of BdLinOps(X, X) and is defined as follows:

(Def. 2) g · f = modetrans(g, X, X) · modetrans(f, X, X).

Let X be a complex normed space, let f be an element of BdLinOps(X, X), and let z be a complex number. The functor z · f yields an element of BdLinOps(X, X) and is defined by:

(Def. 3) z · f = (Mult (BdLinOps(X, X), CVSpLinOps(X, X)))(z, f).

Let X be a complex normed space. The functor FuncMult(X) yields a binary operation on BdLinOps(X, X) and is defined as follows:

(Def. 4) For all elements f, g of BdLinOps(X, X) holds (FuncMult(X))(f, g) =

f · g. The following proposition is true

(3) For every complex normed space X holds id_{the carrier of X} is a bounded linear operator from X into X.

Let X be a complex normed space. The functor FuncUnit(X) yielding an element of BdLinOps(X, X) is defined by:

(Def. 5) FuncUnit(X) = id_{the carrier of X}.

The following propositions are true:

(4) Let X be a complex normed space and f, g, h be bounded linear operators from X into X. Then h = f · g if and only if for every vector x of X holds h(x) = f(g(x)).

(5) For every complex normed space X and for all bounded linear operators f, g, h from X into X holds f · (g · h) = (f · g) · h.

(6) Let X be a complex normed space and f be a bounded linear operator from X into X. Then f · id_{the carrier of X} = f and id_{the carrier of X} · f = f.

(7) For every complex normed space X and for all elements f, g, h of BdLinOps(X, X) holds f · (g · h) = (f · g) · h.

(8) For every complex normed space X and for every element f of BdLinOps(X, X) holds f · FuncUnit(X) = f and FuncUnit(X) · f = f.

(9) For every complex normed space X and for all elements f, g, h of BdLinOps(X, X) holds f · (g + h) = f · g + f · h.

(10) For every complex normed space X and for all elements f, g, h of BdLinOps(X, X) holds (g + h) · f = g · f + h · f.

(11) Let X be a complex normed space, f, g be elements of BdLinOps(X, X), and a, b be complex numbers. Then (a · b) · (f · g) = a · f · (b · g).

(12) Let X be a complex normed space, f, g be elements of BdLinOps(X, X),

and a be a complex number. Then a · (f · g) = (a · f) · g. Let X be a complex normed space. The functor RingOfBoundedLinearOperators(X) yields a double loop struc-

ture and is defined by: (Def. 6) RingOfBoundedLinearOperators(X) = ⟨BdLinOps(X, X), Add (BdLinOps(X, X), CVSpLinOps(X, X)), FuncMult(X), FuncUnit(X), Zero (BdLinOps(X, X), CVSpLinOps(X, X))⟩.

Let X be a complex normed space. Note that RingOfBoundedLinearOperators(X) is non empty and strict. Next we state two propositions:

(13) Let X be a complex normed space and x, y, z be elements of RingOfBoundedLinearOperators(X). Then x + y = y + x and (x +

y) + z = x + (y + z) and x + 0_{RingOfBoundedLinearOperators (X)} = x and there exists an element t of RingOfBoundedLinearOperators(X) such that x + t = 0_{RingOfBoundedLinearOperators (X)} and (x · y) · z = x · (y · z) and x · 1_{RingOfBoundedLinearOperators (X)} = x and 1_{RingOfBoundedLinearOperators (X)} · x = x and x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x.

(14) For every complex normed space X holds

RingOfBoundedLinearOperators(X) is a ring. Let X be a complex normed space. Observe that RingOfBoundedLinearOperators(X) is Abelian, add-associative,

right zeroed, right complementable, associative, left unital, right unital, and distributive.

Let X be a complex normed space. The functor CAlgBdLinOps(X) yields a complex algebra structure and is defined by:

(Def. 7) CAlgBdLinOps(X) = ⟨BdLinOps(X, X), FuncMult(X), Add (BdLinOps (X, X), CVSpLinOps(X, X)), Mult (BdLinOps(X, X), CVSpLinOps(X, X)), FuncUnit(X), Zero (BdLinOps(X, X), CVSpLinOps(X, X))⟩.

Let X be a complex normed space. Note that CAlgBdLinOps(X) is non empty and strict.

The following proposition is true

(15) Let X be a complex normed space, x, y, z be elements of CAlgBdLinOps(X), and a, b be complex numbers. Then x+y = y +x and (x + y) + z = x + (y + z) and x + 0_{CAlgBdLinOps (X)} = x and there exists an element t of CAlgBdLinOps(X) such that x + t = 0_{CAlgBdLinOps (X)} and (x·y)·z = x·(y·z) and x·1_{CAlgBdLinOps (X)} = x and 1_{CAlgBdLinOps (X)}·x = x and x·(y+z) = x·y+x·z and (y+z)·x = y·x+z·x and a·(x·y) = (a·x)·y and a·(x+y) = a·x+a·y and (a+b)·x = a·x+b·x and (a·b)·x = a·(b·x) and (a · b) · (x · y) = a · x · (b · y).

A complex BL algebra is an Abelian add-associative right zeroed right complementable associative complex algebra-like non empty complex algebra structure.

We now state the proposition

(16) For every complex normed space X holds CAlgBdLinOps(X) is a complex BL algebra.

Let us note that Complex-l1-Space is complete. Let us mention that Complex-l1-Space is non trivial. Let us note that there exists a complex Banach space which is non trivial. The following two propositions are true:

(17) For every non trivial complex normed space X there exists a vector w of X such that ∥w∥ = 1.

(18) For every non trivial complex normed space X holds

(BdLinOpsNorm(X, X))(id_{the carrier of X}) = 1. We introduce normed complex algebra structures which are extensions of

complex algebra structure and complex normed space structure and are systems

⟨ a carrier, a multiplication, an addition, an external multiplication, a unity, a zero, a norm ⟩, where the carrier is a set, the multiplication and the addition are binary operations on the carrier, the external multiplication is a function from [: ℂ, the carrier :] into the carrier, the unity and the zero are elements of the carrier, and the norm is a function from the carrier into ℝ.

One can check that there exists a normed complex algebra structure which is non empty.

Let X be a complex normed space. The functor CNAlgBdLinOps(X) yields a normed complex algebra structure and is defined by:

(Def. 8) CNAlgBdLinOps(X) = ⟨BdLinOps(X, X), FuncMult(X), Add (BdLinOps(X, X), CVSpLinOps(X, X)), Mult (BdLinOps(X, X), CVSpLinOps(X, X)), FuncUnit(X), Zero (BdLinOps(X, X), CVSpLinOps(X, X)), BdLinOpsNorm(X, X)⟩.

Let X be a complex normed space. Note that CNAlgBdLinOps(X) is non empty and strict.

The following propositions are true:

(19) Let X be a complex normed space, x, y, z be elements of CNAlgBdLinOps(X), and a, b be complex numbers. Then x+y = y+x and (x+y)+z = x+(y +z) and x+0_{CNAlgBdLinOps (X)} = x and there exists an element t of CNAlgBdLinOps(X) such that x + t = 0_{CNAlgBdLinOps (X)} and (x·y)·z = x·(y·z) and x·1_{CNAlgBdLinOps (X)} = x and 1_{CNAlgBdLinOps (X)}·x = x and x·(y+z) = x·y+x·z and (y+z)·x = y·x+z·x and a·(x·y) = (a·x)·y and (a·b)·(x·y) = a·x·(b·y) and a·(x+y) = a·x+a·y and (a+b)·x = a·x+b·x and (a · b) · x = a · (b · x) and 1_{ℂ} · x = x.

(20) Let X be a complex normed space. Then CNAlgBdLinOps(X) is complex normed space-like, Abelian, add-associative, right zeroed, right complementable, associative, complex algebra-like, and complex linear spacelike.

Let us observe that there exists a non empty normed complex algebra structure which is complex normed space-like, Abelian, add-associative, right zeroed, right complementable, associative, complex algebra-like, complex linear spacelike, and strict.

A normed complex algebra is a complex normed space-like Abelian addassociative right zeroed right complementable associative complex algebra-like complex linear space-like non empty normed complex algebra structure.

Let X be a complex normed space. One can check that CNAlgBdLinOps(X) is complex normed space-like, Abelian, add-associative, right zeroed, right complementable, associative, complex algebra-like, and complex linear space-like.

Let X be a non empty normed complex algebra structure. We say that X is Banach Algebra-like1 if and only if:

(Def. 9) For all elements x, y of X holds ∥x · y∥ ∥x∥ · ∥y∥.

We say that X is Banach Algebra-like2 if and only if: (Def. 10) ∥1_{X}∥ = 1.

We say that X is Banach Algebra-like3 if and only if: (Def. 11) For every complex number a and for all elements x, y of X holds a · (x ·

y) = x · (a · y). Let X be a normed complex algebra. We say that X is Banach Algebra-like

if and only if the condition (Def. 12) is satisfied. (Def. 12) X is Banach Algebra-like1, Banach Algebra-like2, Banach Algebra-like3,

left unital, left distributive, and complete. One can verify that every normed complex algebra which is Banach Algebra-

like is also Banach Algebra-like1, Banach Algebra-like2, Banach Algebra-like3, left distributive, left unital, and complete and every normed complex algebra which is Banach Algebra-like1, Banach Algebra-like2, Banach Algebra-like3, left distributive, left unital, and complete is also Banach Algebra-like.

Let X be a non trivial complex Banach space. One can verify that CNAlgBdLinOps(X) is Banach Algebra-like.

One can check that there exists a normed complex algebra which is Banach Algebra-like.

A complex Banach algebra is a Banach Algebra-like normed complex algebra.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

242 noboru endou

[8] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, [9] Noboru Endou. Complex Banach space of bounded linear operators. Formalized Mathe-

matics, 12(2):201–209, 2004.

[10] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[11] Noboru Endou. Complex linear space of complex sequences. Formalized Mathematics, 12(2):109–117, 2004.

[12] Noboru Endou. Complex valued functions space. Formalized Mathematics, 12(3):231–235, 2004.

[13] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[14] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[15] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[16] Yasunari Shidama. The Banach algebra of bounded linear operators. Formalized Mathematics, 12(2):103–108, 2004.

[17] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[19] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

Formulas and Identities of Trigonometric Functions

Yuzhong Ding QingDao University of Science and Technology

Xiquan Liang QingDao University of Science and Technology

MML Identifier: SIN COS5.

The articles [2], [5], [1], [6], [3], and [4] provide the terminology and notation for this paper.

In this paper t_{1}, t_{2}, t_{3}, t_{4} are real numbers. One can prove the following propositions:

(1) If cost_{1} ≠ 0, then cosect_{1} = \begin{matrix}sec & t_{1} \\ tan & t_{1} \end{matrix} .

(2) If sint_{1} ≠ 0, then cost_{1} = sint_{1} · cott_{1}.

(3) If sint_{2} ≠ 0 and sint_{3} ≠ 0 and sint_{4} ≠ 0, then sin(t_{2} + t_{3} + t_{4}) = sint_{2} · sint_{3} · sint_{4} · ((cott_{3} · cott_{4} + cott_{2} · cott_{4} + cott_{2} · cott_{3}) − 1).

(4) If sint_{2} ≠ 0 and sint_{3} ≠ 0 and sint_{4} ≠ 0, then cos(t_{2} + t_{3} + t_{4}) = − sint_{2} · sint_{3} · sint_{4} · ((cott_{2} + cott_{3} + cott_{4}) − cott_{2} · cott_{3} · cott_{4}) .

(5) sin(2 · t_{1}) = 2 · sint_{1} · cost_{1}.

(6) If cost_{1} ≠ 0, then sin(2 · t_{1}) = \frac{2·tan t_{1}}{1+(tan t_{1})^{2}} .

(7) cos(2 · t_{1}) = (cost_{1})^{2} − (sint_{1})^{2} and cos(2 · t_{1}) = 2 · (cost_{1})^{2} − 1 and cos(2 · t_{1}) = 1 − 2 · (sint_{1})^{2}.

(8) If cost_{1} ≠ 0, then cos(2 · t_{1}) = \frac{1−(tan t_{1})^{2}}{1+(tan t_{1})^{2}} .

(9) If cost_{1} ≠ 0, then tan(2 · t_{1}) = \frac{2·tan t_{1}}{1−(tan t_{1})^{2}} .

(10) If sint_{1} ≠ 0, then cot(2 · t_{1}) = \frac{(cot t_{1})^{2} −1}{2·cot t_{1}} .

(11) If cost_{1} ≠ 0, then (sect_{1})^{2} = 1 + (tant_{1})^{2}.

(12) cott_{1} = \frac{1}{tan t_{1}} .

(13) If cost_{1} ≠ 0 and sint_{1} ≠ 0, then sec(2 · t_{1}) = \frac{(sec t_{1})^{2}}{1−(tan t_{1})^{2}} and sec(2 · t_{1}) = \frac{cot t_{1}+tan t_{1}}{cot t_{1}−tan t_{1}}.

(14) If sint_{1} ≠ 0, then (cosect_{1})^{2} = 1 + (cott_{1})^{2}.

(15) If cost_{1} ≠ 0 and sint_{1} ≠ 0, then cosec(2 · t_{1}) = \frac{sec t_{1}·cosec t_{1}}{2} and cosec(2 · t_{1}) = \frac{tan t_{1}+cot t_{1}}{2} .

(16) sin(3 · t_{1}) = − 4 · (sint_{1})^{3}+ 3 · sint_{1}.

(17) cos(3 · t_{1}) = 4 · (cost_{1})^{3} − 3 · cost_{1}.

(18) If cost_{1} ≠ 0, then tan(3 · t_{1}) = \frac{3·tan t_{1}−(tan t_{1})^{3}}{1−3·(tan t_{1})^{2}} .

(19) If sint_{1} ≠ 0, then cot(3 · t_{1}) = \frac{(cot t_{1})^{3}−3·cot t_{1}}{3·(cot t_{1})^{2} −1} .

(20) (sint_{1})^{2} = \frac{1−cos(2·t_{1})}{2} .

(21) (cost_{1})^{2} = \frac{1+cos(2·t_{1})}{2}.

(22) (sint_{1})^{3} = \frac{3·sin t_{1}−sin(3·t_{1})}{4}.

(23) (cost_{1})^{3} = \frac{3·cos t_{1}+cos(3·t_{1})}{4}.

(24) (sint_{1})^{4} = \frac{(3−4·cos(2·t_{1}))+cos(4·t_{1})}{8}.

(25) (cost_{1})^{4} = \frac{3+4·cos(2·t_{1})+cos(4·t_{1})}{8}.

(26) sin(\frac{t_{1}}{2} ) = \sqrt{\frac{1−cos t_{1}}{2}} or sin(\frac{t_{1}}{2} ) = − \sqrt{\frac{1−cos t_{1}}{2}} .

(27) cos(\frac{t_{1}}{2} ) = \sqrt{\frac{1+cos t_{1}}{2}} or cos(\frac{t_{1}}{2} ) = − \sqrt{\frac{1+cos t_{1}}{2}} .

(28) If sin(\frac{t_{1}}{2} ) ≠ 0, then tan(\frac{t_{1}}{2} ) = \frac{1−cos t_{1}}{sin t_{1}} .

(29) If cos(\frac{t_{1}}{2} ) ≠ 0, then tan(\frac{t_{1}}{2} ) = \frac{sin t_{1}}{1+cos t_{1}} .

(30) tan(\frac{t_{1}}{2} ) = \sqrt{\frac{1−cos t_{1}}{1+cos t_{1}}} or tan(\frac{t_{1}}{2} ) = − \sqrt{\frac{1−cos t_{1}}{1+cos t_{1}}} .

(31) If cos(\frac{t_{1}}{2} ) ≠ 0, then cot(\frac{t_{1}}{2} ) = \frac{1+cos t_{1}}{sin t_{1}} .

(32) If sin(\frac{t_{1}}{2} ) ≠ 0, then cot(\frac{t_{1}}{2} ) = \frac{sin t_{1}}{1−cos t_{1}} .

(33) cot(\frac{t_{1}}{2} ) = \sqrt{\frac{1+cos t_{1}}{1−cos t_{1}}} or cot(\frac{t_{1}}{2} ) = − \sqrt{\frac{1+cos t_{1}}{1−cos t_{1}}}.

(34) If sin(\frac{t_{1}}{2} ) ≠ 0 and cos(\frac{t_{1}}{2} ) ≠ 0 and 1 − (tan(\frac{t_{1}}{2} ))^{2} ≠ 0, then sec(\frac{t_{1}}{2} ) = \sqrt{\frac{2·sec t_{1}}{sec t_{1}+1}} or sec(\frac{t_{1}}{2} ) = − \sqrt{\frac{2·sec t_{1}}{sec t_{1}+1}}.

(35) If sin(\frac{t_{1}}{2} ) ≠ 0 and cos(\frac{t_{1}}{2} ) ≠ 0 and 1 − (tan(\frac{t_{1}}{2} ))^{2} ≠ 0, then cosec(\frac{t_{1}}{2} ) =

\sqrt{\frac{2·sec t_{1}}{sec t_{1}−1}} or cosec(\frac{t_{1}}{2} ) = − \sqrt{\frac{2·sec t_{1}}{sec t_{1}−1}}. Let us consider t_{1}. The functor cotht_{1} yielding a real number is defined as

follows: (Def. 1) cotht_{1} = cosh t_{1} ; sinh t_{1} .

Let us consider t_{1}. The functor secht_{1} yielding a real number is defined by: (Def. 2) secht_{1} = \frac{1}{cosh t_{1}} .

Let us consider t_{1}. The functor cosecht_{1} yields a real number and is defined as follows:

(Def. 3) cosecht_{1} = \frac{1}{sinh t_{1}} .

We now state a number of propositions:

(36) cotht_{1} = \frac{exp t_{1}+exp(− t_{1})}{exp t_{1}−exp(− t_{1})} and secht_{1} = \frac{2}{exp t_{1}+exp(− t_{1})} and cosecht_{1} = \frac{2}{exp t_{1}−exp(− t_{1})} .

(37) If expt_{1} − exp(− t_{1}) ≠ 0, then tanht_{1} · cotht_{1} = 1.

(38) (secht_{1})^{2} + (tanht_{1})^{2} = 1.

(39) If sinht_{1} ≠ 0, then (cotht_{1})^{2} − (cosecht_{1})^{2} = 1.

(40) If sinht_{2} ≠ 0 and sinht_{3} ≠ 0, then coth(t_{2} + t_{3}) = \frac{1+coth t_{2}·coth t_{3}}{coth t_{2}+coth t_{3}}.

(41) If sinht_{2} ≠ 0 and sinht_{3} ≠ 0, then coth(t_{2} − t_{3}) = \frac{1−coth t_{2}·coth t_{3}}{coth t_{2}−coth t_{3}} .

(42) If sinht_{2} ≠ 0 and sinht_{3} ≠ 0, then cotht_{2} + cotht_{3} = \frac{sinh(t_{2}+t_{3})}{sinh t_{2}·sinh t_{3}} and cotht_{2} − cotht_{3} = − \frac{sinh(t_{2}−t_{3})}{sinh t_{2}·sinh t_{3}}.

(43) sinh(3 · t_{1}) = 3 · sinht_{1} + 4 · (sinht_{1})^{3}.

(44) cosh(3 · t_{1}) = 4 · (cosht_{1})^{3} − 3 · cosht_{1}.

(45) If sinht_{1} ≠ 0, then coth(2 · t_{1}) = \frac{1+(coth t_{1})^{2}}{2·coth t_{1}} .

(46) If t_{1} > 0, then sinht_{1} 0.

(47) If t_{1} < 0, then sinht_{1} 0.

(48) cosh(\frac{t_{1}}{2} ) = \sqrt{\frac{cosh t_{1}+1}{2}}.

(49) If sinh(\frac{t_{1}}{2} ) ≠ 0, then tanh(\frac{t_{1}}{2} ) = \frac{cosh t_{1}−1}{sinh t_{1}} .

(50) If cosh(\frac{t_{1}}{2} ) ≠ 0, then tanh(\frac{t_{1}}{2} ) = \frac{sinh t_{1}}{cosh t_{1}+1}.

(51) If sinh(\frac{t_{1}}{2} ) ≠ 0, then coth(\frac{t_{1}}{2} ) = \frac{sinh t_{1}}{cosh t_{1}−1} .

(52) If cosh(\frac{t_{1}}{2} ) ≠ 0, then coth(\frac{t_{1}}{2} ) = \frac{cosh t_{1}+1}{sinh t_{1}}.

References

[1] Pacharapokin Chanapat, Kanchun, and Hiroshi Yamazaki. Formulas and identities of trigonometric functions. Formalized Mathematics, 12(2):139–141, 2004.

[2] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[3] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[4] Takashi Mitsuishi and Yuguang Yang. Properties of the trigonometric function. Formalized Mathematics, 8(1):103–106, 1999.

[5] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[6] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

246 yuzhong ding and xiquan liang

Solving Roots of the Special Polynomial Equation with Real Coefficients

Yuzhong Ding QingDao University of Science and Technology

Xiquan Liang QingDao University of Science and Technology

MML Identifier: POLYEQ 4.

The papers [5], [4], [2], [3], and [1] provide the terminology and notation for this paper.

We follow the rules: x, y, a, b, c, p, q are real numbers and m, n are natural numbers.

We now state a number of propositions:

(1) If a ≠ 0 and \frac{b}{a} < 0 and \frac{c}{a} > 0 and Δ(a, b, c) 0, then ^{\frac{− b+\sqrt{Δ(a,b,c)}}{2·a}} > 0

and ^{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} > 0.

(2) If a ≠ 0 and \frac{b}{a} > 0 and \frac{c}{a} > 0 and Δ(a, b, c) 0, then ^{\frac{− b+\sqrt{Δ(a,b,c)}}{2·a}} < 0

and ^{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} < 0.

(3) If a ≠ 0 and \frac{c}{a} < 0, then ^{\frac{− b+\sqrt{Δ(a,b,c)}}{2·a}} > 0 and ^{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} < 0 or

\frac{− b+\sqrt{Δ(a,b,c)}}{2·a} < 0 and ^{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} > 0.

(4) If a > 0 and there exists m such that n = 2 · m and m 1 and x^{n} = a, then x = ^{n} \sqrt{a} or x = − ^{n} \sqrt{a}.

(5) If a ≠ 0 and Poly2(a, b, 0, x) = 0, then x = 0 or x = − \frac{b}{a}.

(6) If a ≠ 0 and Poly2(a, 0, 0, x) = 0, then x = 0.

(7) If a ≠ 0 and there exists m such that n = 2·m+1 and Δ(a, b, c) 0 and

Poly2(a, b, c, x^{n}) = 0, then x = ^{n} \sqrt{\frac{− b+\sqrt{Δ(a,b,c)}}{2·a}} or x = ^{n} \sqrt{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} .

(8) Suppose a ≠ 0 and \frac{b}{a} < 0 and \frac{c}{a} > 0 and there exists m such that n = 2 · m and m 1 and Δ(a, b, c) 0 and Poly2(a, b, c, x^{n}) = 0. Then x = ^{n} \sqrt{\frac{− b+\sqrt{Δ(a,b,c)}}{2·a}} or x = − ^{n} \sqrt{\frac{− b+\sqrt{Δ(a,b,c)}}{2·a}} or x = ^{n} \sqrt{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} or x = − ^{n} \sqrt{\frac{− b−\sqrt{Δ(a,b,c)}}{2·a}} .

(9) If a ≠ 0 and there exists m such that n = 2·m+1 and Poly2(a, b, 0, x^{n}) = 0, then x = 0 or x = ^{n} \sqrt{− \frac{b}{a}}.

(10) If a ≠ 0 and \frac{b}{a} < 0 and there exists m such that n = 2 · m and m 1 and Poly2(a, b, 0, x^{n}) = 0, then x = 0 or x = ^{n} \sqrt{− \frac{b}{a}} or x = − ^{n} \sqrt{− \frac{b}{a}}.

(11) a^{3} + b^{3} = (a + b) · ((a^{2} − a · b) + b^{2}) and a^{5} + b^{5} = (a + b) · ((((a^{4} − a^{3} · b) + a^{2} · b^{2}) − a · b^{3}) + b^{4}).

(12) Suppose a ≠ 0 and b^{2} − 2 · a · b − 3 · a^{2} 0 and Poly3(a, b, b, a, x) = 0.

Then x = − 1 or x = ^{\frac{(a−b)+\sqrt{b^{2} −2·a·b−3·a^{2}}}{2·a}} or x = \frac{a−b−\sqrt{b^{2} −2·a·b−3·a^{2}}}{2·a} . Let a, b, c, d, e, f, x be real numbers. The functor Poly_{5}(a, b, c, d, e, f, x) is

defined by: (Def. 1) Poly_{5}(a, b, c, d, e, f, x) = a · x^{5} + b · x^{4} + c · x^{3} + d · x^{2} + e · x + f.

We now state a number of propositions:

(13) Suppose a ≠ 0 and (b^{2} + 2 · a · b + 5 · a^{2}) − 4 · a · c > 0 and Poly_{5}(a, b, c, c, b, a, x) = 0. Let y_{1}, y_{2} be real numbers. Suppose y_{1} = \frac{(a−b)+\sqrt{(b^{2} +2·a·b+5·a^{2} )−4·a·c}}{2·a} and y_{2} = ^{\frac{a−b−\sqrt{(b^{2} +2·a·b+5·a^{2} )−4·a·c}}{2·a}}. Then x = − 1 or x = ^{\frac{y_{1}+\sqrt{Δ(1,− y_{1},1)}}{2}} or x = ^{\frac{y_{2}+\sqrt{Δ(1,− y_{2},1)}}{2}} or x = ^{\frac{y_{1}−\sqrt{Δ(1,− y_{1},1)}}{2}} or x = ^{\frac{y_{2}−\sqrt{Δ(1,− y_{2},1)}}{2}} .

(14) Suppose x + y = p and x · y = q and p^{2} − 4 · q 0. Then x = ^{\frac{p+\sqrt{p^{2} −4·q}}{2}} and y = ^{\frac{p−\sqrt{p^{2} −4·q}}{2}} or x = ^{\frac{p−\sqrt{p^{2} −4·q}}{2}} and y = ^{\frac{p+\sqrt{p^{2} −4·q}}{2}} .

(15) Suppose x^{n} + y^{n} = p and x^{n} · y^{n} = q and p^{2} − 4 · q 0 and there exists m such that n = 2 · m + 1. Then x = ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} and y = ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} or x = ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} and y = ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} .

(16) Suppose x^{n}+y^{n} = p and x^{n}·y^{n} = q and p^{2}−4·q 0 and p > 0 and q > 0 and there exists m such that n = 2 · m and m 1. Then x = ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} and y = ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} or x = − ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} and y = ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} or x = ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} and y = − ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} or x = − ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} and y = − ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} or x = ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} and y = ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} or x = − ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} and y = ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} or x = ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} and y = − ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} or x = − ^{n} \sqrt{\frac{p−\sqrt{p^{2} −4·q}}{2}} and y = − ^{n} \sqrt{\frac{p+\sqrt{p^{2} −4·q}}{2}} .

(18)^{1} Suppose x^{n} + y^{n} = a and x^{n} − y^{n} = b and there exists m such that n = 2 · m and m 1 and a > 0 and a + b > 0 and a − b > 0. Then

(i) x = ^{n} \sqrt{\frac{a+b}{2}} and y = ^{n} \sqrt{\frac{a−b}{2}} , or

(ii) x = ^{n} \sqrt{\frac{a+b}{2}} and y = − ^{n} \sqrt{\frac{a−b}{2}} , or

(iii) x = − ^{n} \sqrt{\frac{a+b}{2}} and y = ^{n} \sqrt{\frac{a−b}{2}} , or

(iv) x = − ^{n} \sqrt{\frac{a+b}{2}} and y = − ^{n} \sqrt{\frac{a−b}{2}} .

(19) If a·x^{n} +b·y^{n} = p and x·y = 0 and there exists m such that n = 2·m+1 and a · b ≠ 0, then x = 0 and y = ^{n} \sqrt{\frac{p}{b}} or x = ^{n} \sqrt{\frac{p}{a}} and y = 0.

(20) Suppose a · x^{n} + b · y^{n} = p and x · y = 0 and there exists m such that n = 2 · m and m 1 and \frac{p}{b} > 0 and \frac{p}{a} > 0 and a · b ≠ 0. Then x = 0 and

y = ^{n} \sqrt{\frac{p}{b}} or x = 0 and y = − ^{n} \sqrt{\frac{p}{b}} or x = ^{n} \sqrt{\frac{p}{a}} and y = 0 or x = − ^{n} \sqrt{\frac{p}{a}} and y = 0.

(21) If a · x^{n} = p and x · y = q and there exists m such that n = 2 · m + 1 and p · a ≠ 0, then x = ^{n} \sqrt{\frac{p}{a}} and y = q · ^{n} \sqrt{\frac{a}{p}}.

(22) Suppose a · x^{n} = p and x · y = q and there exists m such that n = 2 · m and m 1 and \frac{p}{a} > 0 and a ≠ 0. Then x = ^{n} \sqrt{\frac{p}{a}} and y = q · ^{n} \sqrt{\frac{a}{p}} or

x = − ^{n} \sqrt{\frac{p}{a}} and y = − q · ^{n} \sqrt{\frac{a}{p}}.

(24)^{2} For all real numbers a, x such that a > 0 and a ≠ 1 and a^{x} = 1 holds x = 0.

(25) For all real numbers a, x such that a > 0 and a ≠ 1 and a^{x} = a holds x = 1.

(27)^{3} For all real numbers a, b, x such that a > 0 and a ≠ 1 and x > 0 and log_{a} x = 0 holds x = 1.

(28) For all real numbers a, b, x such that a > 0 and a ≠ 1 and x > 0 and log_{a} x = 1 holds x = a.

References

[1] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[2] Xiquan Liang. Solving roots of polynomial equations of degree 2 and 3 with real coefficients. Formalized Mathematics, 9(2):347–350, 2001.

[3] Jan Popiołek. Quadratic inequalities. Formalized Mathematics, 2(4):507–509, 1991.

[4] Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[5] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

1The proposition (17) has been removed.

2The proposition (23) has been removed.

3The proposition (26) has been removed.

250 yuzhong ding and xiquan liang Received March 18, 2004

Algebraic Properties of Homotopies

Adam Grabowski^{1} University of Białystok

Artur Korniłowicz^{2} University of Białystok

MML Identifier: BORSUK 6.

The notation and terminology used here are introduced in the following papers: [21], [9], [25], [1], [20], [14], [24], [22], [2], [5], [27], [6], [7], [18], [11], [19], [10], [17], [26], [8], [15], [23], [12], [4], [3], [16], and [13].

1. Preliminaries

The scheme ExFunc3CondD deals with a non empty set A, three unary functors F, G, and H yielding sets, and three unary predicates P, Q, R, and states that:

There exists a function f such that domf = A and for every element c of A holds if P[c], then f(c) = F(c) and if Q[c], then f(c) = G(c) and if R[c], then f(c) = H(c)

provided the parameters meet the following conditions:

• For every element c of A holds if P[c], then not Q[c] and if P[c], then not R[c] and if Q[c], then not R[c], and

• For every element c of A holds P[c] or Q[c] or R[c].

Let n be a natural number. Observe that every element of E_{T}^{n} is function-like and relation-like.

Let n be a natural number. Observe that every element of E_{T}^{n} is finite sequence-like.

We now state a number of propositions:

(1) The carrier of [: 𝕀, 𝕀:] = [: [0, 1], [0, 1] :].

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102 and KBN grant 4 T11C 039 24.

2The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan.

(2) For every real number x such that x \frac{1}{2} holds 2 · x − 1 1 − 2 · x.

(3) For every real number x such that x \frac{1}{2} holds 2 · x − 1 1 − 2 · x.

(4) For all real numbers x, a, b, c, d such that a ≠ b holds \frac{d−c}{b−a} · (x − a) + c = (1 − \frac{x−a}{b−a} ) · c + \frac{x−a}{b−a} · d.

(5) For all real numbers a, b, x such that a x and x b holds \frac{x−a}{b−a} ∈ the carrier of [0, 1]_{T}.

(6) For every point x of 𝕀 such that x \frac{1}{2} holds 2 · x is a point of 𝕀.

(7) For every point x of 𝕀 such that x \frac{1}{2} holds 2 · x − 1 is a point of 𝕀.

(8) For all points p, q of 𝕀 holds p · q is a point of 𝕀.

(9) For every point x of 𝕀 holds \frac{1}{2} · x is a point of 𝕀.

(10) For every point x of 𝕀 such that x \frac{1}{2} holds x − \frac{1}{4} is a point of 𝕀. (12)^{3} id_{𝕀} is a path from 0_{𝕀} to 1_{𝕀}.

(13) For all points a, b, c, d of 𝕀 such that a b and c d holds [: [a, b], [c, d] :] is a compact non empty subset of [: 𝕀, 𝕀:].

2. Affine Maps

One can prove the following four propositions:

(14) Let S, T be subsets of E_{T}^{2}. Suppose S = {p; p ranges over points of E_{T}^{2}: p_{2} 2 · p_{1} − 1} and T = {p; p ranges over points of E_{T}^{2}: p_{2} p_{1}}. Then (AffineMap(1, 0, \frac{1}{2}, \frac{1}{2}))^{∘}S = T.

(15) Let S, T be subsets of E_{T}^{2}. Suppose S = {p; p ranges over points of E_{T}^{2}: p_{2} 2 · p_{1} − 1} and T = {p; p ranges over points of E_{T}^{2}: p_{2} p_{1}}. Then (AffineMap(1, 0, \frac{1}{2}, \frac{1}{2}))^{∘}S = T.

(16) Let S, T be subsets of E_{T}^{2}. Suppose S = {p; p ranges over points of E_{T}^{2}: p_{2} 1 − 2 · p_{1}} and T = {p; p ranges over points of E_{T}^{2}: p_{2} − p_{1}}. Then (AffineMap(1, 0, \frac{1}{2}, − \frac{1}{2}))^{∘}S = T.

(17) Let S, T be subsets of E_{T}^{2}. Suppose S = {p; p ranges over points of E_{T}^{2}: p_{2} 1 − 2 · p_{1}} and T = {p; p ranges over points of E_{T}^{2}: p_{2} − p_{1}}. Then (AffineMap(1, 0, \frac{1}{2}, − \frac{1}{2}))^{∘}S = T.

3. Real-Membered Structures

Let T be a 1-sorted structure. We say that T is real-membered if and only if:

(Def. 1) The carrier of T is real-membered. We now state the proposition (18) For every non empty 1-sorted structure T holds T is real-membered iff

3The proposition (11) has been removed.

every element of T is real. Let us mention that 𝕀 is real-membered. One can verify that there exists a 1-sorted structure which is non empty

and real-membered and there exists a topological space which is non empty and real-membered.

Let T be a real-membered 1-sorted structure. Note that every element of T is real.

Let T be a real-membered topological structure. Note that every subspace of T is real-membered.

Let S, T be real-membered non empty topological spaces and let p be an element of [: S, T :]. One can check that p_{1} is real and p_{2} is real.

Let T be a non empty subspace of [: 𝕀, 𝕀:] and let x be a point of T. One can check that x_{1} is real and x_{2} is real.

One can check that ℝ^{1} is real-membered.

4. Closed Subsets of Euclidean Topological Spaces

The following propositions are true:

(19) {p; p ranges over points of E_{T}^{2}: p_{2} 2 · p_{1} − 1} is a closed subset of E_{T}^{2}.

(20) {p; p ranges over points of E_{T}^{2}: p_{2} 2 · p_{1} − 1} is a closed subset of E_{T}^{2}.

(21) {p; p ranges over points of E_{T}^{2}: p_{2} 1 − 2 · p_{1}} is a closed subset of E_{T}^{2}.

(22) {p; p ranges over points of E_{T}^{2}: p_{2} 1 − 2 · p_{1}} is a closed subset of E_{T}^{2}.

(23) {p; p ranges over points of E_{T}^{2}: p_{2} 1 − 2 · p_{1} ∧ p_{2} 2 · p_{1} − 1} is a closed subset of E_{T}^{2}.

(24) There exists a map f from [: ℝ^{1}, ℝ^{1} :] into E_{T}^{2} such that for all real numbers x, y holds f(⟨⟨x, y⟩⟩) = ⟨x, y⟩.

(25) {p; p ranges over points of [: ℝ^{1}, ℝ^{1} :]: p_{2} 1 − 2 · p_{1}} is a closed subset of [: ℝ^{1}, ℝ^{1} :].

(26) {p; p ranges over points of [: ℝ^{1}, ℝ^{1} :]: p_{2} 2 · p_{1} − 1} is a closed subset of [: ℝ^{1}, ℝ^{1} :].

(27) {p; p ranges over points of [: ℝ^{1}, ℝ^{1} :]: p_{2} 1 − 2 · p_{1} ∧ p_{2} 2 · p_{1} − 1} is a closed subset of [: ℝ^{1}, ℝ^{1} :].

(28) {p; p ranges over points of [: 𝕀, 𝕀:]: p_{2} 1 − 2 · p_{1}} is a closed non empty subset of [: 𝕀, 𝕀:].

(29) {p; p ranges over points of [: 𝕀, 𝕀:]: p_{2} 1 − 2 · p_{1} ∧ p_{2} 2 · p_{1} − 1} is a closed non empty subset of [: 𝕀, 𝕀:].

(30) {p; p ranges over points of [: 𝕀, 𝕀:]: p_{2} 2 · p_{1} − 1} is a closed non empty subset of [: 𝕀, 𝕀:].

(31) Let S, T be non empty topological spaces and p be a point of [: S, T :]. Then p_{1} is a point of S and p_{2} is a point of T.

(32) For all subsets A, B of [: 𝕀, 𝕀:] such that A = [: [0, \frac{1}{2}], [0, 1] :] and B = [: [\frac{1}{2}, 1], [0, 1] :] holds Ω_{[: 𝕀, 𝕀:]↾A} ∪ Ω_{[: 𝕀, 𝕀:]↾B} = Ω_{[: 𝕀, 𝕀:]}.

(33) For all subsets A, B of [: 𝕀, 𝕀:] such that A = [: [0, \frac{1}{2}], [0, 1] :] and B = [: [\frac{1}{2}, 1], [0, 1] :] holds Ω_{[: 𝕀, 𝕀:]↾A} ∩ Ω_{[: 𝕀, 𝕀:]↾B} = [: \{\frac{1}{2}\}, [0, 1] :].

5. Compact Spaces

Let T be a topological structure. Note that ∅_{T} is compact. Let T be a topological structure. Observe that there exists a subset of T

which is empty and compact.

Next we state three propositions:

(34) For every topological structure T holds ∅ is an empty compact subset of T.

(35) Let T be a topological structure and a, b be real numbers. If a > b, then [a, b] is an empty compact subset of T.

(36) For all points a, b, c, d of 𝕀 holds [: [a, b], [c, d] :] is a compact subset of [: 𝕀, 𝕀:].

6. Continuous Maps

Let a, b, c, d be real numbers. The functor L_{01}(a, b, c, d) yielding a map from [a, b]_{T} into [c, d]_{T} is defined by:

(Def. 2) L_{01}(a, b, c, d) = L_{01}(c_{[c,d]_{T}}, d_{[c,d]_{T}}) · P_{01}(a, b, 0_{[0,1]_{T}}, 1_{[0,1]_{T}}).

The following propositions are true:

(37) For all real numbers a, b, c, d such that a < b and c < d holds (L_{01}(a, b, c, d))(a) = c and (L_{01}(a, b, c, d))(b) = d.

(38) For all real numbers a, b, c, d such that a < b and c d holds L_{01}(a, b, c, d) is a continuous map from [a, b]_{T} into [c, d]_{T}.

(39) Let a, b, c, d be real numbers. Suppose a < b and c d. Let x be a real number. If a x and x b, then (L_{01}(a, b, c, d))(x) = \frac{d−c}{b−a} · (x − a) + c.

(40) Let f_{1}, f_{2} be maps from [: 𝕀, 𝕀:] into 𝕀. Suppose f_{1} is continuous and f_{2} is continuous and for every point p of [: 𝕀, 𝕀:] holds f_{1}(p) · f_{2}(p) is a point of 𝕀. Then there exists a map g from [: 𝕀, 𝕀:] into 𝕀 such that

(i) for every point p of [: 𝕀, 𝕀:] and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = r_{1} · r_{2}, and

(ii) g is continuous.

(41) Let f_{1}, f_{2} be maps from [: 𝕀, 𝕀:] into 𝕀. Suppose f_{1} is continuous and f_{2} is continuous and for every point p of [: 𝕀, 𝕀:] holds f_{1}(p) + f_{2}(p) is a point of 𝕀. Then there exists a map g from [: 𝕀, 𝕀:] into 𝕀 such that

(i) for every point p of [: 𝕀, 𝕀:] and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = r_{1} + r_{2}, and

(ii) g is continuous.

(42) Let f_{1}, f_{2} be maps from [: 𝕀, 𝕀:] into 𝕀. Suppose f_{1} is continuous and f_{2} is continuous and for every point p of [: 𝕀, 𝕀:] holds f_{1}(p) − f_{2}(p) is a point of 𝕀. Then there exists a map g from [: 𝕀, 𝕀:] into 𝕀 such that

(i) for every point p of [: 𝕀, 𝕀:] and for all real numbers r_{1}, r_{2} such that f_{1}(p) = r_{1} and f_{2}(p) = r_{2} holds g(p) = r_{1} − r_{2}, and

(ii) g is continuous.

7. Paths

We follow the rules: T denotes a non empty topological space and a, b, c, d denote points of T.

The following three propositions are true:

(43) For every path P from a to b such that P is continuous holds P · L_{01}(1_{[0,1]_{T}}, 0_{[0,1]_{T}}) is a continuous map from 𝕀 into T.

(44) Let X be a non empty topological structure, a, b be points of X, and P be a path from a to b. If P (0) = a and P (1) = b, then (P · L_{01}(1_{[0,1]_{T}}, 0_{[0,1]_{T}}))(0) = b and (P · L_{01}(1_{[0,1]_{T}}, 0_{[0,1]_{T}}))(1) = a.

(45) Let P be a path from a to b. Suppose P is continuous and P (0) = a and

P (1) = b. Then − P is continuous and (− P)(0) = b and (− P)(1) = a. Let T be a topological structure and let a, b be points of T. We say that a,

b are connected if and only if: (Def. 3) There exists a map f from 𝕀 into T such that f is continuous and f(0) = a

and f(1) = b. Let T be a non empty topological space and let a, b be points of T. Let us

notice that the predicate a, b are connected is reflexive and symmetric.

We now state several propositions:

(46) If a, b are connected and b, c are connected, then a, c are connected.

(47) For every arcwise connected topological structure T and for all points a, b of T holds a, b are connected.

(48) For every path A from a to a holds A, A are homotopic.

(49) If a, b are connected, then for every path A from a to b holds A, A are homotopic.

(50) If a, b are connected, then for every path A from a to b holds A = − − A.

(51) Let T be a non empty arcwise connected topological space, a, b be points of T, and A be a path from a to b. Then A = − − A.

(52) If a, b are connected, then every path from a to b is continuous.

8. Reexamination of a Path Concept

Let T be a non empty arcwise connected topological space, let a, b, c be points of T, let P be a path from a to b, and let Q be a path from b to c. Then P + Q can be characterized by the condition:

(Def. 4) For every point t of 𝕀 holds if t \frac{1}{2}, then (P + Q)(t) = P (2 · t) and if

\frac{1}{2} t, then (P + Q)(t) = Q(2 · t − 1). Let T be a non empty arcwise connected topological space, let a, b be points

of T, and let P be a path from a to b. Then − P can be characterized by the condition:

(Def. 5) For every point t of 𝕀 holds (− P)(t) = P (1 − t).

9. Reparametrizations

Let T be a non empty topological space, let a, b be points of T, let P be a path from a to b, and let f be a continuous map from 𝕀 into 𝕀. Let us assume that f(0) = 0 and f(1) = 1 and a, b are connected. The functor RePar(P, f) yields a path from a to b and is defined by:

(Def. 6) RePar(P, f) = P · f.

Next we state two propositions:

(53) Let P be a path from a to b and f be a continuous map from 𝕀 into 𝕀. Suppose f(0) = 0 and f(1) = 1 and a, b are connected. Then RePar(P, f), P are homotopic.

(54) Let T be a non empty arcwise connected topological space, a, b be points of T, P be a path from a to b, and f be a continuous map from 𝕀 into 𝕀. If f(0) = 0 and f(1) = 1, then RePar(P, f), P are homotopic.

The map 1^{st}RP from 𝕀 into 𝕀 is defined as follows: (Def. 7) For every point t of 𝕀 holds if t \frac{1}{2}, then (1^{st}RP)(t) = 2 · t and if t > \frac{1}{2},

then (1^{st}RP)(t) = 1. Let us note that 1^{st}RP is continuous. One can prove the following proposition

(55) (1^{st}RP)(0) = 0 and (1^{st}RP)(1) = 1.

The map 2^{nd}RP from 𝕀 into 𝕀 is defined by: (Def. 8) For every point t of 𝕀 holds if t \frac{1}{2}, then (2^{nd}RP)(t) = 0 and if t > \frac{1}{2},

then (2^{nd}RP)(t) = 2 · t − 1.

One can verify that 2^{nd}RP is continuous. One can prove the following proposition

(56) (2^{nd}RP)(0) = 0 and (2^{nd}RP)(1) = 1.

The map 3^{rd}RP from 𝕀 into 𝕀 is defined by the condition (Def. 9). (Def. 9) Let x be a point of 𝕀. Then

(i) if x \frac{1}{2}, then (3^{rd}RP)(x) = \frac{1}{2} · x, (ii) if x > \frac{1}{2} and x \frac{3}{4}, then (3^{rd}RP)(x) = x − \frac{1}{4}, and (iii) if x > \frac{3}{4}, then (3^{rd}RP)(x) = 2 · x − 1. Let us note that 3^{rd}RP is continuous. We now state four propositions:

(57) (3^{rd}RP)(0) = 0 and (3^{rd}RP)(1) = 1.

(58) Let P be a path from a to b and Q be a constant path from b to b. If a, b are connected, then RePar(P, 1^{st}RP) = P + Q.

(59) Let P be a path from a to b and Q be a constant path from a to a. If a, b are connected, then RePar(P, 2^{nd}RP) = Q + P.

(60) Let P be a path from a to b, Q be a path from b to c, and R be a path from c to d. Suppose a, b are connected and b, c are connected and c, d are connected. Then RePar(P + Q + R, 3^{rd}RP) = P + (Q + R).

10. Decomposition of the Unit Square

The subset LowerLeftUnitTriangle of [: 𝕀, 𝕀:] is defined as follows: (Def. 10) For every set x holds x ∈ LowerLeftUnitTriangle iff there exist points a,

b of 𝕀 such that x = ⟨⟨a, b⟩⟩ and b 1 − 2 · a. We introduce IAA as a synonym of LowerLeftUnitTriangle.

The subset UpperUnitTriangle of [: 𝕀, 𝕀:] is defined by: (Def. 11) For every set x holds x ∈ UpperUnitTriangle iff there exist points a, b

of 𝕀 such that x = ⟨⟨a, b⟩⟩ and b 1 − 2 · a and b 2 · a − 1. We introduce IBB as a synonym of UpperUnitTriangle.

The subset LowerRightUnitTriangle of [: 𝕀, 𝕀:] is defined as follows: (Def. 12) For every set x holds x ∈ LowerRightUnitTriangle iff there exist points

a, b of 𝕀 such that x = ⟨⟨a, b⟩⟩ and b 2 · a − 1. We introduce ICC as a synonym of LowerRightUnitTriangle.

The following propositions are true:

(61) IAA = {p; p ranges over points of [: 𝕀, 𝕀:]: p_{2} 1 − 2 · p_{1}}.

(62) IBB = {p; p ranges over points of [: 𝕀, 𝕀:]: p_{2} 1−2·p_{1} ∧ p_{2} 2·p_{1} −1}.

(63) ICC = {p; p ranges over points of [: 𝕀, 𝕀:]: p_{2} 2 · p_{1} − 1}.

One can check the following observations:

∗ IAA is closed and non empty, ∗ IBB is closed and non empty, and

∗ ICC is closed and non empty.

Next we state a number of propositions:

(64) IAA∪ IBB∪ ICC = [: [0, 1], [0, 1] :].

(65) IAA∩ IBB = {p; p ranges over points of [: 𝕀, 𝕀:]: p_{2} = 1 − 2 · p_{1}}.

(66) ICC∩ IBB = {p; p ranges over points of [: 𝕀, 𝕀:]: p_{2} = 2 · p_{1} − 1}.

(67) For every point x of [: 𝕀, 𝕀:] such that x ∈ IAA holds x_{1} \frac{1}{2}.

(68) For every point x of [: 𝕀, 𝕀:] such that x ∈ ICC holds x_{1} \frac{1}{2}.

(69) For every point x of 𝕀 holds ⟨⟨0, x⟩⟩∈ IAA.

(70) For every set s such that ⟨⟨0, s⟩⟩∈ IBB holds s = 1.

(71) For every set s such that ⟨⟨s, 1⟩⟩∈ ICC holds s = 1.

(72) ⟨⟨0, 1⟩⟩∈ IBB.

(73) For every point x of 𝕀 holds ⟨⟨x, 1⟩⟩∈ IBB.

(74) ⟨⟨\frac{1}{2}, 0⟩⟩∈ ICC and ⟨⟨1, 1⟩⟩∈ ICC.

(75) ⟨⟨\frac{1}{2}, 0⟩⟩∈ IBB.

(76) For every point x of 𝕀 holds ⟨⟨1, x⟩⟩∈ ICC.

(77) For every point x of 𝕀 such that x \frac{1}{2} holds ⟨⟨x, 0⟩⟩∈ ICC.

(78) For every point x of 𝕀 such that x \frac{1}{2} holds ⟨⟨x, 0⟩⟩∈ IAA.

(79) For every point x of 𝕀 such that x < \frac{1}{2} holds ⟨⟨x, 0⟩⟩ ∉ IBB and ⟨⟨x, 0⟩⟩ ∉ ICC.

(80) IAA∩ ICC = \{⟨⟨\frac{1}{2}, 0⟩⟩\}.

11. Properties of a Homotopy

We use the following convention: X denotes a non empty arcwise connected topological space and a_{1}, b_{1}, c_{1}, d_{1} denote points of X.

One can prove the following propositions:

(81) Let P be a path from a to b, Q be a path from b to c, and R be a path from c to d. Suppose a, b are connected and b, c are connected and c, d are connected. Then (P + Q) + R, P + (Q + R) are homotopic.

(82) Let P be a path from a_{1} to b_{1}, Q be a path from b_{1} to c_{1}, and R be a path from c_{1} to d_{1}. Then (P + Q) + R, P + (Q + R) are homotopic.

(83) Let P_{1}, P_{2} be paths from a to b and Q_{1}, Q_{2} be paths from b to c. Suppose a, b are connected and b, c are connected and P_{1}, P_{2} are homotopic and Q_{1}, Q_{2} are homotopic. Then P_{1} + Q_{1}, P_{2} + Q_{2} are homotopic.

(84) Let P_{1}, P_{2} be paths from a_{1} to b_{1} and Q_{1}, Q_{2} be paths from b_{1} to c_{1}. Suppose P_{1}, P_{2} are homotopic and Q_{1}, Q_{2} are homotopic. Then P_{1} + Q_{1}, P_{2} + Q_{2} are homotopic.

(85) Let P, Q be paths from a to b. Suppose a, b are connected and P, Q are homotopic. Then − P, − Q are homotopic.

(86) For all paths P, Q from a_{1} to b_{1} such that P, Q are homotopic holds − P, − Q are homotopic.

(87) Let P, Q, R be paths from a to b. Suppose P, Q are homotopic and Q, R are homotopic. Then P, R are homotopic.

(88) Let P be a path from a to b and Q be a constant path from b to b. If a, b are connected, then P + Q, P are homotopic.

(89) For every path P from a_{1} to b_{1} and for every constant path Q from b_{1} to b_{1} holds P + Q, P are homotopic.

(90) Let P be a path from a to b and Q be a constant path from a to a. If a, b are connected, then Q + P, P are homotopic.

(91) For every path P from a_{1} to b_{1} and for every constant path Q from a_{1} to a_{1} holds Q + P, P are homotopic.

(92) Let P be a path from a to b and Q be a constant path from a to a. If a, b are connected, then P + − P , Q are homotopic.

(93) For every path P from a_{1} to b_{1} and for every constant path Q from a_{1} to a_{1} holds P + − P , Q are homotopic.

(94) Let P be a path from b to a and Q be a constant path from a to a. If b, a are connected, then − P + P, Q are homotopic.

(95) For every path P from b_{1} to a_{1} and for every constant path Q from a_{1} to a_{1} holds − P + P, Q are homotopic.

(96) For all constant paths P, Q from a to a holds P, Q are homotopic. Let T be a non empty topological space, let a, b be points of T, and let P, Q

be paths from a to b. Let us assume that P, Q are homotopic. A map from [: 𝕀, 𝕀:] into T is said to be a homotopy between P and Q if it satisfies the conditions (Def. 13).

(Def. 13)(i) It is continuous, and

(ii) for every point s of 𝕀 holds it(s, 0) = P (s) and it(s, 1) = Q(s) and for every point t of 𝕀 holds it(0, t) = a and it(1, t) = b.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.

[4] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[5] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

260 adam grabowski and artur korniłowicz

[7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[8] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[9] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[10] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 1990.

[11] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[12] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[13] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449–454, 1997.

[14] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[15] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[16] Michał Muzalewski. Categories of groups. Formalized Mathematics, 2(4):563–571, 1991.

[17] Yatsuka Nakamura. On Outside Fashoda Meet Theorem. Formalized Mathematics, 9(4):697–704, 2001.

[18] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[19] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[20] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[22] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[23] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

[24] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341– 347, 2003.

[25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[26] Toshihiko Watanabe. The Brouwer fixed point theorem for intervals. Formalized Mathematics, 3(1):85–88, 1992.

[27] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

The Fundamental Group

Artur Korniłowicz^{1} University of Białystok Yasunari Shidama Shinshu University Nagano

Adam Grabowski^{2} University of Białystok

Summary. This is the next article in a series devoted to the homotopy theory (following [11] and [12]). The concept of fundamental groups of pointed topological spaces has been introduced. Isomorphism of fundamental groups defined with respect to different points belonging to the same component has been stated. Triviality of fundamental group(s) of ℝ^{n} has been shown.

MML Identifier: TOPALG 1.

The articles [22], [7], [26], [27], [19], [4], [6], [5], [28], [2], [21], [1], [18], [20], [16], [8], [3], [15], [13], [17], [29], [9], [14], [24], [23], [10], [11], [25], and [12] provide the terminology and notation for this paper.

1. Preliminaries

We adopt the following convention: p, q, x, y are real numbers and n is a natural number.

Next we state a number of propositions:

(1) Let G, H be groups and h be a homomorphism from G to H. If h·h^{−1} = id_{H} and h^{−1} · h = id_{G}, then h is an isomorphism.

(2) For every subset X of 𝕀 and for every point a of 𝕀 such that X = ]a, 1] holds X^{c} = [0, a].

1The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan.

2This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102 and KBN grant 4 T11C 039 24.

(3) For every subset X of 𝕀 and for every point a of 𝕀 such that X = [0, a[ holds X^{c} = [a, 1].

(4) For every subset X of 𝕀 and for every point a of 𝕀 such that X = ]a, 1] holds X is open.

(5) For every subset X of 𝕀 and for every point a of 𝕀 such that X = [0, a[ holds X is open.

(6) For every element f of ℝ^{n} holds x · − f = − x · f.

(7) For all elements f, g of ℝ^{n} holds x · (f − g) = x · f − x · g.

(8) For every element f of ℝ^{n} holds (x − y) · f = x · f − y · f.

(9) For all elements f, g, h, k of ℝ^{n} holds (f +g)−(h+k) = (f −h)+(g−k).

(10) For every element f of R^{n} such that 0 x and x 1 holds |x · f| |f|.

(11) For every element f of R^{n} and for every point p of 𝕀 holds |p · f| |f|.

(12) Let e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6} be points of E^{n} and p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{n}. Suppose e_{1} = p_{1} and e_{2} = p_{2} and e_{3} = p_{3} and e_{4} = p_{4} and e_{5} = p_{1} +p_{3} and e_{6} = p_{2} + p_{4} and ρ(e_{1}, e_{2}) < x and ρ(e_{3}, e_{4}) < y. Then ρ(e_{5}, e_{6}) < x + y.

(13) Let e_{1}, e_{2}, e_{5}, e_{6} be points of E^{n} and p_{1}, p_{2} be points of E_{T}^{n}. If e_{1} = p_{1} and e_{2} = p_{2} and e_{5} = y · p_{1} and e_{6} = y · p_{2} and ρ(e_{1}, e_{2}) < x and y ≠ 0, then ρ(e_{5}, e_{6}) < |y| · x.

(14) Let e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6} be points of E^{n} and p_{1}, p_{2}, p_{3}, p_{4} be points of E_{T}^{n}. Suppose e_{1} = p_{1} and e_{2} = p_{2} and e_{3} = p_{3} and e_{4} = p_{4} and e_{5} = x·p_{1}+y·p_{3} and e_{6} = x · p_{2} + y · p_{4} and ρ(e_{1}, e_{2}) < p and ρ(e_{3}, e_{4}) < q and x ≠ 0 and y ≠ 0. Then ρ(e_{5}, e_{6}) < |x| · p + |y| · q.

(16)^{3} Let X be a non empty topological space and f, g be maps from X into E_{T}^{n}. Suppose f is continuous and for every point p of X holds g(p) = y·f(p). Then g is continuous.

(17) Let X be a non empty topological space and f_{1}, f_{2}, g be maps from X into E_{T}^{n}. Suppose f_{1} is continuous and f_{2} is continuous and for every point p of X holds g(p) = x · f_{1}(p) + y · f_{2}(p). Then g is continuous.

(18) Let F be a map from [: E_{T}^{n}, 𝕀:] into E_{T}^{n}. Suppose that for every point x of E_{T}^{n} and for every point i of 𝕀 holds F (x, i) = (1 − i) · x. Then F is continuous.

(19) Let F be a map from [: E_{T}^{n}, 𝕀:] into E_{T}^{n}. Suppose that for every point x of E_{T}^{n} and for every point i of 𝕀 holds F (x, i) = i · x. Then F is continuous.

2. Paths

For simplicity, we follow the rules: X denotes a non empty topological space, a, b, c, d, e, f denote points of X, T denotes a non empty arcwise connected topological space, and a_{1}, b_{1}, c_{1}, d_{1}, e_{1}, f_{1} denote points of T.

3The proposition (15) has been removed.

One can prove the following propositions:

(20) Suppose a, b are connected and b, c are connected. Let A be a path from a to b and B be a path from b to c. Then A, A + B + − B are homotopic.

(21) For every path A from a_{1} to b_{1} and for every path B from b_{1} to c_{1} holds A, A + B + − B are homotopic.

(22) Suppose a, b are connected and c, b are connected. Let A be a path from a to b and B be a path from c to b. Then A, A + − B + B are homotopic.

(23) For every path A from a_{1} to b_{1} and for every path B from c_{1} to b_{1} holds A, A + − B + B are homotopic.

(24) Suppose a, b are connected and c, a are connected. Let A be a path from a to b and B be a path from c to a. Then A, − B + B + A are homotopic.

(25) For every path A from a_{1} to b_{1} and for every path B from c_{1} to a_{1} holds A, − B + B + A are homotopic.

(26) Suppose a, b are connected and a, c are connected. Let A be a path from a to b and B be a path from a to c. Then A, B + − B + A are homotopic.

(27) For every path A from a_{1} to b_{1} and for every path B from a_{1} to c_{1} holds A, B + − B + A are homotopic.

(28) Suppose a, b are connected and c, b are connected. Let A, B be paths from a to b and C be a path from b to c. If A + C, B + C are homotopic, then A, B are homotopic.

(29) Let A, B be paths from a_{1} to b_{1} and C be a path from b_{1} to c_{1}. If A + C, B + C are homotopic, then A, B are homotopic.

(30) Suppose a, b are connected and a, c are connected. Let A, B be paths from a to b and C be a path from c to a. If C + A, C + B are homotopic, then A, B are homotopic.

(31) Let A, B be paths from a_{1} to b_{1} and C be a path from c_{1} to a_{1}. If C +A, C + B are homotopic, then A, B are homotopic.

(32) Suppose a, b are connected and b, c are connected and c, d are connected and d, e are connected. Let A be a path from a to b, B be a path from b to c, C be a path from c to d, and D be a path from d to e. Then A + B + C + D, A + (B + C) + D are homotopic.

(33) Let A be a path from a_{1} to b_{1}, B be a path from b_{1} to c_{1}, C be a path from c_{1} to d_{1}, and D be a path from d_{1} to e_{1}. Then A + B + C + D, A + (B + C) + D are homotopic.

(34) Suppose a, b are connected and b, c are connected and c, d are connected and d, e are connected. Let A be a path from a to b, B be a path from b to c, C be a path from c to d, and D be a path from d to e. Then (A + B + C) + D, A + (B + C + D) are homotopic.

(35) Let A be a path from a_{1} to b_{1}, B be a path from b_{1} to c_{1}, C be a path from c_{1} to d_{1}, and D be a path from d_{1} to e_{1}. Then (A + B + C) + D, A + (B + C + D) are homotopic.

(36) Suppose a, b are connected and b, c are connected and c, d are connected and d, e are connected. Let A be a path from a to b, B be a path from b to c, C be a path from c to d, and D be a path from d to e. Then (A + (B + C)) + D, A + B + (C + D) are homotopic.

(37) Let A be a path from a_{1} to b_{1}, B be a path from b_{1} to c_{1}, C be a path from c_{1} to d_{1}, and D be a path from d_{1} to e_{1}. Then (A + (B + C)) + D, A + B + (C + D) are homotopic.

(38) Suppose a, b are connected and b, c are connected and b, d are connected. Let A be a path from a to b, B be a path from d to b, and C be a path from b to c. Then A + − B + B + C, A + C are homotopic.

(39) Let A be a path from a_{1} to b_{1}, B be a path from d_{1} to b_{1}, and C be a path from b_{1} to c_{1}. Then A + − B + B + C, A + C are homotopic.

(40) Suppose a, b are connected and a, c are connected and c, d are connected. Let A be a path from a to b, B be a path from c to d, and C be a path from a to c. Then A + − A + C + B + − B, C are homotopic.

(41) Let A be a path from a_{1} to b_{1}, B be a path from c_{1} to d_{1}, and C be a path from a_{1} to c_{1}. Then A + − A + C + B + − B, C are homotopic.

(42) Suppose a, b are connected and a, c are connected and d, c are connected. Let A be a path from a to b, B be a path from c to d, and C be a path from a to c. Then A + (− A + C + B) + − B, C are homotopic.

(43) Let A be a path from a_{1} to b_{1}, B be a path from c_{1} to d_{1}, and C be a path from a_{1} to c_{1}. Then A + (− A + C + B) + − B, C are homotopic.

(44) Suppose that (i) a, b are connected, (ii) b, c are connected, (iii) c, d are connected, (iv) d, e are connected, and

(v) a, f are connected. Let A be a path from a to b, B be a path from b to c, C be a path from c to d, D be a path from d to e, and E be a path from f to c. Then (A + (B + C)) + D, A + B + − E + (E + C + D) are homotopic.

(45) Let A be a path from a_{1} to b_{1}, B be a path from b_{1} to c_{1}, C be a path from c_{1} to d_{1}, D be a path from d_{1} to e_{1}, and E be a path from f_{1} to c_{1}. Then (A + (B + C)) + D, A + B + − E + (E + C + D) are homotopic.

3. The Fundamental Group

Let T be a topological structure and let t be a point of T. A loop of t is a path from t to t.

Let T be a non empty topological structure and let t be a point of T. The functor Loops(t) is defined by:

(Def. 1) For every set x holds x ∈ Loops(t) iff x is a loop of t.

Let T be a non empty topological structure and let t be a point of T. Observe that Loops(t) is non empty.

Let X be a non empty topological space and let a be a point of X. The functor EqRel(X, a) yielding a binary relation on Loops(a) is defined by:

(Def. 2) For all loops P, Q of a holds ⟨⟨P, Q⟩⟩ ∈ EqRel(X, a) iff P, Q are homo-

topic. Let X be a non empty topological space and let a be a point of X. One can

check that EqRel(X, a) is non empty, total, symmetric, and transitive.

We now state two propositions:

(46) For all loops P, Q of a holds Q ∈ [P]_{EqRel(X,a)} iff P, Q are homotopic.

(47) For all loops P, Q of a holds [P]_{EqRel(X,a)} = [Q]_{EqRel(X,a)} iff P, Q are

homotopic. Let X be a non empty topological space and let a be a point of X. The

functor FundamentalGroup(X, a) yielding a strict groupoid is defined by the conditions (Def. 3).

(Def. 3)(i) The carrier of FundamentalGroup(X, a) = ClassesEqRel(X, a), and

(ii) for all elements x, y of FundamentalGroup(X, a) there exist loops P, Q of a such that x = [P]_{EqRel(X,a)} and y = [Q]_{EqRel(X,a)} and (the multiplication of FundamentalGroup(X, a))(x, y) = [P + Q]_{EqRel(X,a)}.

We introduce π_{1}(X, a) as a synonym of FundamentalGroup(X, a).

Let X be a non empty topological space and let a be a point of X. One can verify that π_{1}(X, a) is non empty.

Next we state the proposition

(48) For every set x holds x ∈ the carrier of π_{1}(X, a) iff there exists a loop P

of a such that x = [P]_{EqRel(X,a)}. Let X be a non empty topological space and let a be a point of X. Note

that π_{1}(X, a) is associative and group-like.

Let T be a non empty topological space, let x_{0}, x_{1} be points of T, and let P be a path from x_{0} to x_{1}. Let us assume that x_{0}, x_{1} are connected. The functor π_{1}-iso(P) yielding a map from π_{1}(T, x_{1}) into π_{1}(T, x_{0}) is defined by:

(Def. 4) For every loop Q of x_{1} holds (π_{1}-iso(P))([Q]_{EqRel(T,x_{1})}) = [P + Q + − P]_{EqRel(T,x_{0})}.

For simplicity, we follow the rules: x_{0}, x_{1} denote points of X, P, Q denote paths from x_{0} to x_{1}, y_{0}, y_{1} denote points of T, and R, V denote paths from y_{0} to y_{1}.

Next we state three propositions:

(49) If x_{0}, x_{1} are connected and P, Q are homotopic, then π_{1}-iso(P) = π_{1}-iso(Q).

(50) If R, V are homotopic, then π_{1}-iso(R) = π_{1}-iso(V).

(51) If x_{0}, x_{1} are connected, then π_{1}-iso(P) is a homomorphism from

π_{1}(X, x_{1}) to π_{1}(X, x_{0}). Let T be a non empty arcwise connected topological space, let x_{0}, x_{1} be po-

ints of T, and let P be a path from x_{0} to x_{1}. Then π_{1}-iso(P) is a homomorphism from π_{1}(T, x_{1}) to π_{1}(T, x_{0}).

The following propositions are true:

(52) If x_{0}, x_{1} are connected, then π_{1}-iso(P) is one-to-one.

(53) If x_{0}, x_{1} are connected, then π_{1}-iso(P) is onto.

Let T be a non empty arcwise connected topological space, let x_{0}, x_{1} be points of T, and let P be a path from x_{0} to x_{1}. One can verify that π_{1}-iso(P) is one-to-one and onto.

One can prove the following propositions:

(54) If x_{0}, x_{1} are connected, then (π_{1}-iso(P))^{−1} = π_{1}-iso(− P).

(55) (π_{1}-iso(R))^{−1} = π_{1}-iso(− R).

(56) If x_{0}, x_{1} are connected, then for every homomorphism h from π_{1}(X, x_{1}) to π_{1}(X, x_{0}) such that h = π_{1}-iso(P) holds h is an isomorphism.

(57) π_{1}-iso(R) is an isomorphism.

(58) If x_{0}, x_{1} are connected, then π_{1}(X, x_{0}) and π_{1}(X, x_{1}) are isomorphic.

(59) π_{1}(T, y_{0}) and π_{1}(T, y_{1}) are isomorphic.

4. Euclidean Topological Space

Let n be a natural number, let a, b be points of E_{T}^{n}, and let P, Q be paths from a to b. The functor RealHomotopy(P, Q) yields a map from [: 𝕀, 𝕀:] into E_{T}^{n} and is defined by:

(Def. 5) For all elements s, t of 𝕀 holds (RealHomotopy(P, Q))(s, t) = (1 − t) ·

P (s) + t · Q(s). The following proposition is true

(60) For all points a, b of E_{T}^{n} and for all paths P, Q from a to b holds P, Q

are homotopic. Let n be a natural number, let a, b be points of E_{T}^{n}, and let P, Q be paths

from a to b. Then RealHomotopy(P, Q) is a homotopy between P and Q.

the fundamental group 267

Let n be a natural number, let a, b be points of E_{T}^{n}, and let P, Q be paths from a to b. One can check that every homotopy between P and Q is continuous.

Next we state the proposition

(61) For every point a of E_{T}^{n} and for every loop C of a holds the carrier of π_{1}(E_{T}^{n}, a) = {[C]_{EqRel(E_{T}^{n},a)}}.

Let n be a natural number and let a be a point of E_{T}^{n}. Note that π_{1}(E_{T}^{n}, a) is trivial.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[3] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[8] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[9] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257–261, 1990.

[10] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[11] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449–454, 1997.

[12] Adam Grabowski and Artur Korniłowicz. Algebraic properties of homotopies. Formalized Mathematics, 12(3):251–260, 2004.

[13] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

[14] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[15] Michał Muzalewski. Categories of groups. Formalized Mathematics, 2(4):563–571, 1991.

[16] Yatsuka Nakamura. Half open intervals in real numbers. Formalized Mathematics, 10(1):21–22, 2002.

[17] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[18] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[19] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[20] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[21] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[22] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[23] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

[24] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[25] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.

[26] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186,

268 artur korniłowicz et al.

[28] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

[29] Mariusz Żynel and Czesław Byliński. Properties of relational structures, posets, lattices and maps. Formalized Mathematics, 6(1):123–130, 1997.

The Continuous Functions on Normed Linear Spaces

Takaya Nishiyama Shinshu University

Nagano

Keiji Ohkubo Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this article, the basic properties of the continuous function on normed linear spaces are described.

MML Identifier: NFCONT 1.

The articles [16], [19], [20], [2], [21], [4], [9], [3], [1], [11], [15], [5], [17], [18], [10], [7], [8], [6], [13], [22], [12], and [14] provide the notation and terminology for this paper.

We use the following convention: n is a natural number, x, X, X_{1} are sets, and s, r, p are real numbers.

Let S, T be 1-sorted structures. A partial function from S to T is a partial function from the carrier of S to the carrier of T.

For simplicity, we adopt the following rules: S, T denote real normed spaces, f, f_{1}, f_{2} denote partial functions from S to T, s_{1} denotes a sequence of S, x_{0}, x_{1}, x_{2} denote points of S, and Y denotes a subset of S.

Let R_{1} be a real linear space and let S_{1} be a sequence of R_{1}. The functor − S_{1} yields a sequence of R_{1} and is defined as follows:

(Def. 1) For every n holds (− S_{1})(n) = − S_{1}(n).

Next we state two propositions:

(1) For all sequences s_{2}, s_{3} of S holds s_{2} − s_{3} = s_{2} + − s_{3}.

(2) For every sequence s_{4} of S holds − s_{4} = (− 1) · s_{4}.

Let us consider S, T and let f be a partial function from S to T. The functor ∥f∥ yielding a partial function from the carrier of S to ℝ is defined as follows:

(Def. 2) dom∥f∥ = domf and for every point c of S such that c ∈ dom∥f∥ holds ∥f∥(c) = ∥f_{c}∥.

Let us consider S, x_{0}. A subset of S is called a neighbourhood of x_{0} if: (Def. 3) There exists a real number g such that 0 < g and {y; y ranges over points

of S: ∥y − x_{0}∥ < g} ⊆ it. The following two propositions are true:

(3) For every real number g such that 0 < g holds {y; y ranges over points of S: ∥y − x_{0}∥ < g} is a neighbourhood of x_{0}.

(4) For every neighbourhood N of x_{0} holds x_{0} ∈ N.

Let us consider S and let X be a subset of S. We say that X is compact if and only if the condition (Def. 4) is satisfied.

(Def. 4) Let s_{1} be a sequence of S. Suppose rngs_{1} ⊆ X. Then there exists a sequence s_{5} of S such that s_{5} is a subsequence of s_{1} and convergent and lims_{5} ∈ X.

Let us consider S and let X be a subset of S. We say that X is closed if and only if:

(Def. 5) For every sequence s_{1} of S such that rngs_{1} ⊆ X and s_{1} is convergent

holds lims_{1} ∈ X. Let us consider S and let X be a subset of S. We say that X is open if and

only if: (Def. 6) X^{c} is closed.

Let us consider S, T, let us consider f, and let s_{4} be a sequence of S. Let us assume that rngs_{4} ⊆ domf. The functor f · s_{4} yields a sequence of T and is defined as follows:

(Def. 7) f · s_{4} = (f qua function) ·(s_{4}).

Let us consider S, let f be a partial function from the carrier of S to ℝ, and let s_{4} be a sequence of S. Let us assume that rngs_{4} ⊆ domf. The functor f · s_{4} yields a sequence of real numbers and is defined as follows:

(Def. 8) f · s_{4} = (f qua function) ·(s_{4}).

Let us consider S, T and let us consider f, x_{0}. We say that f is continuous in x_{0} if and only if:

(Def. 9) x_{0} ∈ domf and for every s_{1} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}). Let us consider S, let f be a partial function from the carrier of S to ℝ, and

let us consider x_{0}. We say that f is continuous in x_{0} if and only if: (Def. 10) x_{0} ∈ domf and for every s_{1} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}). The scheme SeqPointNormSpChoice deals with a non empty normed struc-

ture A and a binary predicate P, and states that:

There exists a sequence s_{1} of A such that for every natural number n holds P[n, s_{1}(n)] provided the following condition is met:

• For every natural number n there exists a point r of A such that P[n, r].

The following propositions are true:

(5) For every sequence s_{4} of S and for every partial function h from S to T such that rngs_{4} ⊆ domh holds s_{4}(n) ∈ domh.

(6) For every sequence s_{4} of S and for every set x holds x ∈ rngs_{4} iff there exists n such that x = s_{4}(n).

(7) For all sequences s_{4}, s_{2} of S such that s_{2} is a subsequence of s_{4} holds rngs_{2} ⊆ rngs_{4}.

(8) For all f, s_{1} such that rngs_{1} ⊆ domf and for every n holds (f · s_{1})(n) = f_{s_{1}(n)}.

(9) Let f be a partial function from the carrier of S to ℝ and given s_{1}. If rngs_{1} ⊆ domf, then for every n holds (f · s_{1})(n) = f_{s_{1}(n)}.

(10) Let h be a partial function from S to T, s_{4} be a sequence of S, and N_{1} be an increasing sequence of naturals. If rngs_{4} ⊆ domh, then (h · s_{4}) · N_{1} = h · (s_{4} · N_{1}).

(11) Let h be a partial function from the carrier of S to ℝ, s_{4} be a sequence of S, and N_{1} be an increasing sequence of naturals. If rngs_{4} ⊆ domh, then (h · s_{4}) · N_{1} = h · (s_{4} · N_{1}).

(12) Let h be a partial function from S to T and s_{2}, s_{3} be sequences of S. If rngs_{2} ⊆ domh and s_{3} is a subsequence of s_{2}, then h · s_{3} is a subsequence of h · s_{2}.

(13) Let h be a partial function from the carrier of S to ℝ and s_{2}, s_{3} be sequences of S. If rngs_{2} ⊆ domh and s_{3} is a subsequence of s_{2}, then h · s_{3} is a subsequence of h · s_{2}.

(14) f is continuous in x_{0} if and only if the following conditions are satisfied: (i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for every

x_{1} such that x_{1} ∈ domf and ∥x_{1} − x_{0}∥ < s holds ∥f_{x_{1}} − f_{x_{0}}∥ < r.

(15) Let f be a partial function from the carrier of S to ℝ. Then f is conti-

nuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for every

x_{1} such that x_{1} ∈ domf and ∥x_{1} − x_{0}∥ < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(16) Let given f, x_{0}. Then f is continuous in x_{0} if and only if the following

conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every neighbourhood N_{2} of f_{x_{0}} there exists a neighbourhood N of x_{0}

such that for every x_{1} such that x_{1} ∈ domf and x_{1} ∈ N holds f_{x_{1}} ∈ N_{2}.

(17) Let given f, x_{0}. Then f is continuous in x_{0} if and only if the following

conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every neighbourhood N_{2} of f_{x_{0}} there exists a neighbourhood N of

x_{0} such that f^{∘}N ⊆ N_{2}.

(18) If x_{0} ∈ domf and there exists a neighbourhood N of x_{0} such that domf ∩ N = {x_{0}}, then f is continuous in x_{0}.

(19) Let h_{1}, h_{2} be partial functions from S to T and s_{4} be a sequence of S. If rngs_{4} ⊆ domh_{1} ∩ domh_{2}, then (h_{1} + h_{2}) · s_{4} = h_{1} · s_{4} + h_{2} · s_{4} and (h_{1} − h_{2}) · s_{4} = h_{1} · s_{4} − h_{2} · s_{4}.

(20) Let h be a partial function from S to T, s_{4} be a sequence of S, and r be a real number. If rngs_{4} ⊆ domh, then (r h) · s_{4} = r · (h · s_{4}).

(21) Let h be a partial function from S to T and s_{4} be a sequence of S. If rngs_{4} ⊆ domh, then ∥h · s_{4}∥ = ∥h∥ · s_{4} and − h · s_{4}= (− h) · s_{4}.

(22) If f_{1} is continuous in x_{0} and f_{2} is continuous in x_{0}, then f_{1} + f_{2} is continuous in x_{0} and f_{1} − f_{2} is continuous in x_{0}.

(23) If f is continuous in x_{0}, then r f is continuous in x_{0}.

(24) If f is continuous in x_{0}, then ∥f∥ is continuous in x_{0} and − f is continuous

in x_{0}. Let us consider S, T and let us consider f, X. We say that f is continuous

on X if and only if: (Def. 11) X ⊆ domf and for every x_{0} such that x_{0} ∈ X holds f↾X is continuous

in x_{0}. Let us consider S, let f be a partial function from the carrier of S to ℝ, and

let us consider X. We say that f is continuous on X if and only if: (Def. 12) X ⊆ domf and for every x_{0} such that x_{0} ∈ X holds f↾X is continuous

in x_{0}. One can prove the following propositions:

(25) Let given X, f. Then f is continuous on X if and only if the following

conditions are satisfied:

(i) X ⊆ domf, and (ii) for every s_{1} such that rngs_{1} ⊆ X and s_{1} is convergent and lims_{1} ∈ X

holds f · s_{1} is convergent and f_{lim s_{1}} = lim(f · s_{1}).

(26) f is continuous on X if and only if the following conditions are satisfied: (i) X ⊆ domf, and (ii) for all x_{0}, r such that x_{0} ∈ X and 0 < r there exists s such that 0 < s

and for every x_{1} such that x_{1} ∈ X and ∥x_{1} −x_{0}∥ < s holds ∥f_{x_{1}} −f_{x_{0}}∥ < r.

(27) Let f be a partial function from the carrier of S to ℝ. Then f is conti-

nuous on X if and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for all x_{0}, r such that x_{0} ∈ X and 0 < r there exists s such that 0 < s

and for every x_{1} such that x_{1} ∈ X and ∥x_{1} − x_{0}∥ < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(28) f is continuous on X iff f↾X is continuous on X.

(29) Let f be a partial function from the carrier of S to ℝ. Then f is continuous on X if and only if f↾X is continuous on X.

(30) If f is continuous on X and X_{1} ⊆ X, then f is continuous on X_{1}.

(31) If x_{0} ∈ domf, then f is continuous on {x_{0}}.

(32) For all X, f_{1}, f_{2} such that f_{1} is continuous on X and f_{2} is continuous on X holds f_{1} + f_{2} is continuous on X and f_{1} − f_{2} is continuous on X.

(33) Let given X, X_{1}, f_{1}, f_{2}. Suppose f_{1} is continuous on X and f_{2} is continuous on X_{1}. Then f_{1} + f_{2} is continuous on X ∩ X_{1} and f_{1} − f_{2} is continuous on X ∩ X_{1}.

(34) For all r, X, f such that f is continuous on X holds r f is continuous on X.

(35) If f is continuous on X, then ∥f∥ is continuous on X and − f is continuous on X.

(36) Suppose f is total and for all x_{1}, x_{2} holds f_{x_{1}+x_{2}} = f_{x_{1}} + f_{x_{2}} and there exists x_{0} such that f is continuous in x_{0}. Then f is continuous on the carrier of S.

(37) For every f such that domf is compact and f is continuous on domf holds rngf is compact.

(38) Let f be a partial function from the carrier of S to ℝ. If domf is compact and f is continuous on domf, then rngf is compact.

(39) If Y ⊆ domf and Y is compact and f is continuous on Y, then f^{∘}Y is compact.

(40) Let f be a partial function from the carrier of S to ℝ. Suppose domf ≠ ∅ and domf is compact and f is continuous on domf. Then there exist x_{1}, x_{2} such that x_{1} ∈ domf and x_{2} ∈ domf and f_{x_{1}} = suprngf and f_{x_{2}} = inf rngf.

(41) Let given f. Suppose domf ≠ ∅ and domf is compact and f is continuous on domf. Then there exist x_{1}, x_{2} such that x_{1} ∈ domf and x_{2} ∈ domf and ∥f∥_{x_{1}} = suprng∥f∥ and ∥f∥_{x_{2}} = inf rng∥f∥.

(42) ∥f∥↾X = ∥f↾X∥.

(43) Let given f, Y. Suppose Y ≠ ∅ and Y ⊆ domf and Y is compact and f is continuous on Y. Then there exist x_{1}, x_{2} such that x_{1} ∈ Y and x_{2} ∈ Y and ∥f∥_{x_{1}} = sup(∥f∥^{∘}Y) and ∥f∥_{x_{2}} = inf(∥f∥^{∘}Y).

(44) Let f be a partial function from the carrier of S to ℝ and given Y. Suppose Y ≠ ∅ and Y ⊆ domf and Y is compact and f is continuous on Y. Then there exist x_{1}, x_{2} such that x_{1} ∈ Y and x_{2} ∈ Y and f_{x_{1}} = sup(f^{∘}Y)

and f_{x_{2}} = inf(f^{∘}Y). Let us consider S, T and let us consider X, f. We say that f is Lipschitzian

on X if and only if: (Def. 13) X ⊆ domf and there exists r such that 0 < r and for all x_{1}, x_{2} such

that x_{1} ∈ X and x_{2} ∈ X holds ∥f_{x_{1}} − f_{x_{2}}∥ r · ∥x_{1} − x_{2}∥. Let us consider S, let us consider X, and let f be a partial function from

the carrier of S to ℝ. We say that f is Lipschitzian on X if and only if: (Def. 14) X ⊆ domf and there exists r such that 0 < r and for all x_{1}, x_{2} such

that x_{1} ∈ X and x_{2} ∈ X holds |f_{x_{1}} − f_{x_{2}}| r · ∥x_{1} − x_{2}∥. The following propositions are true:

(45) If f is Lipschitzian on X and X_{1} ⊆ X, then f is Lipschitzian on X_{1}.

(46) If f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1}, then f_{1} + f_{2} is Lipschitzian on X ∩ X_{1}.

(47) If f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1}, then f_{1} − f_{2} is Lipschitzian on X ∩ X_{1}.

(48) If f is Lipschitzian on X, then p f is Lipschitzian on X.

(49) If f is Lipschitzian on X, then − f is Lipschitzian on X and ∥f∥ is Lipschitzian on X.

(50) If X ⊆ domf and f is a constant on X, then f is Lipschitzian on X.

(51) id_{Y} is Lipschitzian on Y.

(52) If f is Lipschitzian on X, then f is continuous on X.

(53) Let f be a partial function from the carrier of S to ℝ. If f is Lipschitzian on X, then f is continuous on X.

(54) For every f such that there exists a point r of T such that rngf = {r} holds f is continuous on domf.

(55) If X ⊆ domf and f is a constant on X, then f is continuous on X.

(56) For every partial function f from S to S such that for every x_{0} such that x_{0} ∈ domf holds f_{x_{0}} = x_{0} holds f is continuous on domf.

(57) For every partial function f from S to S such that f = id_{dom f} holds f is continuous on domf.

(58) For every partial function f from S to S such that Y ⊆ domf and f↾Y = id_{Y} holds f is continuous on Y.

(59) Let f be a partial function from S to S, r be a real number, and p be a point of S. Suppose X ⊆ domf and for every x_{0} such that x_{0} ∈ X holds f_{x_{0}} = r · x_{0} + p. Then f is continuous on X.

(60) Let f be a partial function from the carrier of S to ℝ. If for every x_{0} such that x_{0} ∈ domf holds f_{x_{0}} = ∥x_{0}∥, then f is continuous on domf.

(61) Let f be a partial function from the carrier of S to ℝ. If X ⊆ domf and for every x_{0} such that x_{0} ∈ X holds f_{x_{0}} = ∥x_{0}∥, then f is continuous on X. the continuous functions on normed linear … 275

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[5] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[6] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[7] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[8] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[9] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[10] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[11] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[12] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[13] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[14] Yasunari Shidama. The series on Banach algebra. Formalized Mathematics, 12(2):131– 138, 2004.

[15] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[16] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[17] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[18] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[21] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[22] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171–175, 1992.

The Uniform Continuity of Functions on Normed Linear Spaces

Takaya Nishiyama Shinshu University

Nagano

Artur Korniłowicz^{1} University of Białystok Yasunari Shidama Shinshu University Nagano

Summary. In this article, the basic properties of uniform continuity of functions on normed linear spaces are described.

MML Identifier: NFCONT 2.

The notation and terminology used in this paper are introduced in the following articles: [15], [18], [19], [1], [20], [3], [2], [7], [14], [16], [9], [13], [4], [17], [6], [5], [11], [21], [10], [12], and [8].

1. The Uniform Continuity of Functions on Normed Linear Spaces

For simplicity, we follow the rules: X, X_{1} are sets, s, r, p are real numbers, S, T are real normed spaces, f, f_{1}, f_{2} are partial functions from S to T, x_{1}, x_{2} are points of S, and Y is a subset of S.

Let us consider X, S, T and let us consider f. We say that f is uniformly continuous on X if and only if the conditions (Def. 1) are satisfied.

(Def. 1)(i) X ⊆ domf, and

(ii) for every r such that 0 < r there exists s such that 0 < s and for all x_{1},

x_{2} such that x_{1} ∈ X and x_{2} ∈ X and ∥x_{1} − x_{2}∥ < s holds ∥f_{x_{1}} − f_{x_{2}}∥ < r. Let us consider X, S and let f be a partial function from the carrier of S

to ℝ. We say that f is uniformly continuous on X if and only if the conditions (Def. 2) are satisfied.

1The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan.

(Def. 2)(i) X ⊆ domf, and

(ii) for every r such that 0 < r there exists s such that 0 < s and for all x_{1},

x_{2} such that x_{1} ∈ X and x_{2} ∈ X and ∥x_{1} − x_{2}∥ < s holds |f_{x_{1}} − f_{x_{2}}| < r. The following propositions are true:

(1) If f is uniformly continuous on X and X_{1} ⊆ X, then f is uniformly continuous on X_{1}.

(2) If f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1}, then f_{1} + f_{2} is uniformly continuous on X ∩ X_{1}.

(3) If f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1}, then f_{1} − f_{2} is uniformly continuous on X ∩ X_{1}.

(4) If f is uniformly continuous on X, then p f is uniformly continuous on X.

(5) If f is uniformly continuous on X, then − f is uniformly continuous on X.

(6) If f is uniformly continuous on X, then ∥f∥ is uniformly continuous on X.

(7) If f is uniformly continuous on X, then f is continuous on X.

(8) Let f be a partial function from the carrier of S to ℝ. If f is uniformly continuous on X, then f is continuous on X.

(9) If f is Lipschitzian on X, then f is uniformly continuous on X.

(10) For all f, Y such that Y is compact and f is continuous on Y holds f is uniformly continuous on Y.

(11) If Y ⊆ domf and Y is compact and f is uniformly continuous on Y, then f^{∘}Y is compact.

(12) Let f be a partial function from the carrier of S to ℝ and given Y. Suppose Y ≠ ∅ and Y ⊆ domf and Y is compact and f is uniformly continuous on Y. Then there exist x_{1}, x_{2} such that x_{1} ∈ Y and x_{2} ∈ Y and f_{x_{1}} = sup(f^{∘}Y) and f_{x_{2}} = inf(f^{∘}Y).

(13) If X ⊆ domf and f is a constant on X, then f is uniformly continuous on X.

2. The Contraction Mapping Principle on Normed Linear Spaces

Let M be a real Banach space. A function from the carrier of M into the carrier of M is said to be a contraction of M if:

(Def. 3) There exists a real number L such that 0 < L and L < 1 and for all

points x, y of M holds ∥it(x) − it(y)∥ L · ∥x − y∥. The following two propositions are true:

the uniform continuity of functions on … 279

(14) Let X be a real Banach space and f be a function from X into X. Suppose f is a contraction of X. Then there exists a point x_{3} of X such that f(x_{3}) = x_{3} and for every point x of X such that f(x) = x holds x_{3} = x.

(15) Let X be a real Banach space and f be a function from X into X. Given a natural number n_{0} such that f^{n_{0}} is a contraction of X. Then there exists a point x_{3} of X such that f(x_{3}) = x_{3} and for every point x of X such that f(x) = x holds x_{3} = x.

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[2] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[3] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[4] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[5] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[6] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[7] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[8] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269–275, 2004.

[9] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[10] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[11] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.

[12] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2003.

[13] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[14] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[15] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[16] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[17] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[20] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[21] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171–175, 1992.

Series on Complex Banach Algebra

Noboru Endou Gifu National College of Technology

Summary. This article is an extension of [20].

MML Identifier: CLOPBAN3.

The articles [22], [24], [25], [5], [6], [3], [2], [21], [11], [1], [23], [4], [15], [16], [17], [14], [12], [13], [19], [18], [10], [8], [9], [7], and [20] provide the notation and terminology for this paper.

1. Basic Properties of Sequences of Norm Space

Let X be a non empty complex normed space structure and let s_{1} be a sequence of X. The functor (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} yielding a sequence of X is defined as follows:

(Def. 1) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(0) = s_{1}(0) and for every natural number n holds

(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n + 1) = (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n) + s_{1}(n + 1). One can prove the following proposition

(1) Let X be an add-associative right zeroed right complementable non empty complex normed space structure and s_{1} be a sequence of X. Suppose that for every natural number n holds s_{1}(n) = 0_{X}. Let m be a natural number. Then (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(m) = 0_{X}.

Let X be a complex normed space and let s_{1} be a sequence of X. We say that s_{1} is summable if and only if:

(Def. 2) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} is convergent.

Let X be a complex normed space. One can verify that there exists a sequence of X which is summable.

Let X be a complex normed space and let s_{1} be a sequence of X. The functor ∑ s_{1} yields an element of X and is defined by:

(Def. 3) ∑ s_{1} = lim((∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}).

Let X be a complex normed space and let s_{1} be a sequence of X. We say that s_{1} is norm-summable if and only if:

(Def. 4) ∥s_{1}∥ is summable.

The following propositions are true:

(2) For every complex normed space X and for every sequence s_{1} of X and for every natural number m holds 0 ∥s_{1}∥(m).

(3) For every complex normed space X and for all elements x, y, z of X holds ∥x − y∥ = ∥(x − z) + (z − y)∥.

(4) Let X be a complex normed space and s_{1} be a sequence of X. Suppose s_{1} is convergent. Let s be a real number. Suppose 0 < s. Then there exists a natural number n such that for every natural number m if n m, then ∥s_{1}(m) − s_{1}(n)∥ < s.

(5) Let X be a complex normed space and s_{1} be a sequence of X. Then s_{1} is Cauchy sequence by norm if and only if for every real number p such that p > 0 there exists a natural number n such that for every natural number m such that n m holds ∥s_{1}(m) − s_{1}(n)∥ < p.

(6) Let X be a complex normed space and s_{1} be a sequence of X. Suppose that for every natural number n holds s_{1}(n) = 0_{X}. Let m be a natural number. Then (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ}(m) = 0.

Let X be a complex normed space and let s_{1} be a sequence of X. Let us observe that s_{1} is constant if and only if:

(Def. 5) There exists an element r of X such that for every natural number n

holds s_{1}(n) = r. Let X be a complex normed space, let s_{1} be a sequence of X, and let k

be a natural number. The functor s_{1} ↑ k yielding a sequence of X is defined as follows:

(Def. 6) For every natural number n holds (s_{1} ↑ k)(n) = s_{1}(n + k).

Let X be a complex normed space and let s_{1}, s_{2} be sequences of X. We say that s_{1} is a subsequence of s_{2} if and only if:

(Def. 7) There exists an increasing sequence N_{1} of naturals such that s_{1} = s_{2}·N_{1}. Next we state a number of propositions:

(7) For every complex normed space X and for every sequence s_{1} of X holds s_{1} ↑ 0 = s_{1}.

(8) For every complex normed space X and for every sequence s_{1} of X and for all natural numbers k, m holds s_{1} ↑ k ↑ m = s_{1} ↑ m ↑ k.

(9) For every complex normed space X and for every sequence s_{1} of X and for all natural numbers k, m holds s_{1} ↑ k ↑ m = s_{1} ↑ (k + m).

(10) Let X be a complex normed space and s_{1}, s_{2} be sequences of X. If s_{2} is a subsequence of s_{1} and s_{1} is convergent, then s_{2} is convergent.

(11) Let X be a complex normed space and s_{1}, s_{2} be sequences of X. If s_{2} is a subsequence of s_{1} and s_{1} is convergent, then lims_{2} = lims_{1}.

(12) Let X be a complex normed space, s_{1} be a sequence of X, and k be a natural number. Then s_{1} ↑ k is a subsequence of s_{1}.

(13) Let X be a complex normed space, s_{1}, s_{2} be sequences of X, and k be a natural number. If s_{1} is convergent, then s_{1} ↑ k is convergent and lim(s_{1} ↑ k) = lims_{1}.

(14) Let X be a complex normed space and s_{1}, s_{2} be sequences of X. Suppose s_{1} is convergent and there exists a natural number k such that s_{1} = s_{2} ↑k. Then s_{2} is convergent.

(15) Let X be a complex normed space and s_{1}, s_{2} be sequences of X. Suppose s_{1} is convergent and there exists a natural number k such that s_{1} = s_{2} ↑k. Then lims_{2} = lims_{1}.

(16) For every complex normed space X and for every sequence s_{1} of X such that s_{1} is constant holds s_{1} is convergent.

(17) Let X be a complex normed space and s_{1} be a sequence of X. If for

every natural number n holds s_{1}(n) = 0_{X}, then s_{1} is norm-summable. Let X be a complex normed space. Observe that there exists a sequence of

X which is norm-summable.

The following three propositions are true:

(18) Let X be a complex normed space and s be a sequence of X. If s is summable, then s is convergent and lims = 0_{X}.

(19) For every complex normed space X and for all sequences s_{3}, s_{4} of X holds (∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ} + (∑_{α=0}^{κ}(s_{4})(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{3} + s_{4})(α))_{κ∈ℕ}.

(20) For every complex normed space X and for all sequences s_{3}, s_{4} of X

holds (∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ} − (∑_{α=0}^{κ}(s_{4})(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{3} − s_{4})(α))_{κ∈ℕ}. Let X be a complex normed space and let s_{1} be a norm-summable sequence

of X. Observe that ∥s_{1}∥ is summable.

Let X be a complex normed space. One can check that every sequence of X which is summable is also convergent.

The following two propositions are true:

(21) Let X be a complex normed space and s_{2}, s_{5} be sequences of X. If s_{2} is summable and s_{5} is summable, then s_{2} +s_{5} is summable and ∑(s_{2} +s_{5}) = ∑ s_{2} + ∑ s_{5}.

(22) Let X be a complex normed space and s_{2}, s_{5} be sequences of X. If s_{2} is summable and s_{5} is summable, then s_{2} −s_{5} is summable and ∑(s_{2} −s_{5}) = ∑ s_{2} − ∑ s_{5}.

Let X be a complex normed space and let s_{2}, s_{5} be summable sequences of X. One can check that s_{2} + s_{5} is summable and s_{2} − s_{5} is summable.

The following propositions are true:

(23) For every complex normed space X and for every sequence s_{1} of X and for every complex number z holds (∑_{α=0}^{κ}(z · s_{1})(α))_{κ∈ℕ} = z · (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}.

(24) Let X be a complex normed space, s_{1} be a summable sequence of X, and

z be a complex number. Then z · s_{1} is summable and ∑(z · s_{1}) = z · ∑ s_{1}. Let X be a complex normed space, let z be a complex number, and let s_{1}

be a summable sequence of X. One can check that z · s_{1} is summable.

Next we state two propositions:

(25) Let X be a complex normed space and s, s_{3} be sequences of X. If for every natural number n holds s_{3}(n) = s(0), then (∑_{α=0}^{κ}(s ↑ 1)(α))_{κ∈ℕ} = (∑_{α=0}^{κ} s(α))_{κ∈ℕ} ↑ 1 − s_{3}.

(26) Let X be a complex normed space and s be a sequence of X. If s is

summable, then for every natural number n holds s ↑ n is summable. Let X be a complex normed space, let s_{1} be a summable sequence of X, and

let n be a natural number. Observe that s_{1} ↑ n is summable.

We now state the proposition

(27) Let X be a complex normed space and s_{1} be a sequence of X. Then

(∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ} is upper bounded if and only if s_{1} is norm-summable. Let X be a complex normed space and let s_{1} be a norm-summable sequence

of X. Note that (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ} is upper bounded.

The following propositions are true:

(28) Let X be a complex Banach space and s_{1} be a sequence of X. Then s_{1} is summable if and only if for every real number p such that 0 < p there exists a natural number n such that for every natural number m such that n m holds ∥(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(n)∥ < p.

(29) Let X be a complex normed space, s be a sequence of X, and n, m be natural numbers. If n m, then ∥(∑_{α=0}^{κ} s(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ} s(α))_{κ∈ℕ}(n)∥ |(∑_{α=0}^{κ}∥s∥(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ}∥s∥(α))_{κ∈ℕ}(n)|.

(30) For every complex Banach space X and for every sequence s_{1} of X such that s_{1} is norm-summable holds s_{1} is summable.

(31) Let X be a complex normed space, r_{1} be a sequence of real numbers, and s_{5} be a sequence of X. Suppose r_{1} is summable and there exists a natural number m such that for every natural number n such that m n holds ∥s_{5}(n)∥ r_{1}(n). Then s_{5} is norm-summable.

(32) Let X be a complex normed space and s_{2}, s_{5} be sequences of X. Suppose for every natural number n holds 0 ∥s_{2}∥(n) and ∥s_{2}∥(n) ∥s_{5}∥(n) and s_{5} is norm-summable. Then s_{2} is norm-summable and ∑∥s_{2}∥ ∑∥s_{5}∥.

(33) Let X be a complex normed space and s_{1} be a sequence of X. Suppose

that

(i) for every natural number n holds ∥s_{1}∥(n) > 0, and (ii) there exists a natural number m such that for every natural number n

such that n m holds \frac{∥s_{1}∥(n+1)}{∥s_{1}∥(n)} 1. Then s_{1} is not norm-summable.

(34) Let X be a complex normed space, s_{1} be a sequence of X, and r_{1} be a sequence of real numbers. Suppose for every natural number n holds r_{1}(n) = ^{n} \sqrt{∥s_{1}∥(n)} and r_{1} is convergent and limr_{1} < 1. Then s_{1} is normsummable.

(35) Let X be a complex normed space, s_{1} be a sequence of X, and r_{1} be a

sequence of real numbers. Suppose that

(i) for every natural number n holds r_{1}(n) = ^{n} \sqrt{∥s_{1}∥(n)}, and (ii) there exists a natural number m such that for every natural number n

such that m n holds r_{1}(n) 1. Then ∥s_{1}∥ is not summable.

(36) Let X be a complex normed space, s_{1} be a sequence of X, and r_{1} be a sequence of real numbers. Suppose for every natural number n holds r_{1}(n) = ^{n} \sqrt{∥s_{1}∥(n)} and r_{1} is convergent and limr_{1} > 1. Then s_{1} is not norm-summable.

(37) Let X be a complex normed space, s_{1} be a sequence of X, and r_{1} be a sequence of real numbers. Suppose ∥s_{1}∥ is non-increasing and for every natural number n holds r_{1}(n) = 2^{n} · ∥s_{1}∥(2^{n}). Then s_{1} is norm-summable if and only if r_{1} is summable.

(38) Let X be a complex normed space, s_{1} be a sequence of X, and p be a real number. Suppose p > 1 and for every natural number n such that n 1 holds ∥s_{1}∥(n) = \frac{1}{n^{p}} . Then s_{1} is norm-summable.

(39) Let X be a complex normed space, s_{1} be a sequence of X, and p be a real number. Suppose p 1 and for every natural number n such that n 1 holds ∥s_{1}∥(n) = \frac{1}{n^{p}} . Then s_{1} is not norm-summable.

(40) Let X be a complex normed space, s_{1} be a sequence of X, and r_{1} be a sequence of real numbers. Suppose for every natural number n holds s_{1}(n) ≠ 0_{X} and r_{1}(n) = \frac{∥s_{1}∥(n+1)}{∥s_{1}∥(n)} and r_{1} is convergent and limr_{1} < 1. Then s_{1} is norm-summable.

(41) Let X be a complex normed space and s_{1} be a sequence of X. Suppose

that

(i) for every natural number n holds s_{1}(n) ≠ 0_{X}, and (ii) there exists a natural number m such that for every natural number n

such that n m holds \frac{∥s_{1}∥(n+1)}{∥s_{1}∥(n)} 1. Then s_{1} is not norm-summable.

Let X be a complex Banach space. One can check that every sequence of X which is norm-summable is also summable.

2. Basic Properties of Sequence of Banach Algebra

The scheme ExNCBCASeq deals with a non empty normed complex algebra structure A and a unary functor F yielding a point of A, and states that:

There exists a sequence S of A such that for every natural number n holds S(n) = F(n)

for all values of the parameters.

We now state the proposition

(42) Let X be a complex Banach algebra, x, y, z be elements of X, and a, b be complex numbers. Then x + y = y + x and (x + y) + z = x + (y + z) and x + 0_{X} = x and there exists an element t of X such that x + t = 0_{X} and (x · y) · z = x · (y · z) and 1_{ℂ} · x = x and 0_{ℂ} · x = 0_{X} and a · 0_{X} = 0_{X} and (− 1_{ℂ}) · x = − x and x · 1_{X} = x and 1_{X} · x = x and x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x and a · (x · y) = (a · x) · y and a · (x + y) = a · x + a · y and (a + b) · x = a · x + b · x and (a · b) · x = a · (b · x) and (a · b) · (x · y) = a · x · (b · y) and a · (x · y) = x · (a · y) and 0_{X} · x = 0_{X} and x · 0_{X} = 0_{X} and x · (y − z) = x · y − x · z and (y − z) · x = y · x − z · x and (x+y)−z = x+(y−z) and (x−y)+z = x−(y−z) and x−y−z = x−(y+z) and x+y = (x−z)+(z+y) and x−y = (x−z)+(z−y) and x = (x−y)+y and x = y − (y − x) and ∥x∥ = 0 iff x = 0_{X} and ∥a · x∥ = |a| · ∥x∥ and ∥x + y∥ ∥x∥ + ∥y∥ and ∥x · y∥ ∥x∥ · ∥y∥ and ∥1_{X}∥ = 1 and X is complete.

Let X be a non empty normed complex algebra structure, let S be a sequence of X, and let a be an element of X. The functor a · S yields a sequence of X and is defined by:

(Def. 8) For every natural number n holds (a · S)(n) = a · S(n).

Let X be a non empty normed complex algebra structure, let S be a sequence of X, and let a be an element of X. The functor S · a yields a sequence of X and is defined by:

(Def. 9) For every natural number n holds (S · a)(n) = S(n) · a.

Let X be a non empty normed complex algebra structure and let s_{2}, s_{5} be sequences of X. The functor s_{2} · s_{5} yielding a sequence of X is defined by:

(Def. 10) For every natural number n holds (s_{2} · s_{5})(n) = s_{2}(n) · s_{5}(n).

Let X be a complex Banach algebra and let x be an element of X. Let us assume that x is invertible. The functor x^{−1} yields an element of X and is defined as follows:

(Def. 11) x · x^{−1} = 1_{X} and x^{−1} · x = 1_{X}.

series on complex banach algebra 287

Let X be a complex Banach algebra and let z be an element of X. The functor (z^{κ})_{κ∈ℕ} yielding a sequence of X is defined as follows:

(Def. 12) (z^{κ})_{κ∈ℕ}(0) = 1_{X} and for every natural number n holds (z^{κ})_{κ∈ℕ}(n+1) = (z^{κ})_{κ∈ℕ}(n) · z.

Let X be a complex Banach algebra, let z be an element of X, and let n be a natural number. The functor z_{ℕ}^{n} yielding an element of X is defined as follows:

(Def. 13) z_{ℕ}^{n} = (z^{κ})_{κ∈ℕ}(n).

The following propositions are true:

(43) For every complex Banach algebra X and for every element z of X holds z_{ℕ}^{0} = 1_{X}.

(44) For every complex Banach algebra X and for every element z of X such that ∥z∥ < 1 holds (z^{κ})_{κ∈ℕ} is summable and norm-summable.

(45) Let X be a complex Banach algebra and x be a point of X. If ∥1_{X} −x∥ < 1, then ((1_{X} −x)^{κ})_{κ∈ℕ} is summable and ((1_{X} −x)^{κ})_{κ∈ℕ} is norm-summable.

(46) For every complex Banach algebra X and for every point x of X such that ∥1_{X} − x∥ < 1 holds x is invertible and x^{−1} = ∑(((1_{X} − x)^{κ})_{κ∈ℕ}).

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[7] Noboru Endou. Banach algebra of bounded complex linear operators. Formalized Mathematics, 12(3):237–242, 2004.

[8] Noboru Endou. Banach space of absolute summable complex sequences. Formalized Mathematics, 12(2):191–194, 2004.

[9] Noboru Endou. Complex Banach space of bounded linear operators. Formalized Mathematics, 12(2):201–209, 2004.

[10] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[12] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[13] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[14] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[15] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[16] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[17] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[18] Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[19] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[20] Yasunari Shidama. The series on Banach algebra. Formalized Mathematics, 12(2):131– 138, 2004.

[21] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[22] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[23] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Exponential Function on Complex Banach Algebra

Noboru Endou Gifu National College of Technology

Summary. This article is an extension of [18].

MML Identifier: CLOPBAN4.

The papers [23], [24], [4], [5], [2], [20], [21], [9], [1], [22], [13], [15], [16], [12], [10], [11], [17], [14], [25], [3], [7], [6], [19], and [8] provide the notation and terminology for this paper.

For simplicity, we adopt the following convention: X denotes a complex Banach algebra, w, z, z_{1}, z_{2} denote elements of X, k, l, m, n denote natural numbers, s_{1}, s_{2}, s_{3}, s, s′ denote sequences of X, and r_{1} denotes a sequence of real numbers.

Let X be a non empty normed complex algebra structure and let x, y be elements of X. We say that x, y are commutative if and only if:

(Def. 1) x · y = y · x.

Let us note that the predicate x, y are commutative is symmetric.

One can prove the following propositions:

(1) If s_{2} is convergent and s_{3} is convergent and lim(s_{2} − s_{3}) = 0_{X}, then lims_{2} = lims_{3}.

(2) For every z such that for every natural number n holds s(n) = z holds lims = z.

(3) If s is convergent and s′ is convergent, then s · s′ is convergent.

(4) If s is convergent, then z · s is convergent.

(5) If s is convergent, then s · z is convergent.

(6) If s is convergent, then lim(z · s) = z · lims.

(7) If s is convergent, then lim(s · z) = lims · z.

(8) If s is convergent and s′ is convergent, then lim(s · s′) = lims · lims′.

(9) (∑_{α=0}^{κ}(z·s_{1})(α))_{κ∈ℕ} = z·(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} and (∑_{α=0}^{κ}(s_{1}·z)(α))_{κ∈ℕ} = (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ} · z.

(10) ∥(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(k)∥ (∑_{α=0}^{κ}∥s_{1}∥(α))_{κ∈ℕ}(k).

(11) If for every n such that n m holds s_{2}(n) = s_{3}(n), then (∑_{α=0}^{κ}(s_{2})(α))_{κ∈ℕ}(m) = (∑_{α=0}^{κ}(s_{3})(α))_{κ∈ℕ}(m).

(12) If for every n holds ∥s_{1}(n)∥ r_{1}(n) and r_{1} is convergent and limr_{1} = 0,

then s_{1} is convergent and lims_{1} = 0_{X}. Let us consider X, z. The functor z ExpSeq yields a sequence of X and is

defined as follows: (Def. 2) For every n holds z ExpSeq(n) = \frac{1_{ℂ}}{n!_{ℂ}} · z_{ℕ}^{n}.

The scheme ExNormSpace CASE deals with a non empty complex Banach algebra A and a binary functor F yielding a point of A, and states that:

For every k there exists a sequence s_{1} of A such that for every n holds if n k, then s_{1}(n) = F(k, n) and if n > k, then s_{1}(n) = 0_{A}

for all values of the parameters.

Let us consider X, s_{1}. The functor Shifts_{1} yielding a sequence of X is defined by:

(Def. 3) (Shifts_{1})(0) = 0_{X} and for every natural number k holds (Shifts_{1})(k +

1) = s_{1}(k). Let us consider n, X, z, w. The functor Expan(n, z, w) yielding a sequence

of X is defined by: (Def. 4) For every natural number k holds if k n, then (Expan(n, z, w))(k) =

(Coef n)(k) · z_{ℕ}^{k} · w_{ℕ}^{n− ′k} and if n < k, then (Expan(n, z, w))(k) = 0_{X}. Let us consider n, X, z, w. The functor Expan e(n, z, w) yields a sequence

of X and is defined as follows: (Def. 5) For every natural number k holds if k n, then (Expan e(n, z, w))(k) =

(Coef en)(k) · z_{ℕ}^{k} · w_{ℕ}^{n− ′k} and if n < k, then (Expan e(n, z, w))(k) = 0_{X}. Let us consider n, X, z, w. The functor Alfa(n, z, w) yielding a sequence of

X is defined by: (Def. 6) For every natural number k holds if k n, then (Alfa(n, z, w))(k) =

z ExpSeq(k) · (∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(n − ′ k) and if n < k, then (Alfa(n, z, w))(k) = 0_{X}.

Let us consider X, z, w, n. The functor Conj(n, z, w) yields a sequence of X and is defined as follows:

(Def. 7) For every natural number k holds if k n, then (Conj(n, z, w))(k) = z ExpSeq(k)·((∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(n)−(∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(n− ′ k)) and if n < k, then (Conj(n, z, w))(k) = 0_{X}.

Next we state several propositions:

(13) z ExpSeq(n + 1) = \frac{1_{ℂ}}{(n+1)+0i} · z · z ExpSeq(n) and z ExpSeq(0) = 1_{X} and ∥z ExpSeq(n)∥ ∥z∥ ExpSeq(n).

(14) If 0 < k, then (Shifts_{1})(k) = s_{1}(k − ′1).

(15) (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Shifts_{1})(α))_{κ∈ℕ}(k) + s_{1}(k).

(16) For all z, w such that z, w are commutative holds (z + w)_{ℕ}^{n} = (∑_{α=0}^{κ}(Expan(n, z, w))(α))_{κ∈ℕ}(n).

(17) Expan e(n, z, w) = \frac{1_{ℂ}}{n!_{ℂ}} · Expan(n, z, w).

(18) For all z, w such that z, w are commutative holds \frac{1_{ℂ}}{n!_{ℂ}} · (z + w)_{ℕ}^{n} = (∑_{α=0}^{κ}(Expan e(n, z, w))(α))_{κ∈ℕ}(n).

(19) 0_{X} ExpSeq is norm-summable and ∑(0_{X} ExpSeq) = 1_{X}.

Let us consider X and let z be an element of X. One can check that z ExpSeq is norm-summable.

We now state a number of propositions:

(20) z ExpSeq(0) = 1_{X} and (Expan(0, z, w))(0) = 1_{X}.

(21) If l k, then (Alfa(k + 1, z, w))(l) = (Alfa(k, z, w))(l) + (Expan e(k + 1, z, w))(l).

(22) (∑_{α=0}^{κ}(Alfa(k + 1, z, w))(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Alfa(k, z, w))(α))_{κ∈ℕ}(k) + (∑_{α=0}^{κ}(Expan e(k + 1, z, w))(α))_{κ∈ℕ}(k).

(23) z ExpSeq(k) = (Expan e(k, z, w))(k).

(24) For all z, w such that z, w are commutative holds (∑_{α=0}^{κ} z + w ExpSeq(α))_{κ∈ℕ}(n) = (∑_{α=0}^{κ}(Alfa(n, z, w))(α))_{κ∈ℕ}(n).

(25) For all z, w such that z, w are commutative holds (∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k) · (∑_{α=0}^{κ} w ExpSeq(α))_{κ∈ℕ}(k) − (∑_{α=0}^{κ} z+ w ExpSeq(α))_{κ∈ℕ}(k) = (∑_{α=0}^{κ}(Conj(k, z, w))(α))_{κ∈ℕ}(k).

(26) 0 ∥z∥ ExpSeq(n).

(27) ∥(∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k)∥ (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(k) and (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(k) ∑(∥z∥ ExpSeq) and ∥(∑_{α=0}^{κ} z ExpSeq(α))_{κ∈ℕ}(k)∥ ∑(∥z∥ ExpSeq).

(28) 1 ∑(∥z∥ ExpSeq).

(29) |(∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(n) and if n m, then |(∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(m)−(∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(m) − (∑_{α=0}^{κ}∥z∥ ExpSeq(α))_{κ∈ℕ}(n).

(30) |(∑_{α=0}^{κ}∥Conj(k, z, w)∥(α))_{κ∈ℕ}(n)| = (∑_{α=0}^{κ}∥Conj(k, z, w)∥(α))_{κ∈ℕ}(n).

(31) For every real number p such that p > 0 there exists n such that for every k such that n k holds |(∑_{α=0}^{κ}∥Conj(k, z, w)∥(α))_{κ∈ℕ}(k)| < p.

(32) For every s_{1} such that for every k holds s_{1}(k) =

(∑_{α=0}^{κ}(Conj(k, z, w))(α))_{κ∈ℕ}(k) holds s_{1} is convergent and lims_{1} = 0_{X}. Let us consider X. The functor expX yields a function from the carrier of

X into the carrier of X and is defined by:

(Def. 8) For every element z of the carrier of X holds (expX)(z) = ∑(z ExpSeq). Let us consider X, z. The functor expz yielding an element of X is defined

as follows: (Def. 9) expz = (expX)(z).

The following propositions are true:

(33) For every z holds expz = ∑(z ExpSeq).

(34) Let given z_{1}, z_{2}. Suppose z_{1}, z_{2} are commutative. Then exp(z_{1} + z_{2}) = expz_{1} ·expz_{2} and exp(z_{2} +z_{1}) = expz_{2} ·expz_{1} and exp(z_{1} +z_{2}) = exp(z_{2} + z_{1}) and expz_{1}, expz_{2} are commutative.

(35) For all z_{1}, z_{2} such that z_{1}, z_{2} are commutative holds z_{1}·expz_{2} = expz_{2}·z_{1}.

(36) exp(0_{X}) = 1_{X}.

(37) expz · exp(− z) = 1_{X} and exp(− z) · expz = 1_{X}.

(38) expz is invertible and (expz)^{−1} = exp(− z) and exp(− z) is invertible and (exp(− z))^{−1} = expz.

(39) For every z and for all complex numbers s, t holds s · z, t · z are commutative.

(40) Let given z and s, t be complex numbers. Then exp(s · z) · exp(t · z) = exp((s+t)·z) and exp(t·z)·exp(s·z) = exp((t+s)·z) and exp((s+t)·z) = exp((t + s) · z) and exp(s · z), exp(t · z) are commutative.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Noboru Endou. Banach algebra of bounded complex linear operators. Formalized Mathematics, 12(3):237–242, 2004.

[7] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[8] Noboru Endou. Series on complex Banach algebra. Formalized Mathematics, 12(3):281– 288, 2004.

[9] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[10] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[11] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[13] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[14] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

exponential function … 293

[15] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[16] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[17] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449– 452, 1991.

[18] Yasunari Shidama. The exponential function on Banach algebra. Formalized Mathematics, 12(2):173–177, 2004.

[19] Yasunari Shidama. The series on Banach algebra. Formalized Mathematics, 12(2):131– 138, 2004.

[20] Andrzej Trybulec. Introduction to arithmetics. To appear in Formalized Mathematics.

[21] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[22] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[24] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[25] Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255–263, 1998.

The Fundamental Group of Convex Subspaces of E_{T}^{n}

Artur Korniłowicz^{1} University of Białystok

Summary. The triviality of the fundamental group of subspaces of E_{T}^{n} and ℝ^{1} have been shown.

MML Identifier: TOPALG 2.

The notation and terminology used in this paper have been introduced in the following articles: [20], [6], [23], [1], [17], [24], [4], [5], [3], [2], [19], [11], [16], [22], [21], [18], [14], [8], [7], [15], [13], [9], [10], and [12].

1. Convex subspaces of E_{T}^{n}

In this paper n denotes a natural number and a, b denote real numbers.

Let us consider n. One can verify that there exists a subset of E_{T}^{n} which is non empty and convex.

Let n be a natural number and let T be a subspace of E_{T}^{n}. We say that T is convex if and only if:

(Def. 1) Ω_{T} is a convex subset of E_{T}^{n}.

Let n be a natural number. Note that every non empty subspace of E_{T}^{n} which is convex is also arcwise connected.

Let n be a natural number. One can verify that there exists a subspace of E_{T}^{n} which is strict, non empty, and convex.

The following proposition is true (1) Let X be a non empty topological space, Y be a non empty subspace of

1The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan. This work has been partially supported by KBN grant 4 T11C 039 24.

X, x_{1}, x_{2} be points of X, y_{1}, y_{2} be points of Y, and f be a path from y_{1} to y_{2}. Suppose x_{1} = y_{1} and x_{2} = y_{2} and y_{1}, y_{2} are connected. Then f is a path from x_{1} to x_{2}.

Let n be a natural number, let T be a non empty convex subspace of E_{T}^{n}, let a, b be points of T, and let P, Q be paths from a to b. The functor ConvexHomotopy(P, Q) yielding a map from [: 𝕀, 𝕀:] into T is defined as follows:

(Def. 2) For all elements s, t of 𝕀 and for all points a_{1}, b_{1} of E_{T}^{n} such that a_{1} = P (s)

and b_{1} = Q(s) holds (ConvexHomotopy(P, Q))(s, t) = (1 − t) · a_{1} + t · b_{1}. Next we state the proposition

(2) Let T be a non empty convex subspace of E_{T}^{n}, a, b be points of T, and P, Q be paths from a to b. Then P, Q are homotopic.

Let n be a natural number, let T be a non empty convex subspace of E_{T}^{n}, let a, b be points of T, and let P, Q be paths from a to b. Then ConvexHomotopy(P, Q) is a homotopy between P and Q.

Let n be a natural number, let T be a non empty convex subspace of E_{T}^{n}, let a, b be points of T, and let P, Q be paths from a to b. Note that every homotopy between P and Q is continuous.

We now state the proposition

(3) Let T be a non empty convex subspace of E_{T}^{n}, a be a point of T, and C be a loop of a. Then the carrier of π_{1}(T, a) = {[C]_{EqRel(T,a)}}.

Let n be a natural number, let T be a non empty convex subspace of E_{T}^{n}, and let a be a point of T. Observe that π_{1}(T, a) is trivial.

2. Convex subspaces of ℝ^{1}

We now state the proposition

(4) Proj(|[a]|, 1) = a. One can verify that every subspace of ℝ^{1} is real-membered. Next we state three propositions:

(5) If a b, then [a, b] = {(1 − l) · a + l · b; l ranges over real numbers: 0 l ∧ l 1}.

(6) Let F be a map from [: ℝ^{1}, 𝕀:] into ℝ^{1}. Suppose that for every point x of ℝ^{1} and for every point i of 𝕀 holds F (x, i) = (1 − i) · x. Then F is continuous.

(7) Let F be a map from [: ℝ^{1}, 𝕀:] into ℝ^{1}. Suppose that for every point x of ℝ^{1} and for every point i of 𝕀 holds F (x, i) = i · x. Then F is continuous.

Let P be a subset of ℝ^{1}. We say that P is convex if and only if: (Def. 3) For all points a, b of ℝ^{1} such that a ∈ P and b ∈ P holds [a, b] ⊆ P.

One can check that there exists a subset of ℝ^{1} which is non empty and convex and every subset of ℝ^{1} which is empty is also convex.

We now state four propositions:

(8) [a, b] is a convex subset of ℝ^{1}.

(9) ]a, b[ is a convex subset of ℝ^{1}.

(10) [a, b[ is a convex subset of ℝ^{1}.

(11) ]a, b] is a convex subset of ℝ^{1}.

Let T be a subspace of ℝ^{1}. We say that T is convex if and only if: (Def. 4) Ω_{T} is a convex subset of ℝ^{1}.

Let us note that there exists a subspace of ℝ^{1} which is strict, non empty, and convex.

ℝ^{1} is a strict convex subspace of ℝ^{1}. The following proposition is true

(12) For every non empty convex subspace T of ℝ^{1} and for all points a, b of

T holds [a, b] ⊆ the carrier of T. Let us note that every non empty subspace of ℝ^{1} which is convex is also

arcwise connected.

One can prove the following propositions:

(13) If a b, then [a, b]_{T} is convex.

(14) 𝕀 is convex.

(15) If a b, then [a, b]_{T} is arcwise connected.

Let T be a non empty convex subspace of ℝ^{1}, let a, b be points of T, and let P, Q be paths from a to b. The functor R1Homotopy(P, Q) yields a map from [: 𝕀, 𝕀:] into T and is defined by:

(Def. 5) For all elements s, t of 𝕀 holds (R1Homotopy(P, Q))(s, t) = (1 − t) ·

P (s) + t · Q(s). Next we state the proposition

(16) Let T be a non empty convex subspace of ℝ^{1}, a, b be points of T, and

P, Q be paths from a to b. Then P, Q are homotopic. Let T be a non empty convex subspace of ℝ^{1}, let a, b be points of T, and

let P, Q be paths from a to b. Then R1Homotopy(P, Q) is a homotopy between P and Q.

Let T be a non empty convex subspace of ℝ^{1}, let a, b be points of T, and let P, Q be paths from a to b. Note that every homotopy between P and Q is continuous.

The following proposition is true

(17) Let T be a non empty convex subspace of ℝ^{1}, a be a point of T, and C be a loop of a. Then the carrier of π_{1}(T, a) = {[C]_{EqRel(T,a)}}.

298 artur korniłowicz

Let T be a non empty convex subspace of ℝ^{1} and let a be a point of T. Observe that π_{1}(T, a) is trivial.

One can prove the following four propositions:

(18) If a b, then for all points x, y of [a, b]_{T} and for all paths P, Q from x to y holds P, Q are homotopic.

(19) If a b, then for every point x of [a, b]_{T} and for every loop C of x holds the carrier of π_{1}([a, b]_{T}, x) = {[C]_{EqRel([a, b]_{T},x)}}.

(20) For all points x, y of 𝕀 and for all paths P, Q from x to y holds P, Q are homotopic.

(21) For every point x of 𝕀 and for every loop C of x holds the carrier of π_{1}(𝕀, x) = {[C]_{EqRel(𝕀,x)}}.

Let x be a point of 𝕀. Observe that π_{1}(𝕀, x) is trivial.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[7] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[8] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[9] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449–454, 1997.

[10] Adam Grabowski and Artur Korniłowicz. Algebraic properties of homotopies. Formalized Mathematics, 12(3):251–260, 2004.

[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[12] Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261–268, 2004.

[13] Roman Matuszewski and Yatsuka Nakamura. Projections in n-dimensional Euclidean space to each coordinates. Formalized Mathematics, 6(4):505–509, 1997.

[14] Yatsuka Nakamura. Half open intervals in real numbers. Formalized Mathematics, 10(1):21–22, 2002.

[15] Yatsuka Nakamura and Jarosław Kotowicz. The Jordan’s property for certain subsets of the plane. Formalized Mathematics, 3(2):137–142, 1992.

[16] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[17] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[18] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[19] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[20] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[21] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

the fundamental group of convex subspaces of … 299

[22] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.

[23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[24] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

Intersections of Intervals and Balls in E_{T}^{n}

Artur Korniłowicz^{1} University of Białystok Yasunari Shidama Shinshu University Nagano

MML Identifier: TOPREAL9.

The terminology and notation used in this paper are introduced in the following papers: [17], [19], [1], [4], [16], [8], [14], [2], [3], [5], [18], [13], [7], [9], [6], [15], [11], [12], and [10].

1. Preliminaries

For simplicity, we follow the rules: n denotes a natural number, a, b, r denote real numbers, x, y, z denote points of E_{T}^{n}, and e denotes a point of E^{n}.

The following propositions are true:

(1) x − y − z = x − z − y.

(2) If x + y = x + z, then y = z.

(3) If n is non empty, then x ≠ x + 1.REALn.

(4) For every set x such that x = (1 − r) · y + r · z holds x = y iff r = 0 or y = z and x = z iff r = 1 or y = z.

(5) For every finite sequence f of elements of ℝ holds |f|^{2} = ∑^{2}f.

(6) For every non empty metric space M and for all points z_{1}, z_{2}, z_{3} of M such that z_{1} ≠ z_{2} and z_{1} ∈ \overline{Ball}(z_{3}, r) and z_{2} ∈ \overline{Ball}(z_{3}, r) holds r > 0.

1The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan. This work has been partially supported by KBN grant 4 T11C 039 24.

2. Subsets of E_{T}^{n}

Let n be a natural number, let x be a point of E_{T}^{n}, and let r be a real number. The functor Ball(x, r) yields a subset of E_{T}^{n} and is defined by:

(Def. 1) Ball(x, r) = {p; p ranges over points of E_{T}^{n}: |p − x| < r}.

The functor \overline{Ball}(x, r) yielding a subset of E_{T}^{n} is defined by: (Def. 2) \overline{Ball}(x, r) = \{p; p ranges over points of E_{T}^{n}: |p − x| r\}.

The functor Sphere(x, r) yielding a subset of E_{T}^{n} is defined as follows: (Def. 3) Sphere(x, r) = {p; p ranges over points of E_{T}^{n}: |p − x| = r}.

We now state a number of propositions:

(7) y ∈ Ball(x, r) iff |y − x| < r.

(8) y ∈ \overline{Ball}(x, r) iff |y − x| r.

(9) y ∈ Sphere(x, r) iff |y − x| = r.

(10) If y ∈ Ball(0_{E_{T}^{n}}, r), then |y| < r.

(11) If y ∈ \overline{Ball}(0_{E_{T}^{n}}, r), then |y| r.

(12) If y ∈ Sphere((0_{E_{T}^{n}}), r), then |y| = r.

(13) If x = e, then Ball(e, r) = Ball(x, r).

(14) If x = e, then \overline{Ball}(e, r) = \overline{Ball}(x, r).

(15) If x = e, then Sphere(e, r) = Sphere(x, r).

(16) Ball(x, r) ⊆ \overline{Ball}(x, r).

(17) Sphere(x, r) ⊆ \overline{Ball}(x, r).

(18) Ball(x, r) ∪ Sphere(x, r) = \overline{Ball}(x, r).

(19) Ball(x, r) misses Sphere(x, r).

Let us consider n, x and let r be a non positive real number. One can check that Ball(x, r) is empty.

Let us consider n, x and let r be a positive real number. Note that Ball(x, r) is non empty.

One can prove the following propositions:

(20) If Ball(x, r) is non empty, then r > 0.

(21) If Ball(x, r) is empty, then r 0.

Let us consider n, x and let r be a negative real number. Observe that \overline{Ball}(x, r) is empty.

Let us consider n, x and let r be a non negative real number. Observe that \overline{Ball}(x, r) is non empty.

The following three propositions are true:

(22) If \overline{Ball}(x, r) is non empty, then r 0.

(23) If \overline{Ball}(x, r) is empty, then r < 0.

(24) If a + b = 1 and |a| + |b| = 1 and b ≠ 0 and x ∈ \overline{Ball}(z, r) and y ∈

Ball(z, r), then a · x + b · y ∈ Ball(z, r). Let us consider n, x, r. One can check the following observations:

∗ Ball(x, r) is open and Bounded,

∗ \overline{Ball}(x, r) is closed and Bounded, and

∗ Sphere(x, r) is closed and Bounded. Let us consider n, x, r. Observe that Ball(x, r) is convex and \overline{Ball}(x, r) is

convex.

Let n be a natural number and let f be a map from E_{T}^{n} into E_{T}^{n}. We say that f is homogeneous if and only if:

(Def. 4) For every real number r and for every point x of E_{T}^{n} holds f(r·x) = r·f(x).

We say that f is additive if and only if: (Def. 5) For all points x, y of E_{T}^{n} holds f(x + y) = f(x) + f(y).

Let us consider n. One can verify that (E_{T}^{n}) ⟼→ 0_{E_{T}^{n}} is homogeneous and additive.

Let us consider n. Observe that there exists a map from E_{T}^{n} into E_{T}^{n} which is homogeneous, additive, and continuous.

Let a, c be real numbers. One can check that AffineMap(a, 0, c, 0) is homogeneous and additive.

One can prove the following proposition

(25) For every homogeneous additive map f from E_{T}^{n} into E_{T}^{n} and for every

convex subset X of E_{T}^{n} holds f^{∘}X is convex. In the sequel p, q are points of E_{T}^{n}. Let n be a natural number and let p, q be points of E_{T}^{n}. The functor HL(p, q)

yields a subset of E_{T}^{n} and is defined by: (Def. 6) HL(p, q) = {(1 − l) · p + l · q; l ranges over real numbers: 0 l}.

One can prove the following proposition

(26) For every set x holds x ∈ HL(p, q) iff there exists a real number l such

that x = (1 − l) · p + l · q and 0 l. Let us consider n, p, q. One can verify that HL(p, q) is non empty. The following propositions are true:

(27) p ∈ HL(p, q).

(28) q ∈ HL(p, q).

(29) HL(p, p) = {p}.

(30) If x ∈ HL(p, q), then HL(p, x) ⊆ HL(p, q).

(31) If x ∈ HL(p, q) and x ≠ p, then HL(p, q) = HL(p, x).

(32) L(p, q) ⊆ HL(p, q).

Let us consider n, p, q. Note that HL(p, q) is convex. One can prove the following propositions:

(33) If y ∈ Sphere(x, r) and z ∈ Ball(x, r), then L(y, z) ∩ Sphere(x, r) = {y}.

(34) If y ∈ Sphere(x, r) and z ∈ Sphere(x, r), then L(y, z)\{y, z} ⊆ Ball(x, r).

(35) If y ∈ Sphere(x, r) and z ∈ Sphere(x, r), then L(y, z) ∩ Sphere(x, r) = {y, z}.

(36) If y ∈ Sphere(x, r) and z ∈ Sphere(x, r), then HL(y, z) ∩ Sphere(x, r) = {y, z}.

(37) If y ≠ z and y ∈ Ball(x, r), then there exists a point e of E_{T}^{n} such that {e} = HL(y, z) ∩ Sphere(x, r).

(38) If y ≠ z and y ∈ Sphere(x, r) and z ∈ \overline{Ball}(x, r), then there exists a

point e of E_{T}^{n} such that e ≠ y and {y, e} = HL(y, z) ∩ Sphere(x, r). Let us consider n, x and let r be a negative real number. Observe that

Sphere(x, r) is empty.

Let n be a non empty natural number, let x be a point of E_{T}^{n}, and let r be a non negative real number. Observe that Sphere(x, r) is non empty.

Next we state two propositions:

(39) If Sphere(x, r) is non empty, then r 0.

(40) If n is non empty and Sphere(x, r) is empty, then r < 0.

3. Subsets of E_{T}^{2}

In the sequel s, t are points of E_{T}^{2}.

The following propositions are true:

(41) (a · s + b · t)_{1}= a · s_{1} + b · t_{1}.

(42) (a · s + b · t)_{2}= a · s_{2} + b · t_{2}.

(43) t ∈ Circle(a, b, r) iff |t − [a, b]| = r.

(44) t ∈ ClosedInsideOfCircle(a, b, r) iff |t − [a, b]| r.

(45) t ∈ InsideOfCircle(a, b, r) iff |t − [a, b]| < r.

Let a, b be real numbers and let r be a positive real number. Observe that InsideOfCircle(a, b, r) is non empty.

Let a, b be real numbers and let r be a non negative real number. Observe that ClosedInsideOfCircle(a, b, r) is non empty.

We now state a number of propositions:

(46) Circle(a, b, r) ⊆ ClosedInsideOfCircle(a, b, r).

(47) For every point x of E^{2} such that x = [a, b] holds \overline{Ball}(x, r) = ClosedInsideOfCircle(a, b, r).

(48) For every point x of E^{2} such that x = [a, b] holds Ball(x, r) = InsideOfCircle(a, b, r).

(49) For every point x of E^{2} such that x = [a, b] holds Sphere(x, r) = Circle(a, b, r).

(50) Ball([a, b], r) = InsideOfCircle(a, b, r).

(51) \overline{Ball}([a, b], r) = ClosedInsideOfCircle(a, b, r).

(52) Sphere([a, b], r) = Circle(a, b, r).

(53) InsideOfCircle(a, b, r) ⊆ ClosedInsideOfCircle(a, b, r).

(54) InsideOfCircle(a, b, r) misses Circle(a, b, r).

(55) InsideOfCircle(a, b, r) ∪ Circle(a, b, r) = ClosedInsideOfCircle(a, b, r).

(56) If s ∈ Sphere((0_{E_{T}^{2}}), r), then (s_{1})^{2} + (s_{2})^{2} = r^{2}.

(57) If s ≠ t and s ∈ ClosedInsideOfCircle(a, b, r) and t ∈ ClosedInsideOfCircle(a, b, r), then r > 0.

(58) If s ≠ t and s ∈ InsideOfCircle(a, b, r), then there exists a point e of E_{T}^{2} such that {e} = HL(s, t) ∩ Circle(a, b, r).

(59) If s ∈ Circle(a, b, r) and t ∈ InsideOfCircle(a, b, r), then L(s, t) ∩ Circle(a, b, r) = {s}.

(60) If s ∈ Circle(a, b, r) and t ∈ Circle(a, b, r), then L(s, t) \ {s, t} ⊆ InsideOfCircle(a, b, r).

(61) If s ∈ Circle(a, b, r) and t ∈ Circle(a, b, r), then L(s, t) ∩ Circle(a, b, r) = {s, t}.

(62) If s ∈ Circle(a, b, r) and t ∈ Circle(a, b, r), then HL(s, t)∩Circle(a, b, r) = {s, t}.

(63) If s ≠ t and s ∈ Circle(a, b, r) and t ∈ ClosedInsideOfCircle(a, b, r), then there exists a point e of E_{T}^{2} such that e ≠ s and {s, e} = HL(s, t) ∩ Circle(a, b, r).

Let a, b, r be real numbers. Observe that InsideOfCircle(a, b, r) is convex and ClosedInsideOfCircle(a, b, r) is convex.

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[6] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[7] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

[8] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[9] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

[10] Yatsuka Nakamura. General Fashoda meet theorem for unit circle and square. Formalized Mathematics, 11(3):213–224, 2003.

306 artur korniłowicz and yasunari shidama

[11] Yatsuka Nakamura and Jarosław Kotowicz. The Jordan’s property for certain subsets of the plane. Formalized Mathematics, 3(2):137–142, 1992.

[12] Yatsuka Nakamura, Andrzej Trybulec, and Czesław Byliński. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1–13, 1999.

[13] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[14] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[15] Agnieszka Sakowicz, Jarosław Gryko, and Adam Grabowski. Sequences in E_{T}^{N}. Formalized Mathematics, 5(1):93–96, 1996.

[16] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[17] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[18] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

Some Properties of Fibonacci Numbers

Magdalena Jastrzȩbska University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. We formalized some basic properties of the Fibonacci numbers using definitions and lemmas from [7] and [23], e.g. Cassini’s and Catalan’s identities. We also showed the connections between Fibonacci numbers and Pythagorean triples as defined in [31]. The main result of this article is a proof of Carmichael’s Theorem on prime divisors of prime-generated Fibonacci numbers. According to it, if we look at the prime factors of a Fibonacci number generated by a prime number, none of them have appeared as a factor in any earlier Fibonacci number. We plan to develop the full proof of the Carmichael Theorem following [33].

MML Identifier: FIB NUM2.

The papers [26], [3], [4], [30], [24], [1], [28], [29], [2], [18], [13], [27], [32], [9], [10], [7], [12], [8], [17], [21], [19], [22], [25], [6], [20], [11], [23], [15], [31], [14], [16], and [5] provide the terminology and notation for this paper.

1. Preliminaries

In this paper n, k, r, m, i, j denote natural numbers. We now state a number of propositions:

(1) For every non empty natural number n holds (n − ′1) + 2 = n + 1.

(2) For every odd integer n and for every non empty real number m holds (− m)^{n} = − m^{n}.

(3) For every odd integer n holds (− 1)^{n} = − 1.

(4) For every even integer n and for every non empty real number m holds (− m)^{n} = m^{n}.

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102.

(5) For every even integer n holds (− 1)^{n} = 1.

(6) For every non empty real number m and for every integer n holds ((− 1)· m)^{n} = (− 1)^{n} · m^{n}.

(7) For every non empty real number a holds a^{k+m} = a^{k} · a^{m}.

(8) For every non empty real number k and for every odd integer m holds (k^{m})^{n} = k^{m·n}.

(9) ((− 1)^{− n})^{2} = 1.

(10) For every non empty real number a holds a^{− k} · a^{− m} = a^{− k−m} .

(11) (− 1)^{− 2·n}= 1.

(12) For every non empty real number a holds a^{k} · a^{− k} = 1.

Let n be an odd integer. One can verify that − n is odd. Let n be an even integer. Note that − n is even. One can prove the following two propositions:

(13) (− 1)^{− n} = (− 1)^{n}.

(14) For all natural numbers k, m, m_{1}, n_{1} such that k | m and k | n holds

k | m · m_{1} + n · n_{1}. One can check that there exists a set which is finite, non empty, and natural-

membered and has non empty elements.

Let f be a function from ℕ into ℕ and let A be a finite natural-membered set with non empty elements. Note that f↾A is finite subsequence-like.

One can prove the following proposition

(15) For every finite subsequence p holds rngSeqp ⊆ rngp.

Let f be a function from ℕ into ℕ and let A be a finite natural-membered set with non empty elements. The functor Prefix(f, A) yields a finite sequence of elements of ℕ and is defined as follows:

(Def. 1) Prefix(f, A) = Seq(f↾A).

The following proposition is true

(16) For every natural number k such that k ≠ 0 holds if k + m n, then

m < n. Let us mention that ℕ is lower bounded. Let us mention that {1, 2, 3} is natural-membered and has non empty ele-

ments.

Let us note that {1, 2, 3, 4} is natural-membered and has non empty elements.

The following propositions are true:

(17) For all sets x, y such that 0 < i and i < j holds {⟨⟨i, x⟩⟩, ⟨⟨j, y⟩⟩} is a finite subsequence.

(18) For all sets x, y and for every finite subsequence q such that i < j and q = {⟨⟨i, x⟩⟩, ⟨⟨j, y⟩⟩} holds Seqq = ⟨x, y⟩.

Let n be a natural number. Observe that Segn has non empty elements. Let A be a set with non empty elements. Note that every subset of A has

non empty elements.

Let A be a set with non empty elements and let B be a set. Observe that A ∩ B has non empty elements and B ∩ A has non empty elements.

We now state four propositions:

(19) For every natural number k and for every set a such that k 1 holds {⟨⟨k, a⟩⟩} is a finite subsequence.

(20) Let i, k be natural numbers, y be a set, and f be a finite subsequence. If f = {⟨⟨1, y⟩⟩}, then Shift^{i} f = {⟨⟨1 + i, y⟩⟩}.

(21) Let q be a finite subsequence and k, n be natural numbers. Suppose domq ⊆ Segk and n > k. Then there exists a finite sequence p such that q ⊆ p and domp = Segn.

(22) For every finite subsequence q there exists a finite sequence p such that q ⊆ p.

2. Fibonacci Numbers

In this article we present several logical schemes. The scheme Fib Ind 1 concerns a unary predicate P, and states that:

For every non empty natural number k holds P[k] provided the parameters have the following properties:

• P[1],

• P[2], and

• For every non empty natural number k such that P[k] and P[k+1] holds P[k + 2].

The scheme Fib Ind 2 concerns a unary predicate P, and states that:

For every non trivial natural number k holds P[k] provided the parameters meet the following conditions:

• P[2],

• P[3], and

• For every non trivial natural number k such that P[k] and P[k+1] holds P[k + 2].

Next we state a number of propositions:

(23) Fib(2) = 1.

(24) Fib(3) = 2.

(25) Fib(4) = 3.

(26) Fib(n + 2) = Fib(n) + Fib(n + 1).

(27) Fib(n + 3) = Fib(n + 2) + Fib(n + 1).

(28) Fib(n + 4) = Fib(n + 2) + Fib(n + 3).

(29) Fib(n + 5) = Fib(n + 3) + Fib(n + 4).

(30) Fib(n + 2) = Fib(n + 3) − Fib(n + 1).

(31) Fib(n + 1) = Fib(n + 2) − Fib(n).

(32) Fib(n) = Fib(n + 2) − Fib(n + 1).

3. Cassini’s and Catalan’s Identities

The following propositions are true:

(33) Fib(n) · Fib(n + 2) − Fib(n + 1)^{2} = (− 1)^{n+1}.

(34) For every non empty natural number n holds Fib(n − ′1) · Fib(n + 1) − Fib(n)^{2} = (− 1)^{n}.

(35) τ > 0.

(36) \overline{τ} = (− τ)^{− 1}.

(37) (− τ)^{(− 1)·n}= ((− τ)^{− 1})^{n}.

(38) − \frac{1}{τ} = \overline{τ}.

(39) ((τ^{r})^{2} − 2 · (− 1)^{r}) + (τ^{− r})^{2} = (τ^{r} − \overline{τ}^{r})^{2}.

(40) For all non empty natural numbers n, r such that r n holds Fib(n)^{2} − Fib(n + r) · Fib(n − ′r) = (− 1)^{n− ′r} · Fib(r)^{2}.

(41) Fib(n)^{2} + Fib(n + 1)^{2} = Fib(2 · n + 1).

(42) For every non empty natural number k holds Fib(n + k) = Fib(k) · Fib(n + 1) + Fib(k − ′1) · Fib(n).

(43) For every non empty natural number n holds Fib(n) | Fib(n · k).

(44) For every non empty natural number k such that k | n holds Fib(k) | Fib(n).

(45) Fib(n) Fib(n + 1).

(46) For every natural number n such that n > 1 holds Fib(n) < Fib(n + 1).

(47) For all natural numbers m, n such that m n holds Fib(m) Fib(n).

(48) For every natural number k such that k > 1 holds if k < n, then Fib(k) < Fib(n).

(49) Fib(k) = 1 iff k = 1 or k = 2.

(50) Let k, n be natural numbers. Suppose n > 1 and k ≠ 0 and k ≠ 1 and k ≠ 1 and n ≠ 2 or k ≠ 2 and n ≠ 1. Then Fib(k) = Fib(n) if and only if k = n.

(51) Let n be a natural number. Suppose n > 1 and n ≠ 4. Suppose n is non prime. Then there exists a non empty natural number k such that k ≠ 1 and k ≠ 2 and k ≠ n and k | n.

(52) For every natural number n such that n > 1 and n ≠ 4 holds if Fib(n) is prime, then n is prime.

4. Sequence of Fibonacci Numbers

The function FIB from ℕ into ℕ is defined as follows: (Def. 2) For every natural number k holds FIB(k) = Fib(k).

The subset ℕ_{even} of ℕ is defined by: (Def. 3) ℕ_{even} = {2 · k : k ranges over natural numbers}.

The subset ℕ_{odd} of ℕ is defined as follows: (Def. 4) ℕ_{odd} = {2 · k + 1 : k ranges over natural numbers}.

One can prove the following two propositions:

(53) For every natural number k holds 2 · k ∈ ℕ_{even} and 2 · k + 1 ∉ ℕ_{even}.

(54) For every natural number k holds 2 · k + 1 ∈ ℕ_{odd} and 2 · k ∉ ℕ_{odd}. Let n be a natural number. The functor EvenFibs(n) yielding a finite sequ-

ence of elements of ℕ is defined by: (Def. 5) EvenFibs(n) = Prefix(FIB, ℕ_{even} ∩ Segn).

The functor OddFibs(n) yields a finite sequence of elements of ℕ and is defined by:

(Def. 6) OddFibs(n) = Prefix(FIB, ℕ_{odd} ∩ Segn).

We now state a number of propositions:

(55) EvenFibs(0) = ∅.

(56) Seq(FIB↾{2}) = ⟨1⟩.

(57) EvenFibs(2) = ⟨1⟩.

(58) EvenFibs(4) = ⟨1, 3⟩.

(59) For every natural number k holds ℕ_{even} ∩ Seg(2 · k + 2) ∪ {2 · k + 4} = ℕ_{even} ∩ Seg(2 · k + 4).

(60) For every natural number k holds FIB↾(ℕ_{even} ∩Seg(2·k+2))∪{⟨⟨2·k+4, FIB(2 · k + 4)⟩⟩} = FIB↾(ℕ_{even} ∩ Seg(2 · k + 4)).

(61) For every natural number n holds EvenFibs(2 · n + 2) = EvenFibs(2 · n)^{⌢} ⟨Fib(2 · n + 2)⟩.

(62) OddFibs(1) = ⟨1⟩.

(63) OddFibs(3) = ⟨1, 2⟩.

(64) For every natural number k holds ℕ_{odd} ∩ Seg(2 · k + 3) ∪ {2 · k + 5} = ℕ_{odd} ∩ Seg(2 · k + 5).

(65) For every natural number k holds FIB↾(ℕ_{odd} ∩Seg(2·k +3))∪{⟨⟨2·k +5, FIB(2 · k + 5)⟩⟩} = FIB↾(ℕ_{odd} ∩ Seg(2 · k + 5)).

(66) For every natural number n holds OddFibs(2 · n + 3) = OddFibs(2 · n + 1)^{⌢} ⟨Fib(2 · n + 3)⟩.

(67) For every natural number n holds ∑ EvenFibs(2·n+2) = Fib(2·n+3)−1.

(68) For every natural number n holds ∑ OddFibs(2 · n + 1) = Fib(2 · n + 2).

312 magdalena jastrzȩbska and adam grabowski

5. Carmichael’s Theorem on Prime Divisors

One can prove the following three propositions:

(69) For every natural number n holds Fib(n) and Fib(n + 1) are relative prime.

(70) For every non empty natural number n and for every natural number m such that m ≠ 1 holds if m | Fib(n), then m ∤ Fib(n − ′1).

(71) Let n be a non empty natural number. Suppose m is prime and n is prime and m | Fib(n). Let r be a natural number. If r < n and r ≠ 0, then m ∤ Fib(r).

6. Fibonacci Numbers and Pythagorean Triples

We now state the proposition

(72) For every non empty natural number n holds {Fib(n) · Fib(n + 3), 2 · Fib(n + 1) · Fib(n + 2), Fib(n + 1)^{2} + Fib(n + 2)^{2}} is a Pythagorean triple.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[4] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[5] Grzegorz Bancerek, Mitsuru Aoki, Akio Matsumoto, and Yasunari Shidama. Processes in Petri nets. Formalized Mathematics, 11(1):125–132, 2003.

[6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[7] Grzegorz Bancerek and Piotr Rudnicki. Two programs for scm. Part I - preliminaries. Formalized Mathematics, 4(1):69–72, 1993.

[8] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[11] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[12] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[14] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepin’s test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317–321, 1998.

[15] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4):573–577, 1997.

[16] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[17] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890,

some properties of fibonacci numbers 313

[18] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990.

[19] Yatsuka Nakamura and Andrzej Trybulec. A mathematical model of CPU. Formalized Mathematics, 3(2):151–160, 1992.

[20] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[21] Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[22] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.

[23] Robert M. Solovay. Fibonacci numbers. Formalized Mathematics, 10(2):81–83, 2002.

[24] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[25] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[26] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[27] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341– 347, 2003.

[28] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[29] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[30] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[31] Freek Wiedijk. Pythagorean triples. Formalized Mathematics, 9(4):809–812, 2001.

[32] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[33] Minoru Yabuta. A simple proof of Carmichael’s theorem of primitive divisors. The Fibonacci Quarterly, 39(5):439–443, 2001.

The Hall Marriage Theorem

Ewa Romanowicz University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. The Marriage Theorem, as credited to Philip Hall [7], gives the necessary and sufficient condition allowing us to select a distinct element from each of a finite collection {A_{i}} of n finite subsets. This selection, called a set of different representatives (SDR), exists if and only if the marriage condition (or Hall condition) is satisfied:

∀_{J⊆{1,...,n}} | ⋃_{i∈J} A_{i}| |J|.

The proof which is given in this article (according to Richard Rado, 1967) is based on the lemma that for finite sequences with non-trivial elements which satisfy Hall property there exists a reduction (see Def. 5) such that Hall property again holds (see Th. 29 for details).

MML Identifier: HALLMAR1.

The notation and terminology used here are introduced in the following papers: [9], [5], [10], [11], [4], [8], [2], [6], [1], and [3].

1. Preliminaries

One can prove the following proposition

(1) For all finite sets X, Y holds card(X∪Y)+card(X∩Y) = cardX+cardY. In this article we present several logical schemes. The scheme Regr11 deals

with a natural number A and a unary predicate P, and states that: For every natural number k such that 1 k and k A holds P[k]

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102.

provided the parameters meet the following conditions: • P[A] and A 2, and

• For every natural number k such that 1 k and k < A and P[k + 1] holds P[k].

The scheme Regr2 concerns a unary predicate P, and states that:

P[1] provided the parameters meet the following requirements:

• There exists a natural number n such that n > 1 and P[n], and

• For every natural number k such that k 1 and P[k + 1] holds P[k].

Let F be a non empty set. One can check that there exists a finite sequence of elements of 2^{F} which is non empty and non-empty.

We now state the proposition

(2) Let F be a non empty set, f be a non-empty finite sequence of elements of 2^{F}, and i be a natural number. If i ∈ domf, then f(i) ≠ ∅.

Let F be a finite set, let A be a finite sequence of elements of 2^{F}, and let i be a natural number. Note that A(i) is finite.

2. Union of Finite Sequences

Let F be a set, let A be a finite sequence of elements of 2^{F}, and let J be a set. The functor ⋃_{J} A yields a set and is defined as follows:

(Def. 1) For every set x holds x ∈ ⋃_{J} A iff there exists a set j such that j ∈ J

and j ∈ domA and x ∈ A(j). Next we state two propositions:

(3) For every set F and for every finite sequence A of elements of 2^{F} and for every set J holds ⋃_{J} A ⊆ F.

(4) Let F be a finite set, A be a finite sequence of elements of 2^{F}, and J, K be sets. If J ⊆ K, then ⋃_{J} A ⊆ ⋃_{K} A.

Let F be a finite set, let A be a finite sequence of elements of 2^{F}, and let J be a set. One can verify that ⋃_{J} A is finite.

The following propositions are true:

(5) Let F be a finite set, A be a finite sequence of elements of 2^{F}, and i be a natural number. If i ∈ domA, then ⋃_{{i}} A = A(i).

(6) Let F be a finite set, A be a finite sequence of elements of 2^{F}, and i, j be natural numbers. If i ∈ domA and j ∈ domA, then ⋃_{{i,j}} A = A(i)∪A(j).

(7) Let J be a set, F be a finite set, A be a finite sequence of elements of 2^{F}, and i be a natural number. If i ∈ J and i ∈ domA, then A(i) ⊆ ⋃_{J} A.

(8) Let J be a set, F be a finite set, i be a natural number, and A be a finite sequence of elements of 2^{F}. If i ∈ J and i ∈ domA, then ⋃_{J} A = ⋃_{J\{i}} A ∪ A(i).

(9) Let J_{1}, J_{2} be sets, F be a finite set, i be a natural number, and A be a finite sequence of elements of 2^{F}. If i ∈ domA, then ⋃_{{i}∪J_{1}∪J_{2}} A = A(i) ∪ ⋃_{J_{1}∪J_{2}} A.

(10) Let F be a finite set, A be a finite sequence of elements of 2^{F}, i be a natural number, and x, y be sets. If x ≠ y and x ∈ A(i) and y ∈ A(i), then (A(i) \ {x}) ∪ (A(i) \ {y}) = A(i).

3. Cut Operation for Finite Sequences

Let F be a finite set, let A be a finite sequence of elements of 2^{F}, let i be a natural number, and let x be a set. The functor Cut(A, i, x) yielding a finite sequence of elements of 2^{F} is defined by the conditions (Def. 2).

(Def. 2)(i) domCut(A, i, x) = domA, and

(ii) for every natural number k such that k ∈ domCut(A, i, x) holds if i = k, then (Cut(A, i, x))(k) = A(k) \ {x} and if i ≠ k, then (Cut(A, i, x))(k) = A(k).

The following propositions are true:

(11) Let F be a finite set, A be a finite sequence of elements of 2^{F}, i be a natural number, and x be a set. If i ∈ domA and x ∈ A(i), then card(Cut(A, i, x))(i) = cardA(i) − 1.

(12) Let F be a finite set, A be a finite sequence of elements of 2^{F}, i be a natural number, and x, J be sets. Then ⋃_{J\{i}} Cut(A, i, x) = ⋃_{J\{i}} A.

(13) Let F be a finite set, A be a finite sequence of elements of 2^{F}, i be a natural number, and x, J be sets. If i ∉ J, then ⋃_{J} A = ⋃_{J} Cut(A, i, x).

(14) Let F be a finite set, A be a finite sequence of elements of 2^{F}, i be a natural number, and x, J be sets. If i ∈ domCut(A, i, x) and J ⊆ domCut(A, i, x) and i ∈ J, then ⋃_{J} Cut(A, i, x) = ⋃_{J\{i}} A ∪ (A(i) \ {x}).

4. System of Different Representatives and Hall Property

Let F be a finite set, let X be a finite sequence of elements of 2^{F}, and let A be a set. We say that A is a system of different representatives of X if and only if the condition (Def. 3) is satisfied.

(Def. 3) There exists a finite sequence f of elements of F such that f = A and domX = domf and for every natural number i such that i ∈ domf holds f(i) ∈ X(i) and f is one-to-one.

Let F be a finite set and let A be a finite sequence of elements of 2^{F}. We say that A satisfies Hall condition if and only if:

(Def. 4) For every finite set J such that J ⊆ domA holds cardJ card⋃_{J} A.

Next we state four propositions:

(15) Let F be a finite set and A be a non empty finite sequence of elements of 2^{F}. If A satisfies Hall condition, then A is non-empty.

(16) Let F be a finite set, A be a finite sequence of elements of 2^{F}, and i be a natural number. If i ∈ domA and A satisfies Hall condition, then cardA(i) 1.

(17) Let F be a non empty finite set and A be a non empty finite sequence of elements of 2^{F}. Suppose for every natural number i such that i ∈ domA holds cardA(i) = 1 and A satisfies Hall condition. Then there exists a set which is a system of different representatives of A.

(18) Let F be a finite set and A be a finite sequence of elements of 2^{F} such that there exists a set which is a system of different representatives of A. Then A satisfies Hall condition.

5. Reductions and Singlifications of Finite Sequences

Let F be a set, let A be a finite sequence of elements of 2^{F}, and let i be a natural number. A finite sequence of elements of 2^{F} is said to be a reduction of A at i-th position if:

(Def. 5) domit = domA and for every natural number j such that j ∈ domA

and j ≠ i holds A(j) = it(j) and it(i) ⊆ A(i). Let F be a set and let A be a finite sequence of elements of 2^{F}. A finite

sequence of elements of 2^{F} is said to be a reduction of A if: (Def. 6) domit = domA and for every natural number i such that i ∈ domA

holds it(i) ⊆ A(i). Let F be a set, let A be a finite sequence of elements of 2^{F}, and let i be a

natural number. Let us assume that i ∈ domA and A(i) ≠ ∅. A reduction of A is called a singlification of A at i-th position if:

(Def. 7) \overline{\overline{it(i)}} = 1.

One can prove the following propositions:

(19) Let F be a finite set, A be a finite sequence of elements of 2^{F}, and i be a natural number. Then every reduction of A at i-th position is a reduction of A.

(20) Let F be a finite set, A be a finite sequence of elements of 2^{F}, i be a natural number, and x be a set. If i ∈ domA and x ∈ A(i), then Cut(A, i, x) is a reduction of A at i-th position.

(21) Let F be a finite set, A be a finite sequence of elements of 2^{F}, i be a natural number, and x be a set. If i ∈ domA and x ∈ A(i), then Cut(A, i, x) is a reduction of A.

(22) Let F be a finite set, A be a finite sequence of elements of 2^{F}, and B be a reduction of A. Then every reduction of B is a reduction of A.

(23) Let F be a non empty finite set, A be a non-empty finite sequence of elements of 2^{F}, i be a natural number, and B be a singlification of A at i-th position. If i ∈ domA, then B(i) ≠ ∅.

(24) Let F be a non empty finite set, A be a non-empty finite sequence of elements of 2^{F}, i, j be natural numbers, B be a singlification of A at i-th position, and C be a singlification of B at j-th position. Suppose i ∈ domA and j ∈ domA and C(i) ≠ ∅ and B(j) ≠ ∅. Then C is a singlification of A at j-th position and a singlification of A at i-th position.

(25) Let F be a set, A be a finite sequence of elements of 2^{F}, and i be a natural number. Then A is a reduction of A at i-th position.

(26) For every set F holds every finite sequence A of elements of 2^{F} is a

reduction of A. Let F be a non empty set and let A be a finite sequence of elements of 2^{F}.

Let us assume that A is non-empty. A reduction of A is called a singlification of A if:

(Def. 8) For every natural number i such that i ∈ domA holds \overline{\overline{it(i)}} = 1.

We now state the proposition

(27) Let F be a non empty finite set, A be a non empty non-empty finite sequence of elements of 2^{F}, and f be a function. Then f is a singlification of A if and only if the following conditions are satisfied:

(i) domf = domA, and (ii) for every natural number i such that i ∈ domA holds f is a singlification

of A at i-th position. Let F be a non empty finite set, let A be a non empty finite sequence of

elements of 2^{F}, and let k be a natural number. Note that every singlification of A at k-th position is non empty.

Let F be a non empty finite set and let A be a non empty finite sequence of elements of 2^{F}. One can check that every singlification of A is non empty.

6. Rado’s Proof of the Hall Marriage Theorem

One can prove the following propositions:

(28) Let F be a non empty finite set, A be a non empty finite sequence of elements of 2^{F}, X be a set, and B be a reduction of A. Suppose X is a 320 ewa romanowicz and adam grabowski

system of different representatives of B. Then X is a system of different representatives of A.

(29) Let F be a finite set and A be a finite sequence of elements of 2^{F}. Suppose A satisfies Hall condition. Let i be a natural number. If cardA(i) 2, then there exists a set x such that x ∈ A(i) and Cut(A, i, x) satisfies Hall condition.

(30) Let F be a finite set, A be a finite sequence of elements of 2^{F}, and i be a natural number. If i ∈ domA and A satisfies Hall condition, then there exists a singlification of A at i-th position which satisfies Hall condition.

(31) Let F be a non empty finite set and A be a non empty finite sequence of elements of 2^{F}. If A satisfies Hall condition, then there exists a singlification of A which satisfies Hall condition.

(32) Let F be a non empty finite set and A be a non empty finite sequence of elements of 2^{F}. Then there exists a set which is a system of different representatives of A if and only if A satisfies Hall condition.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[6] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[7] Philip Hall. On representatives of subsets. Journal of London Mathematical Society, 10:26–30, 1935.

[8] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[9] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[10] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[11] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

The Differentiable Functions on Normed Linear Spaces

Hiroshi Imura Shinshu University

Nagano Morishige Kimura Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. In this article, the basic properties of the differentiable functions on normed linear spaces are described.

MML Identifier: NDIFF 1.

The notation and terminology used in this paper are introduced in the following papers: [20], [23], [4], [24], [6], [5], [19], [3], [10], [1], [18], [7], [21], [22], [11], [8], [9], [25], [13], [15], [16], [17], [12], [14], and [2].

For simplicity, we adopt the following rules: n, k denote natural numbers, x, X, Z denote sets, g, r denote real numbers, S denotes a real normed space, r_{1} denotes a sequence of real numbers, s_{1}, s_{2} denote sequences of S, x_{0} denotes a point of S, and Y denotes a subset of S.

Next we state several propositions:

(1) For every point x_{0} of S and for all neighbourhoods N_{1}, N_{2} of x_{0} there exists a neighbourhood N of x_{0} such that N ⊆ N_{1} and N ⊆ N_{2}.

(2) Let X be a subset of S. Suppose X is open. Let r be a point of S. If r ∈ X, then there exists a neighbourhood N of r such that N ⊆ X.

(3) Let X be a subset of S. Suppose X is open. Let r be a point of S. If r ∈ X, then there exists g such that 0 < g and {y; y ranges over points of S: ∥y − r∥ < g} ⊆ X.

(4) Let X be a subset of S. Suppose that for every point r of S such that r ∈ X there exists a neighbourhood N of r such that N ⊆ X. Then X is open.

(5) Let X be a subset of S. Then for every point r of S such that r ∈ X there exists a neighbourhood N of r such that N ⊆ X if and only if X is open.

Let S be a zero structure and let f be a sequence of S. We say that f is non-zero if and only if:

(Def. 1) rngf ⊆ (the carrier of S) \ {0_{S}}.

We introduce f is non-zero as a synonym of f is non-zero.

We now state two propositions:

(6) s_{1} is non-zero iff for every x such that x ∈ ℕ holds s_{1}(x) ≠ 0_{S}.

(7) s_{1} is non-zero iff for every n holds s_{1}(n) ≠ 0_{S}.

Let R_{1} be a real linear space, let S be a sequence of R_{1}, and let a be a sequence of real numbers. The functor a S yields a sequence of R_{1} and is defined as follows:

(Def. 2) For every n holds (a S)(n) = a(n) · S(n).

Let R_{1} be a real linear space, let z be a point of R_{1}, and let a be a sequence of real numbers. The functor a · z yields a sequence of R_{1} and is defined by:

(Def. 3) For every n holds (a · z)(n) = a(n) · z. Next we state a number of propositions:

(8) For all sequences r_{2}, r_{3} of real numbers holds (r_{2} + r_{3}) s_{1} = r_{2} s_{1} + r_{3} s_{1}.

(9) For every sequence r_{1} of real numbers and for all sequences s_{2}, s_{3} of S holds r_{1} (s_{2} + s_{3}) = r_{1} s_{2} + r_{1} s_{3}.

(10) For every sequence r_{1} of real numbers holds r · (r_{1} s_{1}) = r_{1} (r · s_{1}).

(11) For all sequences r_{2}, r_{3} of real numbers holds (r_{2} − r_{3}) s_{1} = r_{2} s_{1} − r_{3} s_{1}.

(12) For every sequence r_{1} of real numbers and for all sequences s_{2}, s_{3} of S holds r_{1} (s_{2} − s_{3}) = r_{1} s_{2} − r_{1} s_{3}.

(13) If r_{1} is convergent and s_{1} is convergent, then r_{1} s_{1} is convergent.

(14) If r_{1} is convergent and s_{1} is convergent, then lim(r_{1} s_{1}) = limr_{1} · lims_{1}.

(15) (s_{1} + s_{2}) ↑ k = s_{1} ↑ k + s_{2} ↑ k.

(16) (s_{1} − s_{2}) ↑ k = s_{1} ↑ k − s_{2} ↑ k.

(17) If s_{1} is non-zero, then s_{1} ↑ k is non-zero.

(18) s_{1} ↑ k is a subsequence of s_{1}.

(19) If s_{1} is constant and s_{2} is a subsequence of s_{1}, then s_{2} is constant.

(20) If s_{1} is constant and s_{2} is a subsequence of s_{1}, then s_{1} = s_{2}.

Let us consider S and let I_{1} be a sequence of S. We say that I_{1} is convergent to 0 if and only if:

(Def. 4) I_{1} is non-zero and convergent and limI_{1} = 0_{S}.

The following propositions are true:

(21) Let X be a real normed space and s_{1} be a sequence of X. Suppose s_{1} is constant. Then s_{1} is convergent and for every natural number k holds lims_{1} = s_{1}(k).

(22) For every real number r such that 0 < r and for every n holds s_{1}(n) = \frac{1}{n+r} · x_{0} holds s_{1} is convergent.

(23) For every real number r such that 0 < r and for every n holds s_{1}(n) = \frac{1}{n+r} · x_{0} holds lims_{1} = 0_{S}.

(24) Let a be a convergent to 0 sequence of real numbers and z be a point of S. If z ≠ 0_{S}, then a · z is convergent to 0.

(25) For every point r of S holds r ∈ Y iff r ∈ the carrier of S iff Y = the

carrier of S. For simplicity, we adopt the following rules: S, T denote non trivial real

normed spaces, f, f_{1}, f_{2} denote partial functions from S to T, s_{4}, s_{1} denote sequences of S, and x_{0} denotes a point of S.

Let S be a non trivial real normed space. Note that there exists a sequence of S which is convergent to 0.

Let us consider S. Note that there exists a sequence of S which is constant. In the sequel h is a convergent to 0 sequence of S and c is a constant sequence

of S.

Let us consider S, T and let I_{1} be a partial function from S to T. We say that I_{1} is rest-like if and only if:

(Def. 5) I_{1} is total and for every h holds ∥h∥^{−1} (I_{1} · h) is convergent and

lim(∥h∥^{−1} (I_{1} · h)) = 0_{T} . Let us consider S, T. Observe that there exists a partial function from S to

T which is rest-like.

Let us consider S, T. A rest of S, T is a rest-like partial function from S to T.

We now state two propositions:

(26) Let R be a partial function from S to T. Suppose R is total. Then R is rest-like if and only if for every real number r such that r > 0 there exists a real number d such that d > 0 and for every point z of S such that z ≠ 0_{S} and ∥z∥ < d holds ∥z∥^{−1} · ∥R_{z}∥ < r.

(27) For every rest R of S, T and for every convergent to 0 sequence s of S

holds R · s is convergent and lim(R · s) = 0_{T} . In the sequel R, R_{2}, R_{3} are rests of S, T and L is a point of

RNormSpaceOfBoundedLinearOperators(S, T).

Next we state several propositions:

(28) rng(s_{1} ↑ n) ⊆ rngs_{1}.

(29) For every partial function h from S to T and for every sequence s_{1} of S such that rngs_{1} ⊆ domh holds (h · s_{1}) ↑ n = h · (s_{1} ↑ n).

(30) Let h_{1}, h_{2} be partial functions from S to T and s_{1} be a sequence of S. If h_{1} is total and h_{2} is total, then (h_{1} + h_{2}) · s_{1} = h_{1} · s_{1} + h_{2} · s_{1} and (h_{1} − h_{2}) · s_{1} = h_{1} · s_{1} − h_{2} · s_{1}.

(31) Let h be a partial function from S to T, s_{1} be a sequence of S, and r be a real number. If h is total, then (r h) · s_{1} = r · (h · s_{1}).

(32) f is continuous in x_{0} if and only if the following conditions are satisfied: (i) x_{0} ∈ domf, and (ii) for every sequence s_{4} of S such that rngs_{4} ⊆ domf and s_{4} is convergent

and lims_{4} = x_{0} and for every n holds s_{4}(n) ≠ x_{0} holds f · s_{4} is convergent and f_{x_{0}} = lim(f · s_{4}).

(33) For all R_{2}, R_{3} holds R_{2} + R_{3} is a rest of S, T and R_{2} − R_{3} is a rest of S, T.

(34) For all r, R holds r R is a rest of S, T.

Let us consider S, T, let f be a partial function from S to T, and let x_{0} be a point of S. We say that f is differentiable in x_{0} if and only if the condition (Def. 6) is satisfied.

(Def. 6) There exists a neighbourhood N of x_{0} such that N ⊆ domf and there exist L, R such that for every point x of S such that x ∈ N holds f_{x}−f_{x_{0}} = L(x − x_{0}) + R_{x−x_{0}} .

Let us consider S, T, let f be a partial function from S to T, and let x_{0} be a point of S. Let us assume that f is differentiable in x_{0}. The functor f′(x_{0}) yielding a point of RNormSpaceOfBoundedLinearOperators(S, T) is defined by the condition (Def. 7).

(Def. 7) There exists a neighbourhood N of x_{0} such that N ⊆ domf and there exists R such that for every point x of S such that x ∈ N holds f_{x} − f_{x_{0}} = f′(x_{0})(x − x_{0}) + R_{x−x_{0}} .

Let us consider X, let us consider S, T, and let f be a partial function from S to T. We say that f is differentiable on X if and only if:

(Def. 8) X ⊆ domf and for every point x of S such that x ∈ X holds f↾X is

differentiable in x. Next we state three propositions:

(35) Let f be a partial function from S to T. If f is differentiable on X, then X is a subset of the carrier of S.

(36) Let f be a partial function from S to T and Z be a subset of S. Suppose Z is open. Then f is differentiable on Z if and only if the following conditions are satisfied:

(i) Z ⊆ domf, and (ii) for every point x of S such that x ∈ Z holds f is differentiable in x.

(37) Let f be a partial function from S to T and Y be a subset of S. If f is

differentiable on Y, then Y is open. Let us consider S, T, let f be a partial function from S to T, and let X

be a set. Let us assume that f is differentiable on X. The functor f′_{↾X} yielding a partial function from S to RNormSpaceOfBoundedLinearOperators(S, T) is defined by:

(Def. 9) dom(f′_{↾X}) = X and for every point x of S such that x ∈ X holds (f′_{↾X})_{x} =

f′(x). One can prove the following proposition

(38) Let f be a partial function from S to T and Z be a subset of S. Suppose Z is open and Z ⊆ domf and there exists a point r of T such that rngf = {r}. Then f is differentiable on Z and for every point x of S such that x ∈ Z holds (f′_{↾Z})_{x} = 0_{RNormSpaceOfBoundedLinearOperators (S,T)}.

Let us consider S and let us consider h, n. Observe that h ↑ n is convergent to 0.

Let us consider S and let us consider c, n. Observe that c ↑ n is constant. The following propositions are true:

(39) Let x_{0} be a point of S and N be a neighbourhood of x_{0}. Suppose f is differentiable in x_{0} and N ⊆ domf. Let h be a convergent to 0 sequence of S and given c. If rngc = {x_{0}} and rng(h + c) ⊆ N, then f · (h + c) − f · c is convergent and lim(f · (h + c) − f · c) = 0_{T} .

(40) Let given f_{1}, f_{2}, x_{0}. Suppose f_{1} is differentiable in x_{0} and f_{2} is differentiable in x_{0}. Then f_{1} + f_{2} is differentiable in x_{0} and (f_{1} + f_{2})′(x_{0}) = f_{1}′(x_{0}) + f_{2}′(x_{0}).

(41) Let given f_{1}, f_{2}, x_{0}. Suppose f_{1} is differentiable in x_{0} and f_{2} is differentiable in x_{0}. Then f_{1} − f_{2} is differentiable in x_{0} and (f_{1} − f_{2})′(x_{0}) = f_{1}′(x_{0}) − f_{2}′(x_{0}).

(42) For all r, f, x_{0} such that f is differentiable in x_{0} holds r f is differentiable in x_{0} and (r f)′(x_{0}) = r · f′(x_{0}).

(43) Let f be a partial function from S to S and Z be a subset of S. Suppose Z is open and Z ⊆ domf and f↾Z = id_{Z}. Then f is differentiable on Z and for every point x of S such that x ∈ Z holds (f′_{↾Z})_{x} = id_{the carrier of S}.

(44) Let Z be a subset of S. Suppose Z is open. Let given f_{1}, f_{2}. Suppose Z ⊆ dom(f_{1} + f_{2}) and f_{1} is differentiable on Z and f_{2} is differentiable on Z. Then f_{1} + f_{2} is differentiable on Z and for every point x of S such that x ∈ Z holds ((f_{1} + f_{2})′_{↾Z})_{x} = f_{1}′(x) + f_{2}′(x).

(45) Let Z be a subset of S. Suppose Z is open. Let given f_{1}, f_{2}. Suppose Z ⊆ dom(f_{1} − f_{2}) and f_{1} is differentiable on Z and f_{2} is differentiable on Z. Then f_{1} − f_{2} is differentiable on Z and for every point x of S such that x ∈ Z holds ((f_{1} − f_{2})′_{↾Z})_{x} = f_{1}′(x) − f_{2}′(x).

(46) Let Z be a subset of S. Suppose Z is open. Let given r, f. Suppose Z ⊆ dom(r f) and f is differentiable on Z. Then r f is differentiable on Z and for every point x of S such that x ∈ Z holds ((r f)′_{↾Z})_{x} = r · f′(x).

326 hiroshi imura et al.

(47) Let Z be a subset of S. Suppose Z is open. Suppose Z ⊆ domf and f is a constant on Z. Then f is differentiable on Z and for every point x of S such that x ∈ Z holds (f′_{↾Z})_{x} = 0_{RNormSpaceOfBoundedLinearOperators (S,T)}.

(48) Let f be a partial function from S to S, r be a real number, p be a point of S, and Z be a subset of S. Suppose Z is open. Suppose Z ⊆ domf and for every point x of S such that x ∈ Z holds f_{x} = r · x + p. Then f is differentiable on Z and for every point x of S such that x ∈ Z holds (f′_{↾Z})_{x} = r · FuncUnit(S).

(49) For every point x_{0} of S such that f is differentiable in x_{0} holds f is continuous in x_{0}.

(50) If f is differentiable on X, then f is continuous on X.

(51) For every subset Z of S such that Z is open holds if f is differentiable on X and Z ⊆ X, then f is differentiable on Z.

(52) Suppose f is differentiable in x_{0}. Then there exists a neighbourhood N of x_{0} such that

(i) N ⊆ domf, and (ii) there exists R such that R_{0_{S}} = 0_{T} and R is continuous in 0_{S} and for

every point x of S such that x ∈ N holds f_{x}−f_{x_{0}} = f′(x_{0})(x−x_{0})+R_{x−x_{0}} .

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[7] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[8] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[9] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[10] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[11] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[12] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269–275, 2004.

[13] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[14] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

[15] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2003.

[16] Yasunari Shidama. The Banach algebra of bounded linear operators. Formalized Mathematics, 12(2):103–108, 2004.

[17] Yasunari Shidama. The series on Banach algebra. Formalized Mathematics, 12(2):131– 138, 2004.

[18] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

the differentiable functions on normed … 327

[19] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[20] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[21] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[22] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[24] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[25] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171–175, 1992.

Lucas Numbers and Generalized Fibonacci Numbers

Piotr Wojtecki University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. The recursive definition of Fibonacci sequences [3] is a good starting point for various variants and generalizations. We can here point out e.g. Lucas (with 2 and 1 as opening values) and the so-called generalized Fibonacci numbers (starting with arbitrary integers a and b).

In this paper, we introduce Lucas and G-numbers and we state their basic properties analogous to those proven in [10] and [5].

MML Identifier: FIB NUM3.

The papers [15], [14], [11], [2], [6], [1], [13], [12], [8], [9], [4], [7], [3], and [10] provide the notation and terminology for this paper.

1. Preliminaries

In this paper a, b, k, n denote natural numbers. The following propositions are true:

(1) For every real number a and for every natural number n such that a^{n} = 0 holds a = 0.

(2) For every non negative real number a holds \sqrt{a} · \sqrt{a} = a.

(3) For every non empty real number a holds a^{2} = (− a)^{2}.

(4) For every non empty natural number k holds (k − ′1) + 2 = (k + 2) − ′1.

(5) (a + b)^{2} = a · a + a · b + a · b + b · b.

(6) For every non empty real number a holds (a^{n})^{2} = a^{2·n}.

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102.

(7) For all real numbers a, b holds (a + b) · (a − b) = a^{2} − b^{2}.

(8) For all non empty real numbers a, b holds (a · b)^{n} = a^{n} · b^{n}.

Let us mention that τ is positive and \overline{τ} is negative.

The following propositions are true:

(9) For every natural number n holds τ^{n} + τ^{n+1} = τ^{n+2}.

(10) For every natural number n holds \overline{τ}^{n} + \overline{τ}^{n+1} = \overline{τ}^{n+2}.

2. Lucas Numbers

Let n be a natural number. The functor Luc(n) yielding a natural number is defined by the condition (Def. 1).

(Def. 1) There exists a function L from ℕ into [: ℕ, ℕ:] such that Luc(n) = L(n)_{1} and L(0) = ⟨⟨2, 1⟩⟩and for every natural number n holds L(n+1) = ⟨⟨L(n)_{2}, L(n)_{1} + L(n)_{2} ⟩⟩.

The following propositions are true:

(11) Luc(0) = 2 and Luc(1) = 1 and for every natural number n holds Luc(n+ 1 + 1) = Luc(n) + Luc(n + 1).

(12) For every natural number n holds Luc(n + 2) = Luc(n) + Luc(n + 1).

(13) For every natural number n holds Luc(n + 1) + Luc(n + 2) = Luc(n + 3).

(14) Luc(2) = 3.

(15) Luc(3) = 4.

(16) Luc(4) = 7.

(17) For every natural number k holds Luc(k) k.

(18) For every non empty natural number m holds Luc(m + 1) Luc(m). Let n be a natural number. Note that Luc(n) is positive. Next we state a number of propositions:

(19) For every natural number n holds 2 · Luc(n + 2) = Luc(n) + Luc(n + 3).

(20) For every natural number n holds Luc(n + 1) = Fib(n) + Fib(n + 2).

(21) For every natural number n holds Luc(n) = τ^{n} + \overline{τ}^{n}.

(22) For every natural number n holds 2·Luc(n)+Luc(n+1) = 5·Fib(n+1).

(23) For every natural number n holds Luc(n + 3) − 2 · Luc(n) = 5 · Fib(n).

(24) For every natural number n holds Luc(n) + Fib(n) = 2 · Fib(n + 1).

(25) For every natural number n holds 3 · Fib(n) + Luc(n) = 2 · Fib(n + 2).

(26) For all natural numbers n, m holds 2 · Luc(n + m) = Luc(n) · Luc(m) + 5 · Fib(n) · Fib(m).

(27) For every natural number n holds Luc(n + 3) · Luc(n) = Luc(n + 2)^{2} − Luc(n + 1)^{2}.

(28) For every natural number n holds Fib(2 · n) = Fib(n) · Luc(n).

(29) For every natural number n holds 2·Fib(2·n+1) = Luc(n+1)·Fib(n)+ Luc(n) · Fib(n + 1).

(30) For every natural number n holds 5 · Fib(n)^{2} − Luc(n)^{2} = 4 · (− 1)^{n+1}.

(31) For every natural number n holds Fib(2 · n + 1) = Fib(n + 1) · Luc(n + 1) − Fib(n) · Luc(n).

3. Generalized Fibonacci Numbers

Let a, b, n be natural numbers. The functor GFib(a, b, n) yielding a natural number is defined by the condition (Def. 2).

(Def. 2) There exists a function L from ℕ into [: ℕ, ℕ:] such that GFib(a, b, n) = L(n)_{1} and L(0) = ⟨⟨a, b⟩⟩and for every natural number n holds L(n + 1) = ⟨⟨L(n)_{2}, L(n)_{1} + L(n)_{2} ⟩⟩.

Next we state a number of propositions:

(32) For all natural numbers a, b holds GFib(a, b, 0) = a and GFib(a, b, 1) = b and for every natural number n holds GFib(a, b, n+1+1) = GFib(a, b, n)+ GFib(a, b, n + 1).

(33) (GFib(a, b, k + 1) + GFib(a, b, k + 1 + 1))^{2} = GFib(a, b, k + 1)^{2} + 2 · GFib(a, b, k + 1) · GFib(a, b, k + 1 + 1) + GFib(a, b, k + 1 + 1)^{2}.

(34) For all natural numbers a, b, n holds GFib(a, b, n) + GFib(a, b, n + 1) = GFib(a, b, n + 2).

(35) For all natural numbers a, b, n holds GFib(a, b, n+1)+GFib(a, b, n+2) = GFib(a, b, n + 3).

(36) For all natural numbers a, b, n holds GFib(a, b, n+2)+GFib(a, b, n+3) = GFib(a, b, n + 4).

(37) For every natural number n holds GFib(0, 1, n) = Fib(n).

(38) For every natural number n holds GFib(2, 1, n) = Luc(n).

(39) For all natural numbers a, b, n holds GFib(a, b, n) + GFib(a, b, n + 3) = 2 · GFib(a, b, n + 2).

(40) For all natural numbers a, b, n holds GFib(a, b, n) + GFib(a, b, n + 4) = 3 · GFib(a, b, n + 2).

(41) For all natural numbers a, b, n holds GFib(a, b, n + 3) − GFib(a, b, n) = 2 · GFib(a, b, n + 1).

(42) For all non empty natural numbers a, b, n holds GFib(a, b, n) = GFib(a, b, 0) · Fib(n − ′1) + GFib(a, b, 1) · Fib(n).

(43) For all natural numbers n, m holds Fib(m) · Luc(n) + Luc(m) · Fib(n) = GFib(Fib(0), Luc(0), n + m).

(44) For every natural number n holds Luc(n) + Luc(n + 3) = 2 · Luc(n + 2).

332 piotr wojtecki and adam grabowski

(45) For all natural numbers a, n holds GFib(a, a, n) = \frac{GFib(a,a,0)}{2} · (Fib(n) + Luc(n)).

(46) For all natural numbers a, b, n holds GFib(b, a+b, n) = GFib(a, b, n+1).

(47) For all natural numbers a, b, n holds GFib(a, b, n + 2) · GFib(a, b, n) − GFib(a, b, n + 1)^{2} = (− 1)^{n} · (GFib(a, b, 2)^{2} − GFib(a, b, 1) · GFib(a, b, 3)).

(48) For all natural numbers a, b, k, n holds GFib(GFib(a, b, k), GFib(a, b, k+ 1), n) = GFib(a, b, n + k).

(49) For all natural numbers a, b, n holds GFib(a, b, n + 1) = a · Fib(n) + b · Fib(n + 1).

(50) For all natural numbers a, b, n holds GFib(0, b, n) = b · Fib(n).

(51) For all natural numbers a, b, n holds GFib(a, 0, n + 1) = a · Fib(n).

(52) For all natural numbers a, b, c, d, n holds GFib(a, b, n) + GFib(c, d, n) = GFib(a + c, b + d, n).

(53) For all natural numbers a, b, k, n holds GFib(k · a, k · b, n) = k · GFib(a, b, n).

(54) For all natural numbers a, b, n holds GFib(a, b, n) = \frac{(a·− \overline{τ}+b)·τ^{n}+(a·τ−b)·\overline{τ}^{n}}{\sqrt{5}}.

(55) For all natural numbers a, n holds GFib(2 · a + 1, 2 · a + 1, n + 1) = (2 · a + 1) · Fib(n + 2).

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Grzegorz Bancerek and Piotr Rudnicki. Two programs for scm. Part I - preliminaries. Formalized Mathematics, 4(1):69–72, 1993.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Magdalena Jastrzȩbska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307–313, 2004.

[6] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[7] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[8] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

[9] Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

[10] Robert M. Solovay. Fibonacci numbers. Formalized Mathematics, 10(2):81–83, 2002.

[11] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[12] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[13] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[14] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[15] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

lucas numbers and generalized fibonacci … 333 Received May 24, 2004

The Operation of Addition of Relational Structures

Katarzyna Romanowicz University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. The article contains the formalization of the addition operator on relational structures as defined by A. Wroński [8] (as a generalization of Troelstra’s sum or Jaśkowski’s star addition). The ordering relation of A ⊗ B is given by

A⊗B = _{A} ∪ _{B} ∪ ( _{A} ∘ _{B}),

where the carrier is defined as the set-theoretical union of carriers of A and B. Main part – Section 3 – is devoted to the Mizar translation of Theorem 1 (iv–xiii), p. 66 of [8].

MML Identifier: LATSUM 1.

The terminology and notation used in this paper are introduced in the following articles: [4], [6], [7], [5], [2], [3], and [1].

1. Preliminaries

One can prove the following proposition

(1) Let x, y, A, B be sets. Suppose x ∈ A∪B and y ∈ A∪B. Then x ∈ A\B and y ∈ A \ B or x ∈ B and y ∈ B or x ∈ A \ B and y ∈ B or x ∈ B and y ∈ A \ B.

Let R, S be relational structures. The predicate R ≈ S is defined by the condition (Def. 1).

(Def. 1) Let x, y be sets. Suppose x ∈ (the carrier of R) ∩ (the carrier of S) and y ∈ (the carrier of R) ∩ (the carrier of S). Then ⟨⟨x, y⟩⟩ ∈ the internal relation of R if and only if ⟨⟨x, y⟩⟩∈ the internal relation of S.

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102.

2. The Wroński’s Operation

Let R, S be relational structures. The functor R⊗S yields a strict relational structure and is defined by the conditions (Def. 2).

(Def. 2)(i) The carrier of R ⊗ S = (the carrier of R) ∪ (the carrier of S), and

(ii) the internal relation of R ⊗ S = (the internal relation of R) ∪ (the internal relation of S) ∪ (the internal relation of R) · (the internal relation of S).

Let R be a relational structure and let S be a non empty relational structure. Observe that R ⊗ S is non empty.

Let R be a non empty relational structure and let S be a relational structure. Observe that R ⊗ S is non empty.

One can prove the following two propositions:

(2) Let R, S be relational structures. Then

(i) the internal relation of R ⊆ the internal relation of R ⊗ S, and (ii) the internal relation of S ⊆ the internal relation of R ⊗ S.

(3) For all relational structures R, S such that R is reflexive and S is reflexive holds R ⊗ S is reflexive.

3. Properties of the Addition

Next we state a number of propositions:

(4) Let R, S be relational structures and a, b be sets. Suppose that

(i) ⟨⟨a, b⟩⟩∈ the internal relation of R ⊗ S, (ii) a ∈ the carrier of R, (iii) b ∈ the carrier of R, (iv) R ≈ S, and

(v) R is transitive. Then ⟨⟨a, b⟩⟩∈ the internal relation of R.

(5) Let R, S be relational structures and a, b be sets. Suppose that

(i) ⟨⟨a, b⟩⟩∈ the internal relation of R ⊗ S, (ii) a ∈ the carrier of S, (iii) b ∈ the carrier of S, (iv) R ≈ S, and

(v) S is transitive. Then ⟨⟨a, b⟩⟩∈ the internal relation of S.

(6) Let R, S be relational structures and a, b be sets. Then

(i) if ⟨⟨a, b⟩⟩∈ the internal relation of R, then ⟨⟨a, b⟩⟩∈ the internal relation of R ⊗ S, and

(ii) if ⟨⟨a, b⟩⟩∈ the internal relation of S, then ⟨⟨a, b⟩⟩∈ the internal relation of R ⊗ S.

(7) Let R, S be non empty relational structures and x be an element of R ⊗ S. Then x ∈ the carrier of R or x ∈ (the carrier of S) \ (the carrier of R).

(8) Let R, S be non empty relational structures, x, y be elements of R, and a, b be elements of R ⊗ S. Suppose x = a and y = b and R ≈ S and R is transitive. Then x y if and only if a b.

(9) Let R, S be non empty relational structures, a, b be elements of R ⊗ S, and c, d be elements of S. Suppose a = c and b = d and R ≈ S and S is transitive. Then a b if and only if c d.

(10) Let R, S be antisymmetric reflexive transitive non empty relational structures with l.u.b.’s and x be a set. If x ∈ the carrier of R, then x is an element of R ⊗ S.

(11) Let R, S be antisymmetric reflexive transitive non empty relational structures with l.u.b.’s and x be a set. If x ∈ the carrier of S, then x is an element of R ⊗ S.

(12) Let R, S be non empty relational structures and x be a set. Suppose x ∈ (the carrier of R) ∩ (the carrier of S). Then x is an element of R.

(13) Let R, S be non empty relational structures and x be a set. Suppose x ∈ (the carrier of R) ∩ (the carrier of S). Then x is an element of S.

(14) Let R, S be antisymmetric reflexive transitive non empty relational structures with l.u.b.’s and x, y be elements of R ⊗ S. Suppose x ∈ the carrier of R and y ∈ the carrier of S and R ≈ S. Then x y if and only if there exists an element a of R ⊗ S such that a ∈ (the carrier of R) ∩ (the carrier of S) and x a and a y.

(15) Let R, S be non empty relational structures, a, b be elements of R, and c, d be elements of S. Suppose a = c and b = d and R ≈ S and R is transitive and S is transitive. Then a b if and only if c d.

(16) Let R be an antisymmetric reflexive transitive non empty relational structure with l.u.b.’s, D be a lower directed subset of R, and x, y be elements of R. If x ∈ D and y ∈ D, then x ⊔ y ∈ D.

(17) Let R, S be relational structures and a, b be sets. Suppose that (i) (the carrier of R) ∩ (the carrier of S) is an upper subset of R, (ii) ⟨⟨a, b⟩⟩∈ the internal relation of R ⊗ S, and (iii) a ∈ the carrier of S.

Then b ∈ the carrier of S.

(18) Let R, S be relational structures and a, b be elements of R ⊗ S. Suppose (the carrier of R) ∩ (the carrier of S) is an upper subset of R and a b and a ∈ the carrier of S. Then b ∈ the carrier of S.

(19) Let R, S be antisymmetric reflexive transitive non empty relational structures with l.u.b.’s, x, y be elements of R, and a, b be elements of

S. Suppose that

(i) (the carrier of R) ∩ (the carrier of S) is a lower directed subset of S, (ii) (the carrier of R) ∩ (the carrier of S) is an upper subset of R, (iii) R ≈ S, (iv) x = a, and

(v) y = b.

Then x ⊔ y = a ⊔ b.

(20) Let R, S be lower-bounded antisymmetric reflexive transitive non empty relational structures with l.u.b.’s. Suppose (the carrier of R) ∩ (the carrier of S) is a non empty lower directed subset of S. Then ⊥_{S} ∈ the carrier of R.

(21) Let R, S be relational structures and a, b be sets. Suppose that (i) (the carrier of R) ∩ (the carrier of S) is a lower subset of S, (ii) ⟨⟨a, b⟩⟩∈ the internal relation of R ⊗ S, and (iii) b ∈ the carrier of R.

Then a ∈ the carrier of R.

(22) Let x, y be sets and R, S be relational structures. Suppose ⟨⟨x, y⟩⟩∈ the internal relation of R ⊗ S and (the carrier of R) ∩ (the carrier of S) is an upper subset of R. Then

(i) x ∈ the carrier of R and y ∈ the carrier of R, or (ii) x ∈ the carrier of S and y ∈ the carrier of S, or (iii) x ∈ (the carrier of R)\(the carrier of S) and y ∈ (the carrier of S)\(the

carrier of R).

(23) Let R, S be relational structures and a, b be elements of R ⊗ S. Suppose (the carrier of R) ∩ (the carrier of S) is a lower subset of S and a b and b ∈ the carrier of R. Then a ∈ the carrier of R.

(24) Let R, S be relational structures. Suppose that (i) R ≈ S, (ii) (the carrier of R) ∩ (the carrier of S) is an upper subset of R, (iii) (the carrier of R) ∩ (the carrier of S) is a lower subset of S, (iv) R is transitive and antisymmetric, and

(v) S is transitive and antisymmetric. Then R ⊗ S is antisymmetric.

(25) Let R, S be relational structures. Suppose that (i) (the carrier of R) ∩ (the carrier of S) is an upper subset of R, (ii) (the carrier of R) ∩ (the carrier of S) is a lower subset of S, (iii) R ≈ S, (iv) R is transitive, and

(v) S is transitive.

Then R ⊗ S is transitive. the operation of addition of relational … 339

References

[1] Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719–725, 1991.

[2] Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81–91, 1997.

[3] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93–107, 1997.

[4] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[5] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 1990.

[6] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[7] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990.

[8] Andrzej Wroński. Remarks on intermediate logics with axioms containing only one variable. Reports on Mathematical Logic, 2:63–76, 1974.

The Nagata-Smirnov Theorem. Part I^{1}

Karol Pa̧k University of Białystok

Summary. In this paper we define a discrete subset family of a topological space and basis sigma locally finite and sigma discrete. First, we prove an auxiliary fact for discrete family and sigma locally finite and sigma discrete basis. We also show the necessary condition for the Nagata Smirnov theorem: every metrizable space is T_{3} and has a sigma locally finite basis. Also, we define a sufficient condition for a T_{3} topological space to be T_{4}. We introduce the concept of pseudo metric.

MML Identifier: NAGATA 1.

The terminology and notation used in this paper have been introduced in the following articles: [9], [27], [28], [32], [20], [5], [12], [8], [21], [15], [2], [17], [14], [18], [19], [6], [10], [11], [24], [23], [4], [33], [1], [3], [25], [16], [26], [7], [13], [29], [31], [34], [30], and [22].

In this paper T, T_{1} denote non empty topological spaces and P_{1} denotes a non empty metric structure.

Let T be a topological space and let F be a family of subsets of T. We say that F is discrete if and only if the condition (Def. 1) is satisfied.

(Def. 1) Let p be a point of T. Then there exists an open subset O of T such that p ∈ O and for all subsets A, B of T such that A ∈ F and B ∈ F holds if O meets A and O meets B, then A = B.

Let T be a non empty topological space. Note that there exists a family of subsets of T which is discrete.

Let us consider T. One can check that there exists a family of subsets of T which is empty and discrete.

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102 and KBN grant 4 T11C 039 24.

For simplicity, we adopt the following convention: F, G, H denote families of subsets of T, A, B denote subsets of T, O, U denote open subsets of T, p denotes a point of T, and x, X denote sets.

The following propositions are true:

(1) For every F such that there exists A such that F = {A} holds F is discrete.

(2) For all F, G such that F ⊆ G and G is discrete holds F is discrete.

(3) For all F, G such that F is discrete holds F ∩ G is discrete.

(4) For all F, G such that F is discrete holds F \ G is discrete.

(5) For all F, G, H such that F is discrete and G is discrete and F ⋒ G = H holds H is discrete.

(6) For all F, A, B such that F is discrete and A ∈ F and B ∈ F holds A = B or A misses B.

(7) If F is discrete, then for every p there exists O such that p ∈ O and {O} ⋒ F \ {∅} is trivial.

(8) F is discrete if and only if the following conditions are satisfied:

(i) for every p there exists O such that p ∈ O and {O} ⋒ F \ {∅} is trivial, and

(ii) for all A, B such that A ∈ F and B ∈ F holds A = B or A misses B. Let us consider T and let F be a discrete family of subsets of T. Observe

that clf F is discrete.

Next we state three propositions:

(9) For every F such that F is discrete and for all A, B such that A ∈ F and B ∈ F holds \overline{A ∩ B} = \overline{A} ∩ \overline{B}.

(10) For every F such that F is discrete holds \overline{⋃ F} = ⋃ clf F.

(11) For every F such that F is discrete holds F is locally finite. Let T be a topological space. A family sequence of T is a function from ℕ

into 2^{2^{the carrier of T}}.

In the sequel U_{1} denotes a family sequence of T, r denotes a real number, n denotes a natural number, and f denotes a function.

Let us consider T, U_{1}, n. Then U_{1}(n) is a family of subsets of T. Let us consider T, U_{1}. Then ⋃ U_{1} is a family of subsets of T. Let T be a non empty topological space and let U_{1} be a family sequence of

T. We say that U_{1} is sigma-discrete if and only if: (Def. 2) For every natural number n holds U_{1}(n) is discrete.

Let T be a non empty topological space. Note that there exists a family sequence of T which is sigma-discrete.

Let T be a non empty topological space and let U_{1} be a family sequence of T. We say that U_{1} is sigma-locally-finite if and only if:

(Def. 3) For every natural number n holds U_{1}(n) is locally finite.

Let us consider T and let F be a family of subsets of T. We say that F is sigma-discrete if and only if:

(Def. 4) There exists a sigma-discrete family sequence f of T such that F = ⋃ f.

Let X be a set. We introduce X is uncountable as an antonym of X is countable.

One can verify that every set which is uncountable is also non empty. Let T be a non empty topological space. One can check that there exists a

family sequence of T which is sigma-locally-finite.

Next we state two propositions:

(12) For every U_{1} such that U_{1} is sigma-discrete holds U_{1} is sigma-locally-finite.

(13) Let A be an uncountable set. Then there exists a family F of subsets of

{[: A, A:]}_{top} such that F is locally finite and F is not sigma-discrete. Let T be a non empty topological space and let U_{1} be a family sequence of

T. We say that U_{1} is Basis-sigma-discrete if and only if: (Def. 5) U_{1} is sigma-discrete and ⋃ U_{1} is a basis of T.

Let T be a non empty topological space and let U_{1} be a family sequence of T. We say that U_{1} is Basis-sigma-locally finite if and only if:

(Def. 6) U_{1} is sigma-locally-finite and ⋃ U_{1} is a basis of T.

The following propositions are true:

(14) Let r be a real number. Suppose P_{1} is a non empty metric space. Let x be an element of P_{1}. Then Ω_{(P_{1})} \ \overline{Ball}(x, r) ∈ the open set family of P_{1}.

(15) For every T such that T is metrizable holds T is a T_{3} space and a T_{1} space.

(16) For every T such that T is metrizable holds there exists a family sequence of T which is Basis-sigma-locally finite.

(17) For every function U from ℕ into 2^{the carrier of T} such that for every n holds U(n) is open holds ⋃ U is open.

(18) Suppose that for all A, U such that A is closed and U is open and A ⊆ U there exists a function W from ℕ into 2^{the carrier of T} such that A ⊆ ⋃ W and ⋃ W ⊆ U and for every n holds \overline{W (n)} ⊆ U and W (n) is open. Then T is a T_{4} space.

(19) Let given T. Suppose T is a T_{3} space. Let B_{1} be a family sequence of T. Suppose ⋃ B_{1} is a basis of T. Let U be a subset of T and p be a point of T. Suppose U is open and p ∈ U. Then there exists a subset O of T such that p ∈ O and \overline{O} ⊆ U and O ∈ ⋃ B_{1}.

(20) For every T such that T is a T_{3} space and a T_{1} space and there exists a family sequence of T which is Basis-sigma-locally finite holds T is a T_{4}

space. Let us consider T and let F, G be real maps of T. The functor F +G yielding

a real map of T is defined as follows: (Def. 7) For every element t of T holds (F + G)(t) = F (t) + G(t).

Next we state four propositions:

(21) Let f be a real map of T. Suppose f is continuous. Let F be a real map of [: T, T :]. Suppose that for all elements x, y of the carrier of T holds F (⟨⟨x, y⟩⟩) = |f(x) − f(y)|. Then F is continuous.

(22) For all real maps F, G of T such that F is continuous and G is continuous holds F + G is continuous.

(23) Let A_{1} be a binary operation on ℝ^{the carrier of T}. Suppose that for all real maps f_{1}, f_{2} of T holds A_{1}(f_{1}, f_{2}) = f_{1} + f_{2}. Then A_{1} is commutative and associative and has a unity.

(24) Let A_{1} be a binary operation on ℝ^{the carrier of T}. Suppose that for all real maps f_{1}, f_{2} of T holds A_{1}(f_{1}, f_{2}) = f_{1} + f_{2}. Let m′_{1} be an element of ℝ^{the carrier of T}. If m′_{1} is a unity w.r.t. A_{1}, then m′_{1} is continuous.

Let T, T_{1} be non empty topological spaces, let S_{1} be a function from the carrier of T into 2^{the carrier of T}, and let F_{1} be a function from the carrier of T into (the carrier of T_{1})^{the carrier of T}. The functor F_{1} ≈ S_{1} yields a map from T into T_{1} and is defined by:

(Def. 8) For every point p of T holds (F_{1} ≈ S_{1})(p) = F_{1}(p)(p).

The following propositions are true:

(25) Let A_{1} be a binary operation on ℝ^{the carrier of T}. Suppose that for all real maps f_{1}, f_{2} of T holds A_{1}(f_{1}, f_{2}) = f_{1} + f_{2}. Let F be a finite sequence of elements of ℝ^{the carrier of T}. Suppose that for every n such that 0 ≠ n and n lenF holds F (n) is a continuous real map of T. Then A_{1} ⊙ F is a continuous real map of T.

(26) Let F be a function from the carrier of T into (the carrier of T_{1})^{the carrier of T}. Suppose that for every point p of T holds F (p) is a continuous map from T into T_{1}. Let S be a function from the carrier of T into 2^{the carrier of T}. Suppose that for every point p of T holds p ∈ S(p) and S(p) is open and for all points p, q of T such that p ∈ S(q) holds F (p)(p) = F (q)(p). Then F ≈ S is continuous.

In the sequel m denotes a function from [: the carrier of T, the carrier of T :] into ℝ.

Let us consider X, r and let f be a function from X into ℝ. The functor min(r, f) yielding a function from X into ℝ is defined as follows:

(Def. 9) For every x such that x ∈ X holds (min(r, f))(x) = min(r, f(x)). One can prove the following proposition

the nagata-smirnov theorem. part i 345

(27) For every real number r and for every real map f of T such that f is continuous holds min(r, f) is continuous.

Let X be a set and let f be a function from [: X, X:] into ℝ. We say that f is a pseudometric of if and only if:

(Def. 10) f is Reflexive, symmetric, and triangle. One can prove the following propositions:

(28) Let f be a function from [: X, X:] into ℝ. Then f is a pseudometric of if and only if for all elements a, b, c of X holds f(a, a) = 0 and f(a,

c) f(a, b) + f(c, b).

(29) For every function f from [: X, X:] into ℝ such that f is a pseudometric of and for all elements x, y of X holds f(x, y) 0.

(30) For all r, m such that r > 0 and m is a pseudometric of holds min(r, m) is a pseudometric of.

(31) For all r, m such that r > 0 and m is a metric of the carrier of T holds min(r, m) is a metric of the carrier of T.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Grzegorz Bancerek. Countable sets and Hessenberg’s theorem. Formalized Mathematics, 2(1):65–69, 1991.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[5] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[6] Józef Białas and Yatsuka Nakamura. Dyadic numbers and T_{4} topological spaces. Formalized Mathematics, 5(3):361–366, 1996.

[7] Józef Białas and Yatsuka Nakamura. The Urysohn lemma. Formalized Mathematics, 9(3):631–636, 2001.

[8] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

[9] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[10] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[11] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[13] Czesław Byliński and Piotr Rudnicki. Bounding boxes for compact sets in E^{2}. Formalized Mathematics, 6(3):427–440, 1997.

[14] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 1990.

[15] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[16] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[17] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[18] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

[19] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

[20] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

346 karol pa̧k

[21] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[22] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[23] Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233–236, 1996.

[24] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[25] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[26] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

[27] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

[28] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[29] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

[30] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[31] Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1(5):979–981, 1990.

[32] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[33] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[34] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

Properties of Groups^{1}

Gijs Geleijnse^{2} Eindhoven University of Technology

Grzegorz Bancerek Białystok Technical University

Summary. In this article we formalize theorems from Chapter 1 of [7]. Our article covers Theorems 1.5.4, 1.5.5 (inequality on indices), 1.5.6 (equality of indices), Lemma 1.6.1 and several other supporting theorems needed to complete the formalization.

MML Identifier: GROUP 8.

The articles [1], [12], [5], [19], [20], [3], [4], [13], [16], [6], [14], [15], [10], [8], [17], [18], [11], [2], and [9] provide the terminology and notation for this paper.

For simplicity, we adopt the following rules: G is a strict group, a, b, x, y, z are elements of the carrier of G, H, K are strict subgroups of G, p is a natural number, and A is a subset of the carrier of G.

We now state a number of propositions:

(1) If p is prime and ord(G) = p and G is finite, then there exists a such that ord(a) = p.

(2) Let a_{1}, a_{2} be elements of the carrier of H and b_{1}, b_{2} be elements of the carrier of G. If a_{1} = b_{1} and a_{2} = b_{2}, then a_{1} · a_{2} = b_{1} · b_{2}.

(3) Let a be an element of the carrier of H and b be an element of the carrier of G. If a = b, then for every natural number n holds a^{n} = b^{n}.

(4) Let a be an element of the carrier of H and b be an element of the carrier of G. If a = b, then for every integer i holds a^{i} = b^{i}.

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102.

2The author visited the University of Białystok as a guest.

(5) Let a be an element of the carrier of H and b be an element of the carrier of G. If a = b and G is finite, then ord(a) = ord(b).

(6) For every element h of the carrier of G such that h ∈ H holds H ·h ⊆ the carrier of H.

(7) For every a such that a ≠ 1_{G} holds gr({a}) ≠ {1}_{G}.

(8) For every integer m holds (1_{G})^{m} = 1_{G}.

(9) For every integer m holds a^{m·ord(a)}= 1_{G}.

(10) For every a such that a is not of order 0 and for every integer m holds a^{m} = a^{m mod ord(a)}.

(11) If b is not of order 0, then gr({b}) is finite.

(12) If b is of order 0, then b^{−1} is of order 0.

(13) b is of order 0 iff for every integer n such that b^{n} = 1_{G} holds n = 0.

(14) Let given G. Given a such that a ≠ 1_{G}. Then for every H holds H = G

or H = {1}_{G} if and only if the following conditions are satisfied:

(i) G is a cyclic group and finite, and (ii) there exists a natural number p such that ord(G) = p and p is prime.

(15) Let x, y, z be elements of the carrier of G and A be a subset of the carrier of G. Then z ∈ x · A · y if and only if there exists an element a of the carrier of G such that z = x · a · y and a ∈ A.

(16) For every non empty subset A of G and for every element x of the carrier

of G holds \overline{\overline{A}} = \overline{\overline{x^{−1} · A · x}}. Let us consider G, H, K. The functor DoubleCosets(H, K) yielding a family

of subsets of the carrier of G is defined as follows: (Def. 1) A ∈ DoubleCosets(H, K) iff there exists a such that A = H · a · K.

We now state two propositions:

(17) z ∈ H · x · K iff there exist elements g, h of the carrier of G such that z = g · x · h and g ∈ H and h ∈ K.

(18) For all H, K holds H · x · K = H · y · K or it is not true that there exists

z such that z ∈ H · x · K and z ∈ H · y · K. In the sequel B, A denote strict subgroups of G and D denotes a strict

subgroup of A.

Let us consider G, A. Observe that the left cosets of A is non empty.

Let us consider G and let H be a subgroup of G. We introduce [G : H]_{ℕ} as a synonym of |• : H|_{ℕ}.

Next we state several propositions:

(19) If G = A ⊔ B and D = A ∩ B and G is finite, then [G : B]_{ℕ} [A : D]_{ℕ}.

(20) If G is finite, then [G : H]_{ℕ} > 0.

(21) Let G be a strict group. Suppose G is finite. Let C be a strict subgroup of G and A, B be strict subgroups of C. Suppose C = A ⊔ B. Let D be a

properties of groups 349

strict subgroup of A. Suppose D = A∩B. Let E be a strict subgroup of B. Suppose E = A∩B. Let F be a strict subgroup of C. Suppose F = A∩B. Suppose the left cosets of B is finite and the left cosets of A is finite and [A : C]_{ℕ} and [B : C]_{ℕ} are relative prime. Then [B : C]_{ℕ} = [D : A]_{ℕ} and [A : C]_{ℕ} = [E : B]_{ℕ}.

(22) For every element a of the carrier of G such that a ∈ H and for every integer j holds a^{j} ∈ H.

(23) For every strict group G such that G ≠ {1}_{G} there exists an element b of the carrier of G such that b ≠ 1_{G}.

(24) Let G be a strict group and a be an element of the carrier of G. Suppose G = gr({a}) and G ≠ {1}_{G}. Let H be a strict subgroup of G. If H ≠ {1}_{G}, then there exists a natural number k such that 0 < k and a^{k} ∈ H.

(25) Let G be a strict cyclic group. Suppose G ≠ {1}_{G}. Let H be a strict subgroup of G. If H ≠ {1}_{G}, then H is a cyclic group.

Acknowledgments

Thanks to the Mizar Group for their help and hospitality.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[6] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[7] Marshall Hall Jr. The Theory of Groups. The Macmillan Company, New York, 1959.

[8] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[9] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990.

[10] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[11] Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623–627, 1991.

[12] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[13] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[14] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[15] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855–864, 1990.

[16] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[17] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991.

[18] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.

[19] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

350 gijs geleijnse and grzegorz bancerek

[20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics,

1(1):73–83, 1990.

Received May 31, 2004

Catalan Numbers

Dorota Czȩstochowska University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. In this paper, we define Catalan sequence (starting from 0) and prove some of its basic properties. The Catalan numbers (0, 1, 1, 2, 5, 14, 42, …) arise in a number of problems in combinatorics. They can be computed e.g. using the formula

C_{n} = \frac{\binom{2n}{n}}{n + 1} ,

their recursive definition is also well known:

C_{1} = 1, C_{n} = Σ_{i=1}^{n−1} C_{i}C_{n−i} , n 2.

Among other things, the Catalan numbers describe the number of ways in which parentheses can be placed in a sequence of numbers to be multiplied, two at a time.

MML Identifier: CATALAN1.

The articles [2], [3], [4], [1], [5], [8], [6], and [7] provide the terminology and notation for this paper.

1. Preliminaries

One can prove the following propositions:

(1) For every natural number n such that n > 1 holds n − ′1 2 · n − ′3.

(2) For every natural number n such that n 1 holds n − ′1 2 · n − ′2.

(3) For every natural number n such that n > 1 holds n < 2 · n − ′1.

(4) For every natural number n such that n > 1 holds (n − ′2) + 1 = n − ′1.

(5) For every natural number n such that n > 1 holds \frac{4·n·n−2·n}{n+1} > 1.

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102.

(6) For every natural number n such that n > 1 holds (2·n− ′2)!·n·(n+1) < (2 · n)!.

(7) For every natural number n holds 2 · (2 − \frac{3}{n+1}) < 4.

2. Definition of Catalan Numbers

Let n be a natural number. The functor Catalan(n) yields a real number and is defined as follows:

(Def. 1) Catalan(n) = \frac{\binom{2·n− ′2}{n− ′1}}{n}. The following propositions are true:

(8) For every natural number n such that n > 1 holds Catalan(n) = \frac{(2·n− ′2)!}{(n− ′1)!·n!}.

(9) For every natural number n such that n > 1 holds Catalan(n) = 4 · \binom{2·n− ′3}{n− ′1} − \binom{2·n− ′1}{n− ′1}.

(10) Catalan(0) = 0.

(11) Catalan(1) = 1.

(12) Catalan(2) = 1.

(13) For every natural number n holds Catalan(n) is an integer.

(14) For every natural number k such that k 1 holds Catalan(k + 1) = \frac{(2·k)!}{k!·(k+1)!}.

3. Basic Properties of Catalan Numbers

We now state several propositions:

(15) For every natural number n such that n > 1 holds Catalan(n) < Catalan(n + 1).

(16) For every natural number n holds Catalan(n) Catalan(n + 1).

(17) For every natural number n holds Catalan(n) 0.

(18) For every natural number n holds Catalan(n) is a natural number.

(19) For every natural number n such that n > 0 holds Catalan(n + 1) =

2 · (2 − \frac{3}{n+1}) · Catalan(n). Let n be a natural number. Note that Catalan(n) is natural. Next we state the proposition

(20) For every natural number n such that n > 0 holds Catalan(n) > 0. Let n be a non empty natural number. One can verify that Catalan(n) is

non empty.

One can prove the following proposition

catalan numbers 353

(21) For every natural number n such that n > 0 holds Catalan(n + 1) < 4 · Catalan(n).

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281–290, 1990.

[4] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[5] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.

[6] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[7] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559–564, 2001.

[8] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

Axiomatization of Boolean Algebras Based on Sheffer Stroke

Violetta Kozarkiewicz University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. We formalized another axiomatization of Boolean algebras. The classical one is introduced in [9], “the fourth set of postulates” due to Huntington [3] ([2] in Mizar) and the single axiom in terms of disjunction and negation is codified recently in [7]. In this article, we aimed at the description of Boolean algebras using Sheffer stroke according to [6], namely by the following three axioms:

(x|x)|(x|x) = x
x|(y|(y|y)) = x|x
(x|(y|z))|(x|(y|z)) = ((y|y)|x)|((z|z)|x)

(↾ is used instead of | in the translation of our Mizar article). Since Sheffer in his original paper proved its equivalence and Huntington’s “first set of postulates”, we have also introduced this axiomatization of BAs.

MML Identifier: SHEFFER1.

The terminology and notation used here are introduced in the following articles: [8], [9], [5], [1], [4], and [2].

1. Preliminaries

The following two propositions are true:

(1) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of L. Then (a + b)^{c} = a^{c} ∗ b^{c}.

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102.

(2) Let L be a join-commutative join-associative Huntington non empty complemented lattice structure and a, b be elements of L. Then (a ∗ b)^{c} = a^{c} + b^{c}.

2. Huntington’s First Axiomatization of Boolean Algebras

Let I_{1} be a non empty lattice structure. We say that I_{1} is upper-bounded’ if and only if:

(Def. 1) There exists an element c of I_{1} such that for every element a of I_{1} holds

c ⊓ a = a and a ⊓ c = a. Let L be a non empty lattice structure. Let us assume that L is upper-

bounded’. The functor ⊤′_{L} yields an element of L and is defined by: (Def. 2) For every element a of L holds ⊤′_{L} ⊓ a = a and a ⊓ ⊤′_{L} = a.

Let I_{1} be a non empty lattice structure. We say that I_{1} is lower-bounded’ if and only if:

(Def. 3) There exists an element c of I_{1} such that for every element a of I_{1} holds

c ⊔ a = a and a ⊔ c = a. Let L be a non empty lattice structure. Let us assume that L is lower-

bounded’. The functor ⊥′_{L} yields an element of L and is defined as follows: (Def. 4) For every element a of L holds ⊥′_{L} ⊔ a = a and a ⊔ ⊥′_{L} = a.

Let I_{1} be a non empty lattice structure. We say that I_{1} is distributive’ if and only if:

(Def. 5) For all elements a, b, c of I_{1} holds a ⊔ (b ⊓ c) = (a ⊔ b) ⊓ (a ⊔ c).

Let L be a non empty lattice structure and let a, b be elements of L. We say that a is a complement’ of b if and only if:

(Def. 6) b ⊔ a = ⊤′_{L} and a ⊔ b = ⊤′_{L} and b ⊓ a = ⊥′_{L} and a ⊓ b = ⊥′_{L}.

Let I_{1} be a non empty lattice structure. We say that I_{1} is complemented’ if and only if:

(Def. 7) For every element b of I_{1} holds there exists an element of I_{1} which is a

complement’ of b. Let L be a non empty lattice structure and let x be an element of L. Let

us assume that L is complemented’, distributive, upper-bounded’, and meetcommutative. The functor x^{c′} yields an element of L and is defined as follows:

(Def. 8) x^{c′} is a complement’ of x.

Let us mention that there exists a non empty lattice structure which is Boolean, join-idempotent, upper-bounded’, complemented’, distributive’, lowerbounded’, and lattice-like.

Next we state several propositions:

(3) Let L be a complemented’ join-commutative meet-commutative distributive upper-bounded’ distributive’ non empty lattice structure and x be an element of L. Then x ⊔ x^{c′} = ⊤′_{L}.

(4) Let L be a complemented’ join-commutative meet-commutative distributive upper-bounded’ distributive’ non empty lattice structure and x be an element of L. Then x ⊓ x^{c′} = ⊥′_{L}.

(5) Let L be a complemented’ join-commutative meet-commutative joinidempotent distributive upper-bounded’ distributive’ non empty lattice structure and x be an element of L. Then x ⊔ ⊤′_{L} = ⊤′_{L}.

(6) Let L be a complemented’ join-commutative meet-commutative joinidempotent distributive upper-bounded’ lower-bounded’ distributive’ non empty lattice structure and x be an element of L. Then x ⊓ ⊥′_{L} = ⊥′_{L}.

(7) Let L be a join-commutative meet-absorbing meet-commutative joinabsorbing join-idempotent distributive non empty lattice structure and x, y, z be elements of L. Then (x ⊔ y ⊔ z) ⊓ x = x.

(8) Let L be a join-commutative meet-absorbing meet-commutative joinabsorbing join-idempotent distributive’ non empty lattice structure and x, y, z be elements of L. Then (x ⊓ y ⊓ z) ⊔ x = x.

Let G be a non empty ⊓-semi lattice structure. We say that G is meetidempotent if and only if:

(Def. 9) For every element x of G holds x ⊓ x = x. Next we state a number of propositions:

(9) Every complemented’ join-commutative meet-commutative distributive upper-bounded’ lower-bounded’ distributive’ non empty lattice structure is meet-idempotent.

(10) Every complemented’ join-commutative meet-commutative distributive upper-bounded’ lower-bounded’ distributive’ non empty lattice structure is join-idempotent.

(11) Every complemented’ join-commutative meet-commutative join-idempotent distributive upper-bounded’ distributive’ non empty lattice structure is meet-absorbing.

(12) Every complemented’ join-commutative upper-bounded’ meet-commutative join-idempotent distributive distributive’ lowerbounded’ non empty lattice structure is join-absorbing.

(13) Every complemented’ join-commutative meet-commutative upperbounded’ lower-bounded’ join-idempotent distributive distributive’ non empty lattice structure is upper-bounded.

(14) Every Boolean lattice-like non empty lattice structure is upper-bounded’.

(15) Every complemented’ join-commutative meet-commutative upperbounded’ lower-bounded’ join-idempotent distributive distributive’ non

empty lattice structure is lower-bounded.

(16) Every Boolean lattice-like non empty lattice structure is lower-bounded’.

(17) Every join-commutative meet-commutative meet-absorbing join-absorbing join-idempotent distributive non empty lattice structure is join-associative.

(18) Every join-commutative meet-commutative meet-absorbing join-absorbing join-idempotent distributive’ non empty lattice structure is meet-associative.

(19) Let L be a complemented’ join-commutative meet-commutative lowerbounded’ upper-bounded’ join-idempotent distributive distributive’ non empty lattice structure. Then ⊤_{L} = ⊤′_{L}.

(20) Let L be a complemented’ join-commutative meet-commutative lowerbounded’ upper-bounded’ join-idempotent distributive distributive’ non empty lattice structure. Then ⊥_{L} = ⊥′_{L}.

(21) For every Boolean distributive’ lattice-like non empty lattice structure L holds ⊤_{L} = ⊤′_{L}.

(22) Let L be a Boolean complemented lower-bounded upper-bounded distributive distributive’ lattice-like non empty lattice structure. Then ⊥_{L} = ⊥′_{L}.

(23) Let L be a complemented’ lower-bounded’ upper-bounded’ joincommutative meet-commutative join-idempotent distributive distributive’ non empty lattice structure and x, y be elements of L. Then x is a complement’ of y if and only if x is a complement of y.

(24) Every complemented’ join-commutative meet-commutative lowerbounded’ upper-bounded’ join-idempotent distributive distributive’ non empty lattice structure is complemented.

(25) Every Boolean lower-bounded’ upper-bounded’ distributive’ lattice-like non empty lattice structure is complemented’.

(26) Let L be a non empty lattice structure. Then L is a Boolean lattice if and only if L is lower-bounded’, upper-bounded’, join-commutative, meetcommutative, distributive, distributive’, and complemented’.

Let us note that every non empty lattice structure which is Boolean and lattice-like is also lower-bounded’, upper-bounded’, complemented’, joincommutative, meet-commutative, distributive, and distributive’ and every non empty lattice structure which is lower-bounded’, upper-bounded’, complemented’, join-commutative, meet-commutative, distributive, and distributive’ is also Boolean and lattice-like.

3. Axiomatization Based on Sheffer Stroke

We introduce Sheffer structures which are extensions of 1-sorted structure and are systems

⟨ a carrier, a Sheffer stroke ⟩, where the carrier is a set and the Sheffer stroke is a binary operation on the carrier.

We consider Sheffer lattice structures as extensions of Sheffer structure and lattice structure as systems

⟨ a carrier, a join operation, a meet operation, a Sheffer stroke ⟩, where the carrier is a set, the join operation is a binary operation on the carrier, the meet operation is a binary operation on the carrier, and the Sheffer stroke is a binary operation on the carrier.

We consider Sheffer ortholattice structures as extensions of Sheffer structure and ortholattice structure as systems

⟨ a carrier, a join operation, a meet operation, a complement operation, a Sheffer stroke ⟩, where the carrier is a set, the join operation is a binary operation on the carrier, the meet operation is a binary operation on the carrier, the complement operation is a unary operation on the carrier, and the Sheffer stroke is a binary operation on the carrier.

The Sheffer ortholattice structure TrivShefferOrthoLattStr is defined by: (Def. 10) TrivShefferOrthoLattStr = ⟨{∅}, op_{2}, op_{2}, op_{1}, op_{2}⟩.

One can verify the following observations:

∗ there exists a Sheffer structure which is non empty,

∗ there exists a Sheffer lattice structure which is non empty, and

∗ there exists a Sheffer ortholattice structure which is non empty.

Let L be a non empty Sheffer structure and let x, y be elements of L. The functor x↾y yields an element of L and is defined as follows:

(Def. 11) x↾y = (the Sheffer stroke of L)(x, y).

Let L be a non empty Sheffer ortholattice structure. We say that L is properly defined if and only if the conditions (Def. 12) are satisfied.

(Def. 12)(i) For every element x of L holds x↾x = x^{c}, (ii) for all elements x, y of L holds x ⊔ y = x↾x↾(y↾y), (iii) for all elements x, y of L holds x ⊓ y = x↾y↾(x↾y), and (iv) for all elements x, y of L holds x↾y = x^{c} + y^{c}. Let L be a non empty Sheffer structure. We say that L satisfies (Sheffer_{1}) if

and only if: (Def. 13) For every element x of L holds x↾x↾(x↾x) = x.

We say that L satisfies (Sheffer_{2}) if and only if:

(Def. 14) For all elements x, y of L holds x↾(y↾(y↾y)) = x↾x.

We say that L satisfies (Sheffer_{3}) if and only if: (Def. 15) For all elements x, y, z of L holds (x↾(y↾z))↾(x↾(y↾z)) = y↾y↾x↾(z↾z↾x).

Let us note that every non empty Sheffer structure which is trivial satisfies also (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}).

One can verify that every non empty ⊔-semi lattice structure which is trivial is also join-commutative and join-associative and every non empty ⊓-semi lattice structure which is trivial is also meet-commutative and meet-associative.

Let us note that every non empty lattice structure which is trivial is also join-absorbing, meet-absorbing, and Boolean.

One can check the following observations:

∗ TrivShefferOrthoLattStr is non empty,

∗ TrivShefferOrthoLattStr is trivial, and

∗ TrivShefferOrthoLattStr is properly defined and well-complemented. Let us mention that there exists a non empty Sheffer ortholattice structure

which is properly defined, Boolean, well-complemented, and lattice-like and satisfies (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}).

Next we state three propositions:

(27) Every properly defined Boolean well-complemented lattice-like non empty Sheffer ortholattice structure satisfies (Sheffer_{1}).

(28) Every properly defined Boolean well-complemented lattice-like non empty Sheffer ortholattice structure satisfies (Sheffer_{2}).

(29) Every properly defined Boolean well-complemented lattice-like non

empty Sheffer ortholattice structure satisfies (Sheffer_{3}). Let L be a non empty Sheffer structure and let a be an element of L. The

functor a^{−1} yielding an element of L is defined as follows: (Def. 16) a^{−1} = a↾a.

One can prove the following propositions:

(30) Let L be a non empty Sheffer ortholattice structure satisfying (Sheffer_{3}) and x, y, z be elements of L. Then (x↾(y↾z))^{−1} = y^{−1} ↾x↾(z^{−1} ↾x).

(31) For every non empty Sheffer ortholattice structure L satisfying (Sheffer_{1}) and for every element x of L holds x = (x^{−1} )^{−1} .

(32) Let L be a properly defined non empty Sheffer ortholattice structure satisfying (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) and x, y be elements of L. Then x↾y = y↾x.

(33) Let L be a properly defined non empty Sheffer ortholattice structure satisfying (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) and x, y be elements of L. Then x↾(x↾x) = y↾(y↾y).

axiomatization of boolean algebras based … 361

(34) Every properly defined non empty Sheffer ortholattice structure satisfying (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) is join-commutative.

(35) Every properly defined non empty Sheffer ortholattice structure satisfying (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) is meet-commutative.

(36) Every properly defined non empty Sheffer ortholattice structure satisfying (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) is distributive.

(37) Every properly defined non empty Sheffer ortholattice structure satisfying (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) is distributive’.

(38) Every properly defined non empty Sheffer ortholattice structure satisfying (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) is a Boolean lattice.

References

[1] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[2] Adam Grabowski. Robbins algebras vs. Boolean algebras. Formalized Mathematics, 9(4):681–690, 2001.

[3] E. V. Huntington. Sets of independent postulates for the algebra of logic. Trans. AMS, 5:288–309, 1904.

[4] Michał Muzalewski. Midpoint algebras. Formalized Mathematics, 1(3):483–488, 1990.

[5] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[6] Henry Maurice Sheffer. A set of five independent postulates for Boolean algebras, with application to logical constants. Transactions of American Mathematical Society, 14(4):481– 488, 1913.

[7] Wioletta Truszkowska and Adam Grabowski. On the two short axiomatizations of ortholattices. Formalized Mathematics, 11(3):335–340, 2003.

[8] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[9] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215–222, 1990.

Short Sheffer Stroke-Based Single Axiom for Boolean Algebras

Aneta Łukaszuk University of Białystok

Adam Grabowski^{1} University of Białystok

Summary. We continue the description of Boolean algebras in terms of the Sheffer stroke as defined in [2]. The single axiomatization for BAs in terms of disjunction and negation was shown in [3]. As was checked automatically with the help of automated theorem prover Otter, single axiom of the form

(x|((y|x)|x))|(y|(z|x)) = y (Sh1)

is enough to axiomatize the class of all Boolean algebras (↾ is used instead of | in translation of our Mizar article). Many theorems in Section 2 were automatically translated from the Otter proof object.

MML Identifier: SHEFFER2.

The terminology and notation used in this paper are introduced in the following papers: [4], [1], and [2].

1. First Implication

Let L be a non empty Sheffer structure. We say that L satisfies (Sh_{1}) if and only if:

(Def. 1) For all elements x, y, z of L holds x↾(y↾x↾x)↾(y↾(z↾x)) = y.

Let us observe that every non empty Sheffer structure which is trivial satisfies also (Sh_{1}).

Let us observe that there exists a non empty Sheffer structure which satisfies (Sh_{1}), (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}).

In the sequel L is a non empty Sheffer structure satisfying (Sh_{1}). One can prove the following propositions:

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102.

(1) For all elements x, y, z, u of L holds (x↾(y↾z)↾(x↾(x↾(y↾z))))↾(z↾(x↾z↾z)↾(u↾(x↾(y↾z)))) = z↾(x↾z↾z).

(2) For all elements x, y, z of L holds (x↾y↾(y↾(z↾y↾y)↾(x↾y)↾(x↾y)))↾z = y↾(z↾y↾y).

(3) For all elements x, y, z of L holds x↾(y↾x↾x)↾(y↾(z↾(x↾z↾z))) = y.

(4) For all elements x, y of L holds x↾(x↾(x↾x↾x)↾(y↾(x↾(x↾x↾x)))) = x↾(x↾x↾x).

(5) For every element x of L holds x↾(x↾x↾x) = x↾x.

(6) For every element x of L holds x↾(x↾x↾x)↾(x↾x) = x.

(7) For all elements x, y, z of L holds x↾x↾(x↾(y↾x)) = x.

(8) For all elements x, y of L holds x↾(y↾y↾x↾x)↾y = y↾y.

(9) For all elements x, y of L holds (x↾y↾(x↾y↾(x↾y)↾(x↾y)))↾(x↾y↾(x↾y)) = y↾(x↾y↾(x↾y)↾y↾y).

(10) For all elements x, y of L holds x↾(y↾x↾(y↾x)↾x↾x) = y↾x.

(11) For all elements x, y of L holds x↾x↾(y↾x) = x.

(12) For all elements x, y of L holds x↾(y↾(x↾x)) = x↾x.

(13) For all elements x, y of L holds x↾y↾(x↾y)↾y = x↾y.

(14) For all elements x, y of L holds x↾(y↾x↾x) = y↾x.

(15) For all elements x, y, z of L holds x↾y↾(x↾(z↾y)) = x.

(16) For all elements x, y, z of L holds x↾(y↾z)↾(x↾z) = x.

(17) For all elements x, y, z of L holds x↾(x↾y↾(z↾y)) = x↾y.

(18) For all elements x, y, z of L holds (x↾(y↾z)↾z)↾x = x↾(y↾z).

(19) For all elements x, y of L holds x↾(y↾x↾x) = x↾y.

(20) For all elements x, y of L holds x↾y = y↾x.

(21) For all elements x, y of L holds x↾y↾(x↾x) = x.

(22) For all elements x, y, z of L holds x↾y↾(y↾(z↾x)) = y.

(23) For all elements x, y, z of L holds x↾(y↾z)↾(z↾x) = x.

(24) For all elements x, y, z of L holds x↾y↾(y↾(x↾z)) = y.

(25) For all elements x, y, z of L holds x↾(y↾z)↾(y↾x) = x.

(26) For all elements x, y, z of L holds x↾y↾(x↾z)↾z = x↾z.

(27) For all elements x, y, z of L holds x↾(y↾(x↾(y↾z))) = x↾(y↾z).

(28) For all elements x, y, z of L holds (x↾(y↾(x↾z)))↾y = y↾(x↾z).

(29) For all elements x, y, z, u of L holds (x↾(y↾z))↾(x↾(u↾(y↾x))) = x↾(y↾z)↾(y↾x).

(30) For all elements x, y, z of L holds (x↾(y↾(x↾z)))↾y = y↾(z↾x).

(31) For all elements x, y, z, u of L holds x↾(y↾z)↾(x↾(u↾(y↾x))) = x.

(32) For all elements x, y of L holds x↾(y↾(x↾y)) = x↾x.

(33) For all elements x, y, z of L holds x↾(y↾z) = x↾(z↾y).

(34) For all elements x, y, z of L holds x↾(y↾(x↾(z↾(y↾x)))) = x↾x.

(35) For all elements x, y, z of L holds (x↾(y↾z))↾(y↾x↾x) = x↾(y↾z)↾(x↾(y↾z)).

(36) For all elements x, y, z of L holds x↾(y↾x)↾y = y↾y.

(37) For all elements x, y, z of L holds (x↾y)↾z = z↾(y↾x).

(38) For all elements x, y, z of L holds x↾(y↾(z↾(x↾y))) = x↾(y↾y).

(39) For all elements x, y, z of L holds (x↾y↾y)↾(y↾(z↾x)) = y↾(z↾x)↾(y↾(z↾x)).

(40) For all elements x, y, z, u of L holds (x↾y)↾(z↾u) = u↾z↾(y↾x).

(41) For all elements x, y, z of L holds x↾(y↾(y↾x↾z)) = x↾(y↾y).

(42) For all elements x, y of L holds x↾(y↾x) = x↾(y↾y).

(43) For all elements x, y of L holds (x↾y)↾y = y↾(x↾x).

(44) For all elements x, y, z of L holds x↾(y↾y) = x↾(x↾y).

(45) For all elements x, y, z of L holds (x↾(y↾y))↾(x↾(z↾y)) = x↾(z↾y)↾(x↾(z↾y)).

(46) For all elements x, y, z of L holds (x↾(y↾z))↾(x↾(y↾y)) = x↾(y↾z)↾(x↾(y↾z)).

(47) For all elements x, y, z of L holds x↾(y↾y↾(z↾(x↾(x↾y)))) = x↾(y↾y↾(y↾y)).

(48) For all elements x, y, z of L holds (x↾(y↾z)↾(x↾(y↾z)))↾(y↾y) = x↾(y↾y).

(49) For all elements x, y, z of L holds x↾(y↾y↾(z↾(x↾(x↾y)))) = x↾y.

(50) For all elements x, y, z of L holds (x↾y↾(x↾y)↾(z↾(x↾y↾z)↾(x↾y)))↾(x↾x) = z↾(x↾y↾z)↾(x↾x).

(51) For all elements x, y, z of L holds (x↾(y↾z↾x))↾(y↾y) = y↾z↾(y↾y).

(52) For all elements x, y, z of L holds x↾(y↾z↾x)↾(y↾y) = y.

(53) For all elements x, y, z of L holds x↾(y↾(x↾z↾y)↾x) = y↾(x↾z↾y).

(54) For all elements x, y, z of L holds x↾(y↾(y↾(z↾x))↾x) = y↾(x↾(y↾(x↾z))↾y).

(55) For all elements x, y, z of L holds x↾(y↾(y↾(z↾x))↾x) = y↾(y↾(z↾x)).

(56) For all elements x, y, z, u of L holds x↾(y↾(z↾(z↾(u↾(y↾x))))) = x↾(y↾y).

(57) For all elements x, y, z of L holds x↾(y↾(y↾(z↾(x↾y)))) = x↾(y↾(x↾x)).

(58) For all elements x, y, z of L holds x↾(y↾(y↾(z↾(x↾y)))) = x↾x.

(59) For all elements x, y of L holds x↾(y↾(y↾y)) = x↾x.

(60) For all elements x, y, z of L holds x↾(y↾(z↾x)↾(y↾(z↾x))↾(z↾z)) = x↾(y↾(z↾x)).

(61) For all elements x, y, z of L holds x↾(y↾(z↾z)) = x↾(y↾(z↾x)).

(62) For all elements x, y, z of L holds x↾(y↾(z↾z↾x)) = x↾(y↾z).

(63) For all elements x, y, z of L holds (x↾(y↾y))↾(x↾(z↾(y↾y↾x))) = x↾(z↾y)↾(x↾(z↾y)).

(64) For all elements x, y, z of L holds (x↾(y↾y))↾(x↾(z↾(x↾(y↾y)))) = x↾(z↾y)↾(x↾(z↾y)).

(65) For all elements x, y, z of L holds (x↾(y↾y))↾(x↾(z↾z)) = x↾(z↾y)↾(x↾(z↾y)).

(66) For all elements x, y, z of L holds (x↾x↾y)↾(z↾z↾y) = y↾(x↾z)↾(y↾(x↾z)).

(67) For every non empty Sheffer structure L such that L satisfies (Sh_{1}) holds L satisfies (Sheffer_{1}).

(68) For every non empty Sheffer structure L such that L satisfies (Sh_{1}) holds L satisfies (Sheffer_{2}).

(69) For every non empty Sheffer structure L such that L satisfies (Sh_{1}) holds

L satisfies (Sheffer_{3}). Let us mention that there exists a non empty Sheffer ortholattice struc-

ture which is properly defined, Boolean, well-complemented, lattice-like, and de Morgan and satisfies (Sheffer_{1}), (Sheffer_{2}), (Sheffer_{3}), and (Sh_{1}).

Let us mention that every non empty Sheffer ortholattice structure which is properly defined satisfies (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) is also Boolean and lattice-like and every non empty Sheffer ortholattice structure which is Boolean, lattice-like, well-complemented, and properly defined satisfies also (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}).

2. Second Implication

We adopt the following rules: L denotes a non empty Sheffer structure satisfying (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) and v, q, p, w, z, y, x denote elements of L.

One can prove the following propositions:

(70) For all x, w holds w↾(x↾x↾x) = w↾w.

(71) For all p, x holds x = x↾x↾(p↾(p↾p)).

(72) For all y, w holds w↾w↾(w↾(y↾(y↾y))) = w.

(73) For all q, p, y, w holds (w↾(y↾(y↾y))↾p)↾(q↾q↾p) = p↾(w↾q)↾(p↾(w↾q)).

(74) For all q, p, x holds (x↾p)↾(q↾q↾p) = p↾(x↾x↾q)↾(p↾(x↾x↾q)).

(75) For all w, p, y, q holds (w↾w↾p)↾(q↾(y↾(y↾y))↾p) = p↾(w↾q)↾(p↾(w↾q)).

(76) For all p, x holds x = x↾x↾(p↾p↾p).

(77) For all y, w holds w↾w↾(w↾(y↾y↾y)) = w.

(78) For all y, w holds w↾(y↾y↾y)↾(w↾w) = w.

(79) For all p, y, w holds w↾(y↾y↾y)↾(p↾(p↾p)) = w.

(80) For all p, x, y holds y↾(x↾x)↾(y↾(x↾x))↾(p↾(p↾p)) = (x↾x)↾y.

(81) For all x, y holds y↾(x↾x) = (x↾x)↾y.

(82) For all y, w holds w↾y = y↾y↾(y↾y)↾w.

(83) For all y, w holds w↾y = y↾w.

(84) For all x, p holds (p↾x)↾(p↾(x↾x↾(x↾x))) = x↾x↾(x↾x)↾p↾(x↾x↾(x↾x)↾p).

(85) For all x, p holds (p↾x)↾(p↾x) = x↾x↾(x↾x)↾p↾(x↾x↾(x↾x)↾p).

(86) For all x, p holds (p↾x)↾(p↾x) = x↾p↾(x↾x↾(x↾x)↾p).

(87) For all x, p holds (p↾x)↾(p↾x) = x↾p↾(x↾p).

(88) For all y, q, w holds (w↾q↾(y↾y↾y))↾(w↾q↾(w↾q)) = w↾w↾(w↾q)↾(q↾q↾(w↾q)).

(89) For all q, w holds w↾q = w↾w↾(w↾q)↾(q↾q↾(w↾q)).

(90) For all q, p holds (p↾p)↾(p↾(q↾q↾q)) = q↾q↾(q↾q)↾p↾(q↾q↾p).

(91) For all p, q holds p = q↾q↾(q↾q)↾p↾(q↾q↾p).

(92) For all p, q holds p = q↾p↾(q↾q↾p).

(93) For all z, w, x holds (x↾x↾w↾(z↾z↾w))↾(w↾(x↾z)↾(w↾(x↾z))) = w↾w↾(w↾(x↾z))↾(x↾z↾(x↾z)↾(w↾(x↾z))).

(94) For all z, w, x holds (x↾x↾w↾(z↾z↾w))↾(w↾(x↾z)↾(w↾(x↾z))) = w↾(x↾z).

(95) For all w, p holds (p↾p)↾(p↾(w↾(w↾w))) = w↾w↾p↾(w↾w↾(w↾w)↾p).

(96) For all p, w holds p = w↾w↾p↾(w↾w↾(w↾w)↾p).

(97) For all p, w holds p = w↾w↾p↾(w↾p).

(98) For all z, q, x holds (x↾x↾q↾(z↾z↾q))↾(q↾(x↾z)↾(q↾(x↾z))) = z↾z↾(z↾z)↾(x↾x↾q)↾(q↾q↾(x↾x↾q)).

(99) For all q, z, x holds q↾(x↾z) = (z↾z↾(z↾z)↾(x↾x↾q))↾(q↾q↾(x↾x↾q)).

(100) For all q, z, x holds q↾(x↾z) = (z↾(x↾x↾q))↾(q↾q↾(x↾x↾q)).

(101) For all w, y holds w↾w = y↾y↾y↾w.

(102) For all w, p holds p↾w↾(w↾w↾p) = p.

(103) For all y, w holds w↾w↾(w↾w↾(y↾y↾y)) = (y↾y)↾y.

(104) For all y, w holds w↾w↾w = y↾y↾y.

(105) For all p, w holds w↾p↾(p↾(w↾w)) = p.

(106) For all w, p holds p↾(w↾w)↾(w↾p) = p.

(107) For all p, w holds w↾p↾(w↾(p↾p)) = w.

(108) For all x, y holds y↾(y↾(x↾x)↾(y↾(x↾x))↾(x↾y)) = x↾y.

(109) For all p, w holds w↾(p↾p)↾(w↾p) = w.

(110) For all p, w, q, y holds (y↾y↾y↾q)↾(w↾w↾q) = q↾(p↾(p↾p)↾(p↾(p↾p))↾w)↾(q↾(p↾(p↾p)↾(p↾(p↾p))↾w)).

(111) For all q, w, p holds (q↾q)↾(w↾w↾q) = q↾(p↾(p↾p)↾(p↾(p↾p))↾w)↾(q↾(p↾(p↾p)↾(p↾(p↾p))↾w)).

(112) For all w, y, p holds w↾p↾(w↾(p↾(y↾(y↾y)))) = w.

(113) For all w, y, p holds w↾(p↾(y↾(y↾y)))↾(w↾p) = w.

(114) For all q, p, y holds (y↾y↾y↾p)↾(q↾q↾p) = p↾(p↾p↾q)↾(p↾(p↾p↾q)).

(115) For all q, z, x holds (q↾(x↾x↾z)↾(q↾(x↾x↾z)))↾(x↾q↾(z↾z↾q)) = z↾z↾(z↾z)↾(x↾q)↾(q↾q↾(x↾q)).

(116) For all q, z, x holds (q↾(x↾x↾z)↾(q↾(x↾x↾z)))↾(x↾q↾(z↾z↾q)) = z↾(x↾q)↾(q↾q↾(x↾q)).

(117) For all w, q, z holds (w↾w↾(z↾z↾q))↾(q↾(q↾q↾z)↾(q↾(q↾q↾z))) = z↾z↾q↾(w↾q)↾(z↾z↾q↾(w↾q)).

(118) For all q, p, x holds p↾(x↾p)↾(p↾(x↾p))↾(q↾(q↾q)) = (x↾x)↾p.

(119) For all p, x holds p↾(x↾p) = (x↾x)↾p.

(120) For all p, y holds (y↾p)↾(y↾y↾p) = p↾p↾(y↾p).

(121) For all x, y holds x = x↾x↾(y↾x).

(122) For all x, y holds (y↾x)↾x = x↾(y↾y)↾(x↾(y↾y))↾(y↾x).

(123) For all x, z, y holds x↾(y↾y↾z)↾(x↾(y↾y↾z))↾(y↾x↾(z↾z↾x)) = (z↾(y↾x))↾x.

(124) For all x, y, z holds x↾(z↾(z↾z)↾(z↾(z↾z))↾y)↾(x↾(z↾(z↾z)↾(z↾(z↾z))↾y)) = x.

(125) For all x, z, y holds (x↾(y↾y↾z))↾z = z↾(y↾x).

(126) For all x, y holds x↾(y↾x↾x) = y↾x.

(127) For all z, y, x holds y = x↾x↾x↾y↾(z↾z↾y).

(128) For all z, y holds y↾(y↾y↾z)↾(y↾(y↾y↾z)) = y.

(129) For all x, z, y holds y↾y↾z↾(x↾z)↾(y↾y↾z↾(x↾z)) = (x↾x↾(y↾y↾z))↾z.

(130) For all x, z, y holds (y↾y↾z↾(x↾z))↾(y↾y↾z↾(x↾z)) = z↾(y↾(x↾x)).

(131) For all y, x holds x↾y↾(x↾y)↾x = x↾y.

(132) For all p, w holds w↾w↾(w↾p) = w.

(133) For all w, p holds p↾w↾(w↾w) = w.

(134) For all p, y, w holds w↾(y↾(y↾y))↾(w↾p) = w.

(135) For all p, w holds w↾p↾(w↾w) = w.

(136) For all y, p, w holds w↾p↾(w↾(y↾(y↾y))) = w.

(137) For all p, q, w, y, x holds (x↾(y↾(y↾y))↾w↾(q↾q↾w))↾(w↾(x↾q)↾(w↾(x↾q))) = w↾(p↾(p↾p))↾(w↾(x↾q))↾(x↾q↾(x↾q)↾(w↾(x↾q))).

(138) For all q, w, y, x holds (x↾(y↾(y↾y))↾w↾(q↾q↾w))↾(w↾(x↾q)↾(w↾(x↾q))) = w↾(x↾q↾(x↾q)↾(w↾(x↾q))).

(139) For all q, w, y, x holds (x↾(y↾(y↾y))↾w↾(q↾q↾w))↾(w↾(x↾q)↾(w↾(x↾q))) = w↾(x↾q).

(140) For all z, p, q, y, x holds (x↾(y↾(y↾y))↾q↾(z↾z↾q))↾(q↾(x↾z)↾(q↾(x↾z))) = z↾z↾(p↾(p↾p))↾(x↾(y↾(y↾y))↾q)↾(q↾q↾(x↾(y↾(y↾y))↾q)).

(141) For all z, p, q, y, x holds q↾(x↾z) = (z↾z↾(p↾(p↾p))↾(x↾(y↾(y↾y))↾q))↾(q↾q↾(x↾(y↾(y↾y))↾q)).

(142) For all z, q, y, x holds q↾(x↾z) = (z↾(x↾(y↾(y↾y))↾q))↾(q↾q↾(x↾(y↾(y↾y))↾q)).

(143) For all v, p, y, x holds p↾(x↾v) = (v↾(x↾(y↾(y↾y))↾p))↾p.

(144) For all y, w, z, v, x holds (w↾(z↾(x↾v)))↾(x↾(y↾(y↾y))↾z↾(v↾v↾z)) = z↾(x↾v).

(145) For all y, z, x holds (y↾(x↾x↾z)↾(y↾(x↾x↾z)))↾(x↾y↾(z↾z↾y)) = y↾(x↾x↾z).

(146) For all z, y, x holds (z↾(x↾y))↾y = y↾(x↾x↾z).

(147) For all x, w, y, z holds (x↾x↾w↾(z↾(y↾(y↾y))↾w))↾w = w↾(x↾z).

(148) For all z, w, x holds w↾(z↾(x↾x↾w)) = w↾(x↾z).

(149) For all p, z, y, x holds (z↾(x↾p)↾(z↾(x↾p)))↾(x↾(y↾(y↾y))↾z↾(p↾p↾z)) = p↾p↾z↾(x↾(y↾(y↾y))↾z)↾(p↾p↾z↾(x↾(y↾(y↾y))↾z)).

(150) For all p, z, y, x holds z↾(x↾p) = (p↾p↾z↾(x↾(y↾(y↾y))↾z))↾(p↾p↾z↾(x↾(y↾(y↾y))↾z)).

(151) For all z, p, y, x holds z↾(x↾p) = z↾(p↾(x↾(y↾(y↾y))↾(x↾(y↾(y↾y))))).

(152) For all z, p, x holds z↾(x↾p) = z↾(p↾x).

(153) For all w, q, p holds (p↾q)↾w = w↾(q↾p).

(154) For all w, p, q holds (q↾p↾w)↾q = q↾(p↾p↾w).

(155) For all z, w, y, x holds w↾x = w↾(x↾z↾(x↾(y↾(y↾y))↾(x↾(y↾(y↾y)))↾w)).

(156) For all w, z, x holds w↾x = w↾(x↾z↾(x↾w)).

(157) For all q, x, z, y holds (x↾y)↾(x↾(y↾(z↾(z↾z)))↾q↾x) = x↾y↾(x↾(y↾(z↾(z↾z)))).

(158) For all x, q, z, y holds (x↾y)↾(x↾(y↾(z↾(z↾z))↾(y↾(z↾(z↾z)))↾q)) = x↾y↾(x↾(y↾(z↾(z↾z)))).

(159) For all z, x, q, y holds (x↾y)↾(x↾(y↾q)) = x↾y↾(x↾(y↾(z↾(z↾z)))).

(160) For all x, q, y holds x↾y↾(x↾(y↾q)) = x.

(161) L satisfies (Sh_{1}).

Let us mention that every non empty Sheffer structure which satisfies (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}) satisfies also (Sh_{1}) and every non empty Sheffer structure which satisfies (Sh_{1}) satisfies also (Sheffer_{1}), (Sheffer_{2}), and (Sheffer_{3}).

Let us observe that every non empty Sheffer ortholattice structure which is properly defined satisfies (Sh_{1}) is also Boolean and lattice-like and every non empty Sheffer ortholattice structure which is Boolean, lattice-like, wellcomplemented, and properly defined satisfies also (Sh_{1}).

References

[1] Adam Grabowski. Robbins algebras vs. Boolean algebras. Formalized Mathematics, 9(4):681–690, 2001.

[2] Violetta Kozarkiewicz and Adam Grabowski. Axiomatization of Boolean algebras based on Sheffer stroke. Formalized Mathematics, 12(3):355–361, 2004.

[3] Wioletta Truszkowska and Adam Grabowski. On the two short axiomatizations of ortholattices. Formalized Mathematics, 11(3):335–340, 2003.

[4] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215–222,

370 aneta Łukaszuk and adam grabowski Received May 31, 2004

Differentiable Functions on Normed Linear Spaces. Part II

Hiroshi Imura Shinshu University

Nagano

Yuji Sakai Shinshu University Nagano Yasunari Shidama Shinshu University Nagano

Summary. A continuation of [7], the basic properties of the differentiable functions on normed linear spaces are described.

MML Identifier: NDIFF 2.

The terminology and notation used in this paper have been introduced in the following articles: [16], [3], [19], [5], [4], [1], [15], [6], [17], [18], [9], [8], [2], [20], [12], [14], [10], [13], [7], and [11].

For simplicity, we adopt the following rules: S, T denote non trivial real normed spaces, x_{0} denotes a point of S, f denotes a partial function from S to T, h denotes a convergent to 0 sequence of S, and c denotes a constant sequence of S.

Let X, Y, Z be real normed spaces, let f be an element of BdLinOps(X, Y), and let g be an element of BdLinOps(Y, Z). The functor g·f yielding an element of BdLinOps(X, Z) is defined by:

(Def. 1) g · f = modetrans(g, Y, Z) · modetrans(f, X, Y).

Let X, Y, Z be real normed spaces, let f be a point of RNormSpaceOfBoundedLinearOperators(X, Y), and let g be a point of RNormSpaceOfBoundedLinearOperators(Y, Z). The functor g · f yields a point of RNormSpaceOfBoundedLinearOperators(X, Z) and is defined by:

(Def. 2) g · f = modetrans(g, Y, Z) · modetrans(f, X, Y). Next we state three propositions:

(1) Let x_{0} be a point of S. Suppose f is differentiable in x_{0}. Then there exists a neighbourhood N of x_{0} such that

(i) N ⊆ domf, and (ii) for every point z of S and for every convergent to 0 sequence h of real

numbers and for every c such that rngc = {x_{0}} and rng(h · z + c) ⊆ N holds h^{−1} (f · (h · z + c) − f · c) is convergent and f′(x_{0})(z) = lim(h^{−1} (f · (h · z + c) − f · c)).

(2) Let x_{0} be a point of S. Suppose f is differentiable in x_{0}. Let z be a point of S, h be a convergent to 0 sequence of real numbers, and given c. Suppose rngc = {x_{0}} and rng(h · z + c) ⊆ domf. Then h^{−1} (f · (h · z + c) − f · c) is convergent and f′(x_{0})(z) = lim(h^{−1} (f · (h · z + c) − f · c)).

(3) Let x_{0} be a point of S and N be a neighbourhood of x_{0}. Suppose N ⊆ domf. Let z be a point of S and d_{1} be a point of T. Then the following statements are equivalent

(i) for every convergent to 0 sequence h of real numbers and for every c such that rngc = {x_{0}} and rng(h·z +c) ⊆ N holds h^{−1} (f ·(h·z +c)−f ·c) is convergent and d_{1} = lim(h^{−1} (f · (h · z + c) − f · c)),

(ii) for every real number e such that e > 0 there exists a real number d such that d > 0 and for every real number h such that |h| < d and h ≠ 0 and h · z + x_{0} ∈ N holds ∥h^{−1} · (f_{h·z+x_{0}}− f_{x_{0}}) − d_{1}∥ < e.

Let us consider S, T, let us consider f, let x_{0} be a point of S, and let z be a point of S. We say that f is Gateaux differentiable in x_{0}, z if and only if the condition (Def. 3) is satisfied.

(Def. 3) There exists a neighbourhood N of x_{0} such that (i) N ⊆ domf, and (ii) there exists a point d_{1} of T such that for every real number e such

that e > 0 there exists a real number d such that d > 0 and for every real number h such that |h| < d and h ≠ 0 and h · z + x_{0} ∈ N holds ∥h^{−1} · (f_{h·z+x_{0}}− f_{x_{0}}) − d_{1}∥ < e.

One can prove the following proposition

(4) For every real normed space X and for all points x, y of X holds ∥x−y∥ > 0 iff x ≠ y and for every real normed space X and for all points x, y of X holds ∥x − y∥ = ∥y − x∥ and for every real normed space X and for all points x, y of X holds ∥x − y∥ = 0 iff x = y and for every real normed space X and for all points x, y of X holds ∥x − y∥ ≠ 0 iff x ≠ y and for every real normed space X and for all points x, y, z of X and for every real number e such that e > 0 holds if ∥x − z∥ < \frac{e}{2} and ∥z − y∥ < \frac{e}{2}, then ∥x − y∥ < e and for every real normed space X and for all points x, y, z of X and for every real number e such that e > 0 holds if ∥x − z∥ < \frac{e}{2} and ∥y − z∥ < \frac{e}{2}, then ∥x − y∥ < e and for every real normed space X and for every point x of X such that for every real number e such that e > 0 holds ∥x∥ < e holds x = 0_{X} and for every real normed space X and for all points x, y of X such that for every real number e such that e > 0 holds ∥x − y∥ < e holds x = y.

Let us consider S, T, let us consider f, let x_{0} be a point of S, and let z be a point of S. Let us assume that f is Gateaux differentiable in x_{0}, z. The functor GateauxDiff_{z}(f, x_{0}) yields a point of T and is defined by the condition (Def. 4).

(Def. 4) There exists a neighbourhood N of x_{0} such that (i) N ⊆ domf, and (ii) for every real number e such that e > 0 there exists a real number d

such that d > 0 and for every real number h such that |h| < d and h ≠ 0 and h · z + x_{0} ∈ N holds ∥h^{−1} · (f_{h·z+x_{0}}− f_{x_{0}}) − GateauxDiff_{z}(f, x_{0})∥ < e.

We now state two propositions:

(5) Let x_{0} be a point of S and z be a point of S. Then f is Gateaux differentiable in x_{0}, z if and only if there exists a neighbourhood N of x_{0} such that N ⊆ domf and there exists a point d_{1} of T such that for every convergent to 0 sequence h of real numbers and for every c such that rngc = {x_{0}} and rng(h · z + c) ⊆ N holds h^{−1} (f · (h · z + c) − f · c) is convergent and d_{1} = lim(h^{−1} (f · (h · z + c) − f · c)).

(6) Let x_{0} be a point of S. Suppose f is differentiable in x_{0}. Let z be a point of S. Then

(i) f is Gateaux differentiable in x_{0}, z, (ii) GateauxDiff_{z}(f, x_{0}) = f′(x_{0})(z), and (iii) there exists a neighbourhood N of x_{0} such that N ⊆ domf and for

every convergent to 0 sequence h of real numbers and for every c such that rngc = {x_{0}} and rng(h · z + c) ⊆ N holds h^{−1} (f · (h · z + c) − f · c) is convergent and GateauxDiff_{z}(f, x_{0}) = lim(h^{−1} (f · (h · z + c) − f · c)).

In the sequel U is a non trivial real normed space. Next we state several propositions:

(7) Let R be a rest of S, T. Suppose R_{0_{S}} = 0_{T} . Let e be a real number. Suppose e > 0. Then there exists a real number d such that d > 0 and for every point h of S such that ∥h∥ < d holds ∥R_{h}∥ e · ∥h∥.

(8) Let R be a rest of T, U. Suppose R_{0_{T}} = 0_{U}. Let L be a bounded linear operator from S into T. Then R · L is a rest of S, U.

(9) For every rest R of S, T and for every bounded linear operator L from T into U holds L · R is a rest of S, U.

(10) Let R_{1} be a rest of S, T. Suppose (R_{1})_{0_{S}} = 0_{T} . Let R_{2} be a rest of T, U. If (R_{2})_{0_{T}} = 0_{U}, then R_{2} · R_{1} is a rest of S, U.

(11) Let R_{1} be a rest of S, T. Suppose (R_{1})_{0_{S}} = 0_{T} . Let R_{2} be a rest of T, U. Suppose (R_{2})_{0_{T}} = 0_{U}. Let L be a bounded linear operator from S into T. Then R_{2} · (L + R_{1}) is a rest of S, U.

(12) Let R_{1} be a rest of S, T. Suppose (R_{1})_{0_{S}} = 0_{T} . Let R_{2} be a rest of T, U. Suppose (R_{2})_{0_{T}} = 0_{U}. Let L_{1} be a bounded linear operator from S into T and L_{2} be a bounded linear operator from T into U. Then

L_{2} · R_{1} + R_{2} · (L_{1} + R_{1}) is a rest of S, U.

(13) Let f_{1} be a partial function from S to T. Suppose f_{1} is differentiable in x_{0}. Let f_{2} be a partial function from T to U. Suppose f_{2} is differentiable in (f_{1})_{x_{0}}. Then f_{2} ·f_{1} is differentiable in x_{0} and (f_{2} ·f_{1})′(x_{0}) = f_{2}′((f_{1})_{x_{0}})· f_{1}′(x_{0}).

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[2] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[6] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[7] Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321–327, 2004.

[8] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[9] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[10] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269–275, 2004.

[11] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[12] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[13] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

[14] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2003.

[15] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[16] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[17] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[18] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[19] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[20] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171–175, 1992.

Received June 4, 2004

Logical Correctness of Vector Calculation Programs

Takaya Nishiyama Shinshu University

Nagano Hirofumi Fukura Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. In C-program, vectors of n-dimension are sometimes represented by arrays, where the dimension n is saved in the 0-th element of each array. If we write the program in non-overwriting type, we can give Logical-Model to each program. Here, we give a program calculating inner product of 2 vectors, as an example of such a type, and its Logical-Model. If the Logical-Model is well defined, and theorems tying the model with previous definitions are given, we can say that the program is logically correct. In case the program is given as implicit function form (i.e., the result of calculation is given by a variable of one of arguments of a function), its Logical-Model is given by a definition of a new predicate form. Logical correctness of such a program is shown by theorems following the definition. As examples of such programs, we presented vector calculation of add, sub, minus and scalar product.

MML Identifier: PRGCOR 2.

The articles [16], [18], [14], [20], [8], [4], [5], [11], [3], [10], [2], [6], [19], [17], [12], [9], [13], [1], [15], and [7] provide the terminology and notation for this paper.

In this paper m, n, i are natural numbers and D is a set. The following proposition is true

(1) For all n, m holds n ∈ m iff n < m. Let D be a non empty set. One can check that there exists a finite 0-sequence

of D which is non empty.

The following proposition is true

(2) For every non empty set D and for every non empty finite 0-sequence f of D holds lenf > 0.

Let D be a set and let q be a finite sequence of elements of D. The functor FS2XFS(q) yields a finite 0-sequence of D and is defined by:

(Def. 1) lenFS2XFS(q) = lenq and for every i such that i < lenq holds q(i+1) =

(FS2XFS(q))(i). Let D be a set and let q be a finite 0-sequence of D. The functor XFS2FS(q)

yielding a finite sequence of elements of D is defined as follows:

(Def. 2) lenXFS2FS(q) = lenq and for every i such that 1 i and i lenq holds

q(i − ′1) = (XFS2FS(q))(i). One can prove the following two propositions:

(3) For every natural number k and for every set a holds k ⟼→ a is a finite 0-sequence.

(4) Let D be a set, n be a natural number, and r be a set. Suppose r ∈ D. Then n ⟼→ r is a finite 0-sequence of D and for every finite 0-sequence q_{2} such that q_{2} = n ⟼→ r holds lenq_{2} = n.

Let D be a non empty set, let q be a finite sequence of elements of D, and let n be a natural number. Let us assume that n > lenq and ℕ ⊆ D. The functor FS2XFS^{⋆}(q, n) yields a non empty finite 0-sequence of D and is defined by the conditions (Def. 3).

(Def. 3)(i) lenq = (FS2XFS^{⋆}(q, n))(0), (ii) lenFS2XFS^{⋆}(q, n) = n, (iii) for every i such that 1 i and i lenq holds (FS2XFS^{⋆}(q, n))(i) =

q(i), and (iv) for every natural number j such that lenq < j and j < n holds

(FS2XFS^{⋆}(q, n))(j) = 0. Let D be a non empty set and let p be a non empty finite 0-sequence of D.

Let us assume that ℕ ⊆ D and p(0) is a natural number and p(0) ∈ lenp. The functor XFS2FS^{⋆}(p) yielding a finite sequence of elements of D is defined by:

(Def. 4) For every m such that m = p(0) holds lenXFS2FS^{⋆}(p) = m and for

every i such that 1 i and i m holds (XFS2FS^{⋆}(p))(i) = p(i). The following proposition is true

(5) For every non empty set D and for every non empty finite 0-sequence p of D such that ℕ ⊆ D and p(0) = 0 and 0 < lenp holds XFS2FS^{⋆}(p) = ∅.

Let D be a non empty set, let p be a finite 0-sequence of D, and let q be a finite sequence of elements of D. We say that p is an xrep of q if and only if:

(Def. 5) ℕ ⊆ D and p(0) = lenq and lenq < lenp and for every i such that 1 i

and i lenq holds p(i) = q(i). The following proposition is true

(6) Let D be a non empty set and p be a non empty finite 0-sequence of D. Suppose ℕ ⊆ D and p(0) is a natural number and p(0) ∈ lenp. Then p is an xrep of XFS2FS^{⋆}(p).

Let x, y, a, b, c be sets. The functor IFLGT(x, y, a, b, c) yielding a set is defined by:

(Def. 6) IFLGT(x, y, a, b, c) = \{ \begin{matrix}a, if x ∈ y, \\ b, if x = y, \\ c, otherwise. \end{matrix}

Next we state the proposition

(7) Let D be a non empty set, q be a finite sequence of elements of D, and n be a natural number. Suppose ℕ ⊆ D and n > lenq. Then there exists a finite 0-sequence p of D such that lenp = n and p is an xrep of q.

Let b be a finite 0-sequence of ℝ and let n be a natural number. Then b(n) is a real number.

Let a, b be finite 0-sequences of ℝ. Let us assume that b(0) is a natural number and 0 b(0) and b(0) < lena. The functor InnerPrdPrg(a, b) yielding a real number is defined by the condition (Def. 7).

(Def. 7) There exists a finite 0-sequence s of ℝ and there exists an integer n such

that

(i) lens = lena, (ii) s(0) = 0, (iii) n = b(0), (iv) if n ≠ 0, then for every natural number i such that i < n holds s(i+1) =

s(i) + a(i + 1) · b(i + 1), and

(v) InnerPrdPrg(a, b) = s(n). The following propositions are true:

(8) Let a be a finite sequence of elements of ℝ and s be a finite 0-sequence of ℝ. Suppose lens > lena and s(0) = 0 and for every i such that i < lena holds s(i + 1) = s(i) + a(i + 1). Then ∑ a = s(lena).

(9) Let a be a finite sequence of elements of ℝ. Then there exists a finite 0-sequence s of ℝ such that lens = lena + 1 and s(0) = 0 and for every i such that i < lena holds s(i + 1) = s(i) + a(i + 1) and ∑ a = s(lena).

(10) Let a, b be finite sequences of elements of ℝ and n be a natural number. If lena = lenb and n > lena, then |(a, b)| = InnerPrdPrg(FS2XFS^{⋆}(a, n), FS2XFS^{⋆}(b, n)).

Let b, c be finite 0-sequences of ℝ, let a be a real number, and let m be an integer. We say that m scalar prd prg of c, a, b if and only if the conditions (Def. 8) are satisfied.

(Def. 8)(i) lenc = m, (ii) lenb = m, and (iii) there exists an integer n such that c(0) = b(0) and n = b(0) and if

n ≠ 0, then for every natural number i such that 1 i and i n holds c(i) = a · b(i).

We now state the proposition

(11) Let b be a non empty finite 0-sequence of ℝ, a be a real number, and m be a natural number. Suppose b(0) is a natural number and lenb = m and 0 b(0) and b(0) < m. Then

(i) there exists a finite 0-sequence c of ℝ such that m scalar prd prg of c, a, b, and

(ii) for every non empty finite 0-sequence c of ℝ such that m scalar prd prg of c, a, b holds XFS2FS^{⋆}(c) = a · XFS2FS^{⋆}(b).

Let b, c be finite 0-sequences of ℝ and let m be an integer. We say that m vector minus prg of c, b if and only if the conditions (Def. 9) are satisfied.

(Def. 9)(i) lenc = m, (ii) lenb = m, and (iii) there exists an integer n such that c(0) = b(0) and n = b(0) and if

n ≠ 0, then for every natural number i such that 1 i and i n holds c(i) = − b(i).

The following proposition is true

(12) Let b be a non empty finite 0-sequence of ℝ and m be a natural number. Suppose b(0) is a natural number and lenb = m and 0 b(0) and b(0) < m. Then

(i) there exists a finite 0-sequence c of ℝ such that m vector minus prg of c, b, and

(ii) for every non empty finite 0-sequence c of ℝ such that m vector minus prg of c, b holds XFS2FS^{⋆}(c) = − XFS2FS^{⋆}(b).

Let a, b, c be finite 0-sequences of ℝ and let m be an integer. We say that m vector add prg of c, a, b if and only if the conditions (Def. 10) are satisfied.

(Def. 10)(i) lenc = m, (ii) lena = m, (iii) lenb = m, and (iv) there exists an integer n such that c(0) = b(0) and n = b(0) and if

n ≠ 0, then for every natural number i such that 1 i and i n holds c(i) = a(i) + b(i).

Next we state the proposition

(13) Let a, b be non empty finite 0-sequences of ℝ and m be a natural number. Suppose b(0) is a natural number and lena = m and lenb = m and a(0) = b(0) and 0 b(0) and b(0) < m. Then

(i) there exists a finite 0-sequence c of ℝ such that m vector add prg of c, a, b, and

(ii) for every non empty finite 0-sequence c of ℝ such that m vector add prg of c, a, b holds XFS2FS^{⋆}(c) = XFS2FS^{⋆}(a) + XFS2FS^{⋆}(b).

logical correctness of vector calculation … 379

Let a, b, c be finite 0-sequences of ℝ and let m be an integer. We say that m vector sub prg of c, a, b if and only if the conditions (Def. 11) are satisfied.

(Def. 11)(i) lenc = m, (ii) lena = m, (iii) lenb = m, and (iv) there exists an integer n such that c(0) = b(0) and n = b(0) and if

n ≠ 0, then for every natural number i such that 1 i and i n holds c(i) = a(i) − b(i).

One can prove the following proposition

(14) Let a, b be non empty finite 0-sequences of ℝ and m be a natural number. Suppose b(0) is a natural number and lena = m and lenb = m and a(0) = b(0) and 0 b(0) and b(0) < m. Then

(i) there exists a finite 0-sequence c of ℝ such that m vector sub prg of c, a, b, and

(ii) for every non empty finite 0-sequence c of ℝ such that m vector sub prg of c, a, b holds XFS2FS^{⋆}(c) = XFS2FS^{⋆}(a) − XFS2FS^{⋆}(b).

References

[1] Kanchun and Yatsuka Nakamura. The inner product of finite sequences and of points of n-dimensional topological space. Formalized Mathematics, 11(2):179–183, 2003.

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[5] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[7] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[9] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[13] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[14] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[15] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[16] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[17] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[19] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.

380 takaya nishiyama et al.

[20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics,

1(1):73–83, 1990.

Received July 13, 2004

Continuous Mappings between Finite and One-Dimensional Finite Topological Spaces

Hiroshi Imura Shinshu University

Nagano Masami Tanaka Shinshu University Nagano Yatsuka Nakamura Shinshu University Nagano

Summary. We showed relations between separateness and inflation operation. We also gave some relations between separateness and connectedness defined before. For two finite topological spaces, we defined a continuous function from one to another. Some topological concepts are preserved by such continuous functions. We gave one-dimensional concrete models of finite topological space.

MML Identifier: FINTOPO4.

The notation and terminology used here are introduced in the following articles: [12], [5], [13], [1], [14], [3], [4], [2], [6], [10], [9], [11], [7], and [8].

Let F_{1} be a non empty finite topology space and let A, B be subsets of F_{1}. We say that A and B are separated if and only if:

(Def. 1) A^{b} misses B and A misses B^{b}. Next we state a number of propositions:

(1) Let F_{1} be a filled non empty finite topology space, A be a subset of F_{1}, and n, m be natural numbers. If n m, then Finf(A, n) ⊆ Finf(A, m).

(2) Let F_{1} be a filled non empty finite topology space, A be a subset of F_{1}, and n, m be natural numbers. If n m, then Fcl(A, n) ⊆ Fcl(A, m).

(3) Let F_{1} be a filled non empty finite topology space, A be a subset of F_{1}, and n, m be natural numbers. If n m, then Fdfl(A, m) ⊆ Fdfl(A, n).

(4) Let F_{1} be a filled non empty finite topology space, A be a subset of F_{1}, and n, m be natural numbers. If n m, then Fint(A, m) ⊆ Fint(A, n).

(5) Let F_{1} be a non empty finite topology space and A, B be subsets of F_{1}. If A and B are separated, then B and A are separated.

(6) Let F_{1} be a filled non empty finite topology space and A, B be subsets of F_{1}. If A and B are separated, then A misses B.

(7) Let F_{1} be a non empty finite topology space and A, B be subsets of F_{1}. Suppose F_{1} is symmetric. Then A and B are separated if and only if A^{f} misses B and A misses B^{f}.

(8) Let F_{1} be a filled non empty finite topology space and A, B be subsets of F_{1}. If F_{1} is symmetric and A^{b} misses B, then A misses B^{b}.

(9) Let F_{1} be a filled non empty finite topology space and A, B be subsets of F_{1}. If F_{1} is symmetric and A misses B^{b}, then A^{b} misses B.

(10) Let F_{1} be a filled non empty finite topology space and A, B be subsets of F_{1}. Suppose F_{1} is symmetric. Then A and B are separated if and only if A^{b} misses B.

(11) Let F_{1} be a filled non empty finite topology space and A, B be subsets of F_{1}. Suppose F_{1} is symmetric. Then A and B are separated if and only if A misses B^{b}.

(12) Let F_{1} be a filled non empty finite topology space and I_{1} be a subset of F_{1}. Suppose F_{1} is symmetric. Then I_{1} is connected if and only if for all subsets A, B of F_{1} such that I_{1} = A ∪ B and A and B are separated holds A = I_{1} or B = I_{1}.

(13) Let F_{1} be a filled non empty finite topology space and B be a subset of F_{1}. Suppose F_{1} is symmetric. Then B is connected if and only if it is not true that there exists a subset C of F_{1} such that C ≠ ∅ and B \ C ≠ ∅ and C ⊆ B and C^{b} misses B \ C.

Let F_{2}, F_{3} be non empty finite topology spaces, let f be a function from the carrier of F_{2} into the carrier of F_{3}, and let n be a natural number. We say that f is continuous n if and only if:

(Def. 2) For every element x of F_{2} and for every element y of F_{3} such that x ∈ the

carrier of F_{2} and y = f(x) holds f^{∘}U(x, 0) ⊆ U(y, n). Next we state four propositions:

(14) Let F_{2} be a non empty finite topology space, F_{3} be a filled non empty finite topology space, n be a natural number, and f be a function from the carrier of F_{2} into the carrier of F_{3}. If f is continuous 0, then f is continuous n.

(15) Let F_{2} be a non empty finite topology space, F_{3} be a filled non empty finite topology space, n_{0}, n be natural numbers, and f be a function from the carrier of F_{2} into the carrier of F_{3}. If f is continuous n_{0} and n_{0} n, then f is continuous n.

(16) Let F_{2}, F_{3} be non empty finite topology spaces, A be a subset of F_{2}, B be a subset of F_{3}, and f be a function from the carrier of F_{2} into the carrier of F_{3}. If f is continuous 0 and B = f^{∘}A, then f^{∘}A^{b} ⊆ B^{b}.

(17) Let F_{2}, F_{3} be non empty finite topology spaces, A be a subset of F_{2}, B be a subset of F_{3}, and f be a function from the carrier of F_{2} into the carrier of F_{3}. Suppose A is connected and f is continuous 0 and B = f^{∘}A. Then B is connected.

Let n be a natural number. The functor Nbdl1(n) yielding a function from Segn into 2^{Seg n} is defined as follows:

(Def. 3) domNbdl1(n) = Segn and for every natural number i such that i ∈ Segn

holds (Nbdl1(n))(i) = {i, max(i − ′1, 1), min(i + 1, n)}. Let n be a natural number. Let us assume that n > 0. The functor FTSL1(n)

yielding a non empty finite topology space is defined as follows: (Def. 4) FTSL1(n) = ⟨Segn, Nbdl1(n)⟩.

We now state two propositions:

(18) For every natural number n such that n > 0 holds FTSL1(n) is filled.

(19) For every natural number n such that n > 0 holds FTSL1(n) is symme-

tric. Let n be a natural number. The functor Nbdc1(n) yielding a function from

Segn into 2^{Seg n} is defined by the conditions (Def. 5). (Def. 5)(i) domNbdc1(n) = Segn, and

(ii) for every natural number i such that i ∈ Segn holds if 1 < i and i < n, then (Nbdc1(n))(i) = {i, i − ′1, i + 1} and if i = 1 and i < n, then (Nbdc1(n))(i) = {i, n, i + 1} and if 1 < i and i = n, then (Nbdc1(n))(i) = {i, i − ′1, 1} and if i = 1 and i = n, then (Nbdc1(n))(i) = {i}.

Let n be a natural number. Let us assume that n > 0. The functor FTSC1(n) yielding a non empty finite topology space is defined as follows:

(Def. 6) FTSC1(n) = ⟨Segn, Nbdc1(n)⟩.

We now state two propositions:

(20) For every natural number n such that n > 0 holds FTSC1(n) is filled.

(21) For every natural number n such that n > 0 holds FTSC1(n) is symmetric.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[6] Hiroshi Imura and Masayoshi Eguchi. Finite topological spaces. Formalized Mathematics, 3(2):189–193, 1992.

[7] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

[8] Jarosław Kotowicz and Yuji Sakai. Properties of partial functions from a domain to the set of real numbers. Formalized Mathematics, 3(2):279–288, 1992.

[9] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83–86, 1993.

[10] Masami Tanaka and Yatsuka Nakamura. Some set series in finite topological spaces. Fundamental concepts for image processing. Formalized Mathematics, 12(2):125–129, 2004.

[11] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[12] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[13] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[14] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

The Nagata-Smirnov Theorem. Part II^{1}

Karol Pa̧k University of Białystok

Summary. In this paper we show some auxiliary facts for sequence function to be pseudo-metric. Next we prove the Nagata-Smirnov theorem that every topological space is metrizable if and only if it has σ-locally finite basis. We attach also the proof of the Bing’s theorem that every topological space is metrizable if and only if its basis is σ-discrete.

MML Identifier: NAGATA 2.

The terminology and notation used in this paper have been introduced in the following articles: [9], [27], [28], [32], [20], [5], [12], [8], [21], [15], [2], [17], [14], [18], [19], [6], [10], [11], [24], [23], [4], [33], [1], [3], [25], [16], [26], [7], [13], [29], [31], [34], [30], and [22].

For simplicity, we adopt the following convention: i, k, m, n denote natural numbers, r, s denote real numbers, X denotes a set, T, T_{1}, T_{2} denote non empty topological spaces, p denotes a point of T, A denotes a subset of T, A′ denotes a non empty subset of T, p_{1} denotes an element of [: the carrier of T, the carrier of T :], p_{2} denotes a function from [: the carrier of T, the carrier of T :] into ℝ, p′_{1} denotes a real map of [: T, T :], f denotes a real map of T, F_{2} denotes a sequence of partial functions from [: the carrier of T, the carrier of T :] into ℝ, and s_{1} denotes a sequence of real numbers.

The following proposition is true

(1) For every i such that i > 0 there exist n, m such that i = 2^{n} · (2 · m + 1).

The function PairFunc from [: ℕ, ℕ:] into ℕ is defined by: (Def. 1) For all n, m holds PairFunc(⟨⟨n, m⟩⟩) = 2^{n} · (2 · m + 1) − 1.

We now state the proposition (2) PairFunc is bijective. Let X be a set, let f be a function from [: X, X:] into ℝ, and let x be an

1This work has been partially supported by the CALCULEMUS grant HPRN-CT-2000-00102 and KBN grant 4 T11C 039 24.

element of X. The functor ρ(f, x) yielding a function from X into ℝ is defined as follows:

(Def. 2) For every element y of X holds (ρ(f, x))(y) = f(x, y). The following two propositions are true:

(3) Let D be a subset of [: T_{1}, T_{2} :]. Suppose D is open. Let x_{1} be a point of T_{1}, x_{2} be a point of T_{2}, X_{1} be a subset of T_{1}, and X_{2} be a subset of T_{2}. Then

(i) if X_{1} = π_{1}((the carrier of T_{1}) × the carrier of T_{2})^{∘}(D ∩ [: the carrier of T_{1}, {x_{2}}:]), then X_{1} is open, and

(ii) if X_{2} = π_{2}((the carrier of T_{1})×the carrier of T_{2})^{∘}(D∩[: {x_{1}}, the carrier

of T_{2} :]), then X_{2} is open.

(4) For every p_{2} such that for every p′_{1} such that p_{2} = p′_{1} holds p′_{1} is continuous and for every point x of T holds ρ(p_{2}, x) is continuous.

Let X be a non empty set, let f be a function from [: X, X:] into ℝ, and let A be a subset of X. The functor inf(f, A) yielding a function from X into ℝ is defined by:

(Def. 3) For every element x of X holds (inf(f, A))(x) = inf((ρ(f, x))^{∘}A). One can prove the following propositions:

(5) Let X be a non empty set and f be a function from [: X, X:] into ℝ. Suppose f is a pseudometric of. Let A be a non empty subset of X and x be an element of X. Then (inf(f, A))(x) 0.

(6) Let X be a non empty set and f be a function from [: X, X:] into ℝ. Suppose f is a pseudometric of. Let A be a subset of X and x be an element of X. If x ∈ A, then (inf(f, A))(x) = 0.

(7) Let given p_{2}. Suppose p_{2} is a pseudometric of. Let x, y be points of T and A be a non empty subset of T. Then |(inf(p_{2}, A))(x) − (inf(p_{2}, A))(y)| p_{2}(x, y).

(8) Let given p_{2}. Suppose p_{2} is a pseudometric of and for every p holds ρ(p_{2}, p) is continuous. Let A be a non empty subset of T. Then inf(p_{2}, A) is continuous.

(9) For every function f from [: X, X:] into ℝ such that f is a metric of X holds f is a pseudometric of.

(10) Let given p_{2}. Suppose p_{2} is a metric of the carrier of T and for every non empty subset A of T holds \overline{A} = \{p; p ranges over points of T: (inf(p_{2}, A))(p) = 0}. Then T is metrizable.

(11) Let given F_{2}. Suppose for every n there exists p_{2} such that F_{2}(n) = p_{2} and p_{2} is a pseudometric of and for every p_{1} holds F_{2}#p_{1} is summable.

Let given p_{2}. If for every p_{1} holds p_{2}(p_{1}) = ∑(F_{2}#p_{1}), then p_{2} is a pseudometric of.

(12) For all n, s_{1} such that for every m such that m n holds s_{1}(m) r and for every m such that m n holds (∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(m) r · (m + 1).

(13) For every k holds |(∑_{α=0}^{κ}(s_{1})(α))_{κ∈ℕ}(k)| (∑_{α=0}^{κ}|s_{1}|(α))_{κ∈ℕ}(k).

(14) Let F_{1} be a sequence of partial functions from the carrier of T into ℝ.

Suppose that

(i) for every n there exists f such that F_{1}(n) = f and f is continuous and for every p holds f(p) 0, and

(ii) there exists s_{1} such that s_{1} is summable and for all n, p holds (F_{1}#p)(n) s_{1}(n). Let given f. If for every p holds f(p) = ∑(F_{1}#p), then f is continuous.

(15) Let given s, F_{2}. Suppose that for every n there exists p_{2} such that F_{2}(n) = p_{2} and p_{2} is a pseudometric of and for every p_{1} holds p_{2}(p_{1}) s and for every p′_{1} such that p_{2} = p′_{1} holds p′_{1} is continuous. Let given p_{2}. Suppose that for every p_{1} holds p_{2}(p_{1}) = ∑(((\frac{1}{2})^{κ})_{κ∈ℕ} (F_{2}#p_{1})). Then p_{2} is a pseudometric of and for every p′_{1} such that p_{2} = p′_{1} holds p′_{1} is continuous.

(16) Let given p_{2}. Suppose p_{2} is a pseudometric of and for every p′_{1} such that p_{2} = p′_{1} holds p′_{1} is continuous. Let A be a non empty subset of T and given p. If p ∈ \overline{A}, then (inf(p_{2}, A))(p) = 0.

(17) Let given T. Suppose T is a T_{1} space. Let given s, F_{2}. Suppose that

(i) for every n there exists p_{2} such that F_{2}(n) = p_{2} and p_{2} is a pseudometric of and for every p_{1} holds p_{2}(p_{1}) s and for every p′_{1} such that p_{2} = p′_{1} holds p′_{1} is continuous, and

(ii) for all p, A′ such that p ∉ A′ and A′ is closed there exists n such that for every p_{2} such that F_{2}(n) = p_{2} holds (inf(p_{2}, A′))(p) > 0. Then there exists p_{2} such that p_{2} is a metric of the carrier of T and for every p_{1} holds p_{2}(p_{1}) = ∑(((\frac{1}{2})^{κ})_{κ∈ℕ} (F_{2}#p_{1})) and T is metrizable.

(18) Let D be a non empty set, p, q be finite sequences of elements of D, and

B be a binary operation on D. Suppose that

(i) p is one-to-one, (ii) q is one-to-one, (iii) rngq ⊆ rngp, (iv) B is commutative and associative, and

(v) B has a unity or lenq 1 and lenp > lenq. Then there exists a finite sequence r of elements of D such that r is oneto-one and rngr = rngp \ rngq and B ⊙ p = B(B ⊙ q, B ⊙ r).

(19) Let given T. Then T is a T_{3} space and a T_{1} space and there exists a family sequence of T which is Basis-sigma-locally finite if and only if T is metrizable.

388 karol pa̧k

(20) Suppose T is metrizable. Let F_{3} be a family of subsets of T. Suppose F_{3} is a cover of T and open. Then there exists a family sequence U_{1} of T such that ⋃ U_{1} is open and ⋃ U_{1} is a cover of T and ⋃ U_{1} is finer than F_{3} and U_{1} is sigma-discrete.

(21) For every T such that T is metrizable holds there exists a family sequence of T which is Basis-sigma-discrete.

(22) For every T holds T is a T_{3} space and a T_{1} space and there exists a family sequence of T which is Basis-sigma-discrete iff T is metrizable.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Grzegorz Bancerek. Countable sets and Hessenberg’s theorem. Formalized Mathematics, 2(1):65–69, 1991.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[5] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.

[6] Józef Białas and Yatsuka Nakamura. Dyadic numbers and T_{4} topological spaces. Formalized Mathematics, 5(3):361–366, 1996.

[7] Józef Białas and Yatsuka Nakamura. The Urysohn lemma. Formalized Mathematics, 9(3):631–636, 2001.

[8] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

[9] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[10] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[11] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[13] Czesław Byliński and Piotr Rudnicki. Bounding boxes for compact sets in E^{2}. Formalized Mathematics, 6(3):427–440, 1997.

[14] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 1990.

[15] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[16] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605–608, 1991.

[17] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[18] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.

[19] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

[20] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[21] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[22] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[23] Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233–236, 1996.

[24] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[25] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[26] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

the nagata-smirnov theorem. part ii 389

[27] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

[28] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[29] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

[30] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[31] Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1(5):979–981, 1990.

[32] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[33] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[34] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

On the Isomorphism of Fundamental Groups^{1}

Artur Korniłowicz University of Bialystok

MML Identifier: TOPALG 3.

The terminology and notation used here have been introduced in the following articles: [24], [7], [27], [28], [22], [4], [29], [5], [2], [18], [23], [3], [6], [21], [19], [26], [25], [9], [8], [20], [16], [11], [10], [1], [13], [14], [12], [15], and [17].

1. Preliminaries

One can prove the following propositions:

(1) Let A, B, a, b be sets and f be a function from A into B. If a ∈ A and b ∈ B, then f+·(a⟼→ . b) is a function from A into B.

(2) For every function f and for all sets X, x such that f↾X is one-to-one and x ∈ rng(f↾X) holds (f · (f↾X)^{−1} )(x) = x.

(3) Let x, y, X, Y, Z be sets, f be a function from [: X, Y :] into Z, and g be a function. If Z ≠ ∅ and x ∈ X and y ∈ Y, then (g · f)(x, y) = g(f(x, y)).

(4) For all sets X, a, b and for every function f from X into {a, b} holds X = f^{−1} ({a}) ∪ f^{−1} ({b}).

(5) For all non empty 1-sorted structures S, T and for every point s of S and for every point t of T holds (S ⟼→ t)(s) = t.

(6) Let T be a non empty topological structure, t be a point of T, and A be a subset of T. If A = {t}, then Sspace(t) = T ↾A.

1The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan. This work has been partially supported by KBN grant 4 T11C 039 24.

(7) Let T be a topological space, A, B be subsets of T, and C, D be subsets of the topological structure of T. Suppose A = C and B = D. Then A and B are separated if and only if C and D are separated.

(8) For every non empty topological space T holds T is connected iff there exists no map from T into {{0, 1}}_{top} which is continuous and onto.

One can verify that every topological structure which is empty is also connected.

We now state the proposition

(9) For every topological space T such that the topological structure of T is connected holds T is connected.

Let T be a connected topological space. One can check that the topological structure of T is connected.

One can prove the following proposition

(10) Let S, T be non empty topological spaces. Suppose S and T are home-

omorphic and S is arcwise connected. Then T is arcwise connected. One can verify that every non empty topological space which is trivial is also

arcwise connected.

One can prove the following propositions:

(11) For every subspace T of E_{T}^{2} such that the carrier of T is a simple closed curve holds T is arcwise connected.

(12) Let T be a topological space. Then there exists a family F of subsets of

T such that F = {the carrier of T } and F is a cover of T and open. Let T be a topological space. Note that there exists a family of subsets of T

which is non empty, mutually-disjoint, open, and closed.

The following proposition is true

(13) Let T be a topological space, D be a mutually-disjoint open family of subsets of T, A be a subset of T, and X be a set. If A is connected and X ∈ D and X meets A and D is a cover of A, then A ⊆ X.

2. On the Product of Topologies

One can prove the following three propositions:

(14) Let S, T be topological spaces. Then the topological structure of [: S, T :] = [: the topological structure of S, the topological structure of T :].

(15) For all topological spaces S, T and for every subset A of S and for every subset B of T holds \overline{[: A, B:]} = [: \overline{A}, \overline{B}:].

(16) Let S, T be topological spaces, A be a closed subset of S, and B be a closed subset of T. Then [: A, B:] is closed.

Let A, B be connected topological spaces. One can check that [: A, B:] is connected.

One can prove the following propositions:

(17) Let S, T be topological spaces, A be a subset of S, and B be a subset of T. If A is connected and B is connected, then [: A, B:] is connected.

(18) Let S, T be topological spaces, Y be a non empty topological space, A be a subset of S, f be a map from [: S, T :] into Y, and g be a map from [: S↾A, T :] into Y. If g = f↾[: A, the carrier of T :] and f is continuous, then g is continuous.

(19) Let S, T be topological spaces, Y be a non empty topological space, A be a subset of T, f be a map from [: S, T :] into Y, and g be a map from [: S, T ↾A:] into Y. If g = f↾[: the carrier of S, A:] and f is continuous, then g is continuous.

(20) Let S, T, T_{1}, T_{2}, Y be non empty topological spaces, f be a map from [: Y, T_{1} :] into S, g be a map from [: Y, T_{2} :] into S, and F_{1}, F_{2} be closed subsets of T. Suppose that T_{1} is a subspace of T and T_{2} is a subspace of T and F_{1} = Ω_{(T_{1})} and F_{2} = Ω_{(T_{2})} and Ω_{(T_{1})} ∪Ω_{(T_{2})} = Ω_{T} and f is continuous and g is continuous and for every set p such that p ∈ Ω_{[: Y, T_{1} :]} ∩ Ω_{[: Y, T_{2} :]} holds f(p) = g(p). Then there exists a map h from [: Y, T :] into S such that h = f+·g and h is continuous.

(21) Let S, T, T_{1}, T_{2}, Y be non empty topological spaces, f be a map from [: T_{1}, Y :] into S, g be a map from [: T_{2}, Y :] into S, and F_{1}, F_{2} be closed subsets of T. Suppose that T_{1} is a subspace of T and T_{2} is a subspace of T and F_{1} = Ω_{(T_{1})} and F_{2} = Ω_{(T_{2})} and Ω_{(T_{1})} ∪Ω_{(T_{2})} = Ω_{T} and f is continuous and g is continuous and for every set p such that p ∈ Ω_{[: T_{1}, Y :]} ∩ Ω_{[: T_{2}, Y :]} holds f(p) = g(p). Then there exists a map h from [: T, Y :] into S such that h = f+·g and h is continuous.

3. On the Fundamental Groups

Let T be a non empty topological space and let t be a point of T. Observe that every loop of t is continuous.

We now state a number of propositions:

(22) Let T be a non empty topological space, t be a point of T, x be a point of 𝕀, and P be a constant loop of t. Then P (x) = t.

(23) For every non empty topological space T and for every point t of T and for every loop P of t holds P (0) = t and P (1) = t.

(24) Let S, T be non empty topological spaces, f be a continuous map from S into T, and a, b be points of S. If a, b are connected, then f(a), f(b) are connected.

(25) Let S, T be non empty topological spaces, f be a continuous map from S into T, a, b be points of S, and P be a path from a to b. If a, b are connected, then f · P is a path from f(a) to f(b).

(26) Let S be a non empty arcwise connected topological space, T be a non empty topological space, f be a continuous map from S into T, a, b be points of S, and P be a path from a to b. Then f · P is a path from f(a) to f(b).

(27) Let S, T be non empty topological spaces, f be a continuous map from S into T, a be a point of S, and P be a loop of a. Then f · P is a loop of f(a).

(28) Let S, T be non empty topological spaces, f be a continuous map from S into T, a, b be points of S, P, Q be paths from a to b, and P_{1}, Q_{1} be paths from f(a) to f(b). Suppose P, Q are homotopic and P_{1} = f · P and Q_{1} = f · Q. Then P_{1}, Q_{1} are homotopic.

(29) Let S, T be non empty topological spaces, f be a continuous map from S into T, a, b be points of S, P, Q be paths from a to b, P_{1}, Q_{1} be paths from f(a) to f(b), and F be a homotopy between P and Q. Suppose P, Q are homotopic and P_{1} = f · P and Q_{1} = f · Q. Then f · F is a homotopy between P_{1} and Q_{1}.

(30) Let S, T be non empty topological spaces, f be a continuous map from S into T, a, b, c be points of S, P be a path from a to b, Q be a path from b to c, P_{1} be a path from f(a) to f(b), and Q_{1} be a path from f(b) to f(c). Suppose a, b are connected and b, c are connected and P_{1} = f · P and Q_{1} = f · Q. Then P_{1} + Q_{1} = f · (P + Q).

(31) Let S be a non empty topological space, s be a point of S, x, y be elements of π_{1}(S, s), and P, Q be loops of s. If x = [P]_{EqRel(S,s)} and y = [Q]_{EqRel(S,s)}, then x · y = [P + Q]_{EqRel(S,s)}.

Let S, T be non empty topological spaces, let s be a point of S, and let f be a map from S into T. Let us assume that f is continuous. The functor FundGrIso(f, s) yielding a map from π_{1}(S, s) into π_{1}(T, f(s)) is defined by the condition (Def. 1).

(Def. 1) Let x be an element of π_{1}(S, s). Then there exists a loop l_{1} of s and there exists a loop l_{2} of f(s) such that x = [l_{1}]_{EqRel(S,s)} and l_{2} = f · l_{1} and (FundGrIso(f, s))(x) = [l_{2}]_{EqRel(T,f(s))}.

The following proposition is true

(32) Let S, T be non empty topological spaces, s be a point of S, f be a continuous map from S into T, x be an element of π_{1}(S, s), l_{1} be a loop of s, and l_{2} be a loop of f(s). If x = [l_{1}]_{EqRel(S,s)} and l_{2} = f · l_{1}, then (FundGrIso(f, s))(x) = [l_{2}]_{EqRel(T,f(s))}.

Let S, T be non empty topological spaces, let s be a point of S, and let f

on the isomorphism of fundamental groups 395

be a continuous map from S into T. Then FundGrIso(f, s) is a homomorphism from π_{1}(S, s) to π_{1}(T, f(s)).

We now state three propositions:

(33) Let S, T be non empty topological spaces, s be a point of S, and f be a continuous map from S into T. If f is a homeomorphism, then FundGrIso(f, s) is an isomorphism.

(34) Let S, T be non empty topological spaces, s be a point of S, t be a point of T, f be a continuous map from S into T, P be a path from t to f(s), and h be a homomorphism from π_{1}(S, s) to π_{1}(T, t). Suppose f is a homeomorphism and f(s), t are connected and h = π_{1}-iso(P) · FundGrIso(f, s). Then h is an isomorphism.

(35) Let S be a non empty topological space, T be a non empty arcwise connected topological space, s be a point of S, and t be a point of T. If S and T are homeomorphic, then π_{1}(S, s) and π_{1}(T, t) are isomorphic.

References

[1] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

[2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[3] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[8] Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383–386, 1990.

[9] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257–261, 1990.

[10] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.

[11] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Simple closed curves. Formalized Mathematics, 2(5):663–664, 1991.

[12] Mariusz Giero. Hierarchies and classifications of sets. Formalized Mathematics, 9(4):865– 869, 2001.

[13] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449–454, 1997.

[14] Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55–59, 1999.

[15] Adam Grabowski and Artur Korniłowicz. Algebraic properties of homotopies. Formalized Mathematics, 12(3):251–260, 2004.

[16] Zbigniew Karno. Maximal discrete subspaces of almost discrete topological spaces. Formalized Mathematics, 4(1):125–135, 1993.

[17] Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261–268, 2004.

[18] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[19] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[20] Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239–244, 1990.

396 artur korniłowicz

[21] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[22] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[23] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[24] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[25] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

[26] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.

[27] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[28] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[29] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

Algebra of Complex Vector Valued Functions

Noboru Endou Gifu National College of Technology

Summary. This article is an extension of [17].

MML Identifier: VFUNCT 2.

The notation and terminology used here have been introduced in the following papers: [12], [15], [2], [11], [4], [16], [5], [7], [14], [9], [8], [3], [1], [13], [10], and [6].

For simplicity, we follow the rules: M denotes a non empty set, V denotes a complex normed space, f, f_{1}, f_{2}, f_{3} denote partial functions from M to the carrier of V, and z, z_{1}, z_{2} denote complex numbers.

Let M be a non empty set, let V be a complex normed space, and let f_{1}, f_{2} be partial functions from M to the carrier of V. The functor f_{1} + f_{2} yields a partial function from M to the carrier of V and is defined by:

(Def. 1) dom(f_{1} + f_{2}) = domf_{1} ∩ domf_{2} and for every element c of M such that

c ∈ dom(f_{1} + f_{2}) holds (f_{1} + f_{2})_{c} = (f_{1})_{c} + (f_{2})_{c}. The functor f_{1} − f_{2} yields a partial function from M to the carrier of V and is defined as follows:

(Def. 2) dom(f_{1} − f_{2}) = domf_{1} ∩ domf_{2} and for every element c of M such that

c ∈ dom(f_{1} − f_{2}) holds (f_{1} − f_{2})_{c} = (f_{1})_{c} − (f_{2})_{c}. Let M be a non empty set, let V be a complex normed space, let f_{1} be a

partial function from M to ℂ, and let f_{2} be a partial function from M to the carrier of V. The functor f_{1} f_{2} yielding a partial function from M to the carrier of V is defined by:

(Def. 3) dom(f_{1} f_{2}) = domf_{1} ∩ domf_{2} and for every element c of M such that

c ∈ dom(f_{1} f_{2}) holds (f_{1} f_{2})_{c} = (f_{1})_{c} · (f_{2})_{c}. Let X be a non empty set, let V be a complex normed space, let f be a

partial function from X to the carrier of V, and let z be a complex number. The functor z f yields a partial function from X to the carrier of V and is defined as follows:

(Def. 4) dom(z f) = domf and for every element x of X such that x ∈ dom(z f)

holds (z f)_{x} = z · f_{x}. Let X be a non empty set, let V be a complex normed space, and let f be a

partial function from X to the carrier of V. The functor ∥f∥ yielding a partial function from X to ℝ is defined as follows:

(Def. 5) dom∥f∥ = domf and for every element x of X such that x ∈ dom∥f∥

holds ∥f∥(x) = ∥f_{x}∥. The functor − f yields a partial function from X to the carrier of V and is defined by:

(Def. 6) dom(− f) = domf and for every element x of X such that x ∈ dom(− f)

holds (− f)_{x} = − f_{x}. The following propositions are true:

(1) Let f_{1} be a partial function from M to ℂ and f_{2} be a partial function from M to the carrier of V. Then dom(f_{1} f_{2})\(f_{1} f_{2})^{−1} ({0_{V} }) = (domf_{1} \ f_{1}^{−1} ({0})) ∩ (domf_{2} \ f_{2}^{−1} ({0_{V} })).

(2) ∥f∥^{−1} ({0}) = f^{−1} ({0_{V} }) and (− f)^{−1} ({0_{V} }) = f^{−1} ({0_{V} }).

(3) If z ≠ 0_{ℂ}, then (z f)^{−1} ({0_{V} }) = f^{−1} ({0_{V} }).

(4) f_{1} + f_{2} = f_{2} + f_{1}.

(5) (f_{1} + f_{2}) + f_{3} = f_{1} + (f_{2} + f_{3}).

(6) Let f_{1}, f_{2} be partial functions from M to ℂ and f_{3} be a partial function from M to the carrier of V. Then (f_{1} f_{2}) f_{3} = f_{1} (f_{2} f_{3}).

(7) For all partial functions f_{1}, f_{2} from M to ℂ holds (f_{1} + f_{2}) f_{3} = f_{1} f_{3} + f_{2} f_{3}.

(8) For every partial function f_{3} from M to ℂ holds f_{3} (f_{1} + f_{2}) = f_{3} f_{1} + f_{3} f_{2}.

(9) For every partial function f_{1} from M to ℂ holds z (f_{1} f_{2}) = (z f_{1}) f_{2}.

(10) For every partial function f_{1} from M to ℂ holds z (f_{1} f_{2}) = f_{1} (z f_{2}).

(11) For all partial functions f_{1}, f_{2} from M to ℂ holds (f_{1} − f_{2}) f_{3} = f_{1} f_{3} − f_{2} f_{3}.

(12) For every partial function f_{3} from M to ℂ holds f_{3} f_{1} − f_{3} f_{2} = f_{3} (f_{1} − f_{2}).

(13) z (f_{1} + f_{2}) = z f_{1} + z f_{2}.

(14) (z_{1} · z_{2}) f = z_{1} (z_{2} f).

(15) z (f_{1} − f_{2}) = z f_{1} − z f_{2}.

(16) f_{1} − f_{2} = (− 1_{ℂ}) (f_{2} − f_{1}).

(17) f_{1} − (f_{2} + f_{3}) = f_{1} − f_{2} − f_{3}.

(18) 1_{ℂ} f = f.

(19) f_{1} − (f_{2} − f_{3}) = (f_{1} − f_{2}) + f_{3}.

(20) f_{1} + (f_{2} − f_{3}) = (f_{1} + f_{2}) − f_{3}.

(21) For every partial function f_{1} from M to ℂ holds ∥f_{1} f_{2}∥ = |f_{1}| ∥f_{2}∥.

(22) ∥z f∥ = |z| ∥f∥.

(23) − f = (− 1_{ℂ}) f.

(24) − − f = f.

(25) f_{1} − f_{2} = f_{1} + − f_{2}.

(26) f_{1} − − f_{2} = f_{1} + f_{2}.

In the sequel X, Y denote sets.

We now state a number of propositions:

(27) (f_{1}+f_{2})↾X = f_{1}↾X+f_{2}↾X and (f_{1}+f_{2})↾X = f_{1}↾X+f_{2} and (f_{1}+f_{2})↾X = f_{1} + f_{2}↾X.

(28) For every partial function f_{1} from M to ℂ holds (f_{1} f_{2})↾X = (f_{1}↾X) (f_{2}↾X) and (f_{1} f_{2})↾X = (f_{1}↾X) f_{2} and (f_{1} f_{2})↾X = f_{1} (f_{2}↾X).

(29) (− f)↾X = − f↾X and ∥f∥↾X = ∥f↾X∥.

(30) (f_{1}−f_{2})↾X = f_{1}↾X−f_{2}↾X and (f_{1}−f_{2})↾X = f_{1}↾X−f_{2} and (f_{1}−f_{2})↾X = f_{1} − f_{2}↾X.

(31) (z f)↾X = z (f↾X).

(32) f_{1} is total and f_{2} is total iff f_{1} + f_{2} is total and f_{1} is total and f_{2} is total iff f_{1} − f_{2} is total.

(33) For every partial function f_{1} from M to ℂ holds f_{1} is total and f_{2} is total iff f_{1} f_{2} is total.

(34) f is total iff z f is total.

(35) f is total iff − f is total.

(36) f is total iff ∥f∥ is total.

(37) For every element x of M such that f_{1} is total and f_{2} is total holds (f_{1} + f_{2})_{x} = (f_{1})_{x} + (f_{2})_{x} and (f_{1} − f_{2})_{x} = (f_{1})_{x} − (f_{2})_{x}.

(38) Let f_{1} be a partial function from M to ℂ and x be an element of M. If f_{1} is total and f_{2} is total, then (f_{1} f_{2})_{x} = (f_{1})_{x} · (f_{2})_{x}.

(39) For every element x of M such that f is total holds (z f)_{x} = z · f_{x}.

(40) For every element x of M such that f is total holds (− f)_{x} = − f_{x} and

∥f∥(x) = ∥f_{x}∥. Let us consider M, let us consider V, and let us consider f, Y. We say that

f is bounded on Y if and only if: (Def. 7) There exists a real number r such that for every element x of M such

that x ∈ Y ∩ domf holds ∥f_{x}∥ r. One can prove the following propositions:

(41) If Y ⊆ X and f is bounded on X, then f is bounded on Y.

(42) If X misses domf, then f is bounded on X.

(43) 0_{ℂ} f is bounded on Y.

(44) If f is bounded on Y, then z f is bounded on Y.

(45) If f is bounded on Y, then ∥f∥ is bounded on Y and − f is bounded on Y.

(46) If f_{1} is bounded on X and f_{2} is bounded on Y, then f_{1} + f_{2} is bounded on X ∩ Y.

(47) For every partial function f_{1} from M to ℂ such that f_{1} is bounded on X and f_{2} is bounded on Y holds f_{1} f_{2} is bounded on X ∩ Y.

(48) If f_{1} is bounded on X and f_{2} is bounded on Y, then f_{1} − f_{2} is bounded on X ∩ Y.

(49) If f is bounded on X and bounded on Y, then f is bounded on X ∪ Y.

(50) If f_{1} is a constant on X and f_{2} is a constant on Y, then f_{1} + f_{2} is a constant on X ∩ Y and f_{1} − f_{2} is a constant on X ∩ Y.

(51) Let f_{1} be a partial function from M to ℂ. Suppose f_{1} is a constant on X and f_{2} is a constant on Y. Then f_{1} f_{2} is a constant on X ∩ Y.

(52) If f is a constant on Y, then z f is a constant on Y.

(53) If f is a constant on Y, then ∥f∥ is a constant on Y and − f is a constant on Y.

(54) If f is a constant on Y, then f is bounded on Y.

(55) If f is a constant on Y, then for every z holds z f is bounded on Y and − f is bounded on Y and ∥f∥ is bounded on Y.

(56) If f_{1} is bounded on X and f_{2} is a constant on Y, then f_{1} +f_{2} is bounded on X ∩ Y.

(57) If f_{1} is bounded on X and f_{2} is a constant on Y, then f_{1} −f_{2} is bounded on X ∩ Y and f_{2} − f_{1} is bounded on X ∩ Y.

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121–124, 1993.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[6] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

[7] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[8] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

algebra of complex vector valued functions 401

[9] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[10] Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex functions. Formalized Mathematics, 9(1):179–184, 2001.

[11] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[12] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[13] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[14] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[15] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[16] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[17] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171–175, 1992.

Continuous Functions on Real and Complex Normed Linear Spaces

Noboru Endou Gifu National College of Technology

Summary. This article is an extension of [18].

MML Identifier: NCFCONT1.

The notation and terminology used here are introduced in the following papers: [25], [28], [29], [4], [30], [6], [14], [5], [2], [24], [10], [26], [27], [19], [15], [12], [13], [11], [31], [20], [3], [1], [16], [21], [17], [23], [7], [8], [22], [18], and [9].

For simplicity, we use the following convention: n denotes a natural number, r, s denote real numbers, z denotes a complex number, C_{1}, C_{2}, C_{3} denote complex normed spaces, and R_{1} denotes a real normed space.

Let C_{4} be a complex linear space and let s_{1} be a sequence of C_{4}. The functor − s_{1} yields a sequence of C_{4} and is defined by:

(Def. 1) For every n holds (− s_{1})(n) = − s_{1}(n).

The following propositions are true:

(1) For all sequences s_{2}, s_{3} of C_{1} holds s_{2} − s_{3} = s_{2} + − s_{3}.

(2) For every sequence s_{1} of C_{1} holds − s_{1} = (− 1_{ℂ}) · s_{1}.

Let us consider C_{2}, C_{3} and let f be a partial function from C_{2} to C_{3}. The functor ∥f∥ yielding a partial function from the carrier of C_{2} to ℝ is defined by:

(Def. 2) dom∥f∥ = domf and for every point c of C_{2} such that c ∈ dom∥f∥

holds ∥f∥(c) = ∥f_{c}∥. Let us consider C_{1}, R_{1} and let f be a partial function from C_{1} to R_{1}. The

functor ∥f∥ yielding a partial function from the carrier of C_{1} to ℝ is defined as follows:

(Def. 3) dom∥f∥ = domf and for every point c of C_{1} such that c ∈ dom∥f∥ holds ∥f∥(c) = ∥f_{c}∥.

Let us consider R_{1}, C_{1} and let f be a partial function from R_{1} to C_{1}. The functor ∥f∥ yielding a partial function from the carrier of R_{1} to ℝ is defined by:

(Def. 4) dom∥f∥ = domf and for every point c of R_{1} such that c ∈ dom∥f∥

holds ∥f∥(c) = ∥f_{c}∥. Let us consider C_{1} and let x_{0} be a point of C_{1}. A subset of C_{1} is called a

neighbourhood of x_{0} if: (Def. 5) There exists a real number g such that 0 < g and {y; y ranges over points

of C_{1}: ∥y − x_{0}∥ < g} ⊆ it. Next we state two propositions:

(3) Let x_{0} be a point of C_{1} and g be a real number. If 0 < g, then {y; y ranges over points of C_{1}: ∥y − x_{0}∥ < g} is a neighbourhood of x_{0}.

(4) For every point x_{0} of C_{1} and for every neighbourhood N of x_{0} holds x_{0} ∈ N.

Let us consider C_{1} and let X be a subset of C_{1}. We say that X is compact if and only if the condition (Def. 6) is satisfied.

(Def. 6) Let s_{4} be a sequence of C_{1}. Suppose rngs_{4} ⊆ X. Then there exists a sequence s_{5} of C_{1} such that s_{5} is a subsequence of s_{4} and convergent and lims_{5} ∈ X.

Let us consider C_{1} and let X be a subset of C_{1}. We say that X is closed if and only if:

(Def. 7) For every sequence s_{4} of C_{1} such that rngs_{4} ⊆ X and s_{4} is convergent

holds lims_{4} ∈ X. Let us consider C_{1} and let X be a subset of C_{1}. We say that X is open if

and only if: (Def. 8) X^{c} is closed.

Let us consider C_{2}, C_{3}, let f be a partial function from C_{2} to C_{3}, and let s_{1} be a sequence of C_{2}. Let us assume that rngs_{1} ⊆ domf. The functor f · s_{1} yields a sequence of C_{3} and is defined by:

(Def. 9) f · s_{1} = (f qua function) ·(s_{1}).

Let us consider C_{1}, R_{1}, let f be a partial function from C_{1} to R_{1}, and let s_{1} be a sequence of C_{1}. Let us assume that rngs_{1} ⊆ domf. The functor f · s_{1} yielding a sequence of R_{1} is defined by:

(Def. 10) f · s_{1} = (f qua function) ·(s_{1}).

Let us consider C_{1}, R_{1}, let f be a partial function from R_{1} to C_{1}, and let s_{1} be a sequence of R_{1}. Let us assume that rngs_{1} ⊆ domf. The functor f · s_{1} yields a sequence of C_{1} and is defined by:

(Def. 11) f · s_{1} = (f qua function) ·(s_{1}).

Let us consider C_{1}, let f be a partial function from the carrier of C_{1} to ℂ, and let s_{1} be a sequence of C_{1}. Let us assume that rngs_{1} ⊆ domf. The functor

f · s_{1} yields a complex sequence and is defined as follows: (Def. 12) f · s_{1} = (f qua function) ·(s_{1}).

Let us consider R_{1}, let f be a partial function from the carrier of R_{1} to ℂ, and let s_{1} be a sequence of R_{1}. Let us assume that rngs_{1} ⊆ domf. The functor f · s_{1} yielding a complex sequence is defined by:

(Def. 13) f · s_{1} = (f qua function) ·(s_{1}).

Let us consider C_{1}, let f be a partial function from the carrier of C_{1} to ℝ, and let s_{1} be a sequence of C_{1}. Let us assume that rngs_{1} ⊆ domf. The functor f · s_{1} yielding a sequence of real numbers is defined as follows:

(Def. 14) f · s_{1} = (f qua function) ·(s_{1}).

Let us consider C_{2}, C_{3}, let f be a partial function from C_{2} to C_{3}, and let x_{0} be a point of C_{2}. We say that f is continuous in x_{0} if and only if the conditions (Def. 15) are satisfied.

(Def. 15)(i) x_{0} ∈ domf, and

(ii) for every sequence s_{1} of C_{2} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}). Let us consider C_{1}, R_{1}, let f be a partial function from C_{1} to R_{1}, and let x_{0}

be a point of C_{1}. We say that f is continuous in x_{0} if and only if the conditions (Def. 16) are satisfied.

(Def. 16)(i) x_{0} ∈ domf, and

(ii) for every sequence s_{1} of C_{1} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}). Let us consider R_{1}, let us consider C_{1}, let f be a partial function from R_{1}

to C_{1}, and let x_{0} be a point of R_{1}. We say that f is continuous in x_{0} if and only if the conditions (Def. 17) are satisfied.

(Def. 17)(i) x_{0} ∈ domf, and

(ii) for every sequence s_{1} of R_{1} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}). Let us consider C_{1}, let f be a partial function from the carrier of C_{1} to ℂ,

and let x_{0} be a point of C_{1}. We say that f is continuous in x_{0} if and only if the conditions (Def. 18) are satisfied.

(Def. 18)(i) x_{0} ∈ domf, and

(ii) for every sequence s_{1} of C_{1} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}). Let us consider C_{1}, let f be a partial function from the carrier of C_{1} to ℝ,

and let x_{0} be a point of C_{1}. We say that f is continuous in x_{0} if and only if the conditions (Def. 19) are satisfied.

(Def. 19)(i) x_{0} ∈ domf, and

(ii) for every sequence s_{1} of C_{1} such that rngs_{1} ⊆ domf and s_{1} is convergent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}).

Let us consider R_{1}, let f be a partial function from the carrier of R_{1} to ℂ, and let x_{0} be a point of R_{1}. We say that f is continuous in x_{0} if and only if the conditions (Def. 20) are satisfied.

(Def. 20)(i) x_{0} ∈ domf, and

(ii) for every sequence s_{1} of R_{1} such that rngs_{1} ⊆ domf and s_{1} is conver-

gent and lims_{1} = x_{0} holds f · s_{1} is convergent and f_{x_{0}} = lim(f · s_{1}). The following propositions are true:

(5) For every sequence s_{1} of C_{2} and for every partial function h from C_{2} to C_{3} such that rngs_{1} ⊆ domh holds s_{1}(n) ∈ domh.

(6) For every sequence s_{1} of C_{1} and for every partial function h from C_{1} to R_{1} such that rngs_{1} ⊆ domh holds s_{1}(n) ∈ domh.

(7) For every sequence s_{1} of R_{1} and for every partial function h from R_{1} to C_{1} such that rngs_{1} ⊆ domh holds s_{1}(n) ∈ domh.

(8) For every sequence s_{1} of C_{1} and for every set x holds x ∈ rngs_{1} iff there exists n such that x = s_{1}(n).

(9) For all sequences s_{1}, s_{2} of C_{1} such that s_{2} is a subsequence of s_{1} holds rngs_{2} ⊆ rngs_{1}.

(10) Let f be a partial function from C_{2} to C_{3} and C_{5} be a sequence of C_{2}. If rngC_{5} ⊆ domf, then for every n holds (f · C_{5})(n) = f_{C_{5}(n)}.

(11) Let f be a partial function from C_{1} to R_{1} and C_{5} be a sequence of C_{1}. If rngC_{5} ⊆ domf, then for every n holds (f · C_{5})(n) = f_{C_{5}(n)}.

(12) Let f be a partial function from R_{1} to C_{1} and R_{2} be a sequence of R_{1}. If rngR_{2} ⊆ domf, then for every n holds (f · R_{2})(n) = f_{R_{2}(n)}.

(13) Let f be a partial function from the carrier of C_{1} to ℂ and C_{5} be a sequence of C_{1}. If rngC_{5} ⊆ domf, then for every n holds (f · C_{5})(n) = f_{C_{5}(n)}.

(14) Let f be a partial function from the carrier of C_{1} to ℝ and C_{5} be a sequence of C_{1}. If rngC_{5} ⊆ domf, then for every n holds (f · C_{5})(n) = f_{C_{5}(n)}.

(15) Let f be a partial function from the carrier of R_{1} to ℂ and R_{2} be a sequence of R_{1}. If rngR_{2} ⊆ domf, then for every n holds (f · R_{2})(n) = f_{R_{2}(n)}.

(16) Let h be a partial function from C_{2} to C_{3}, C_{5} be a sequence of C_{2}, and N_{1} be an increasing sequence of naturals. If rngC_{5} ⊆ domh, then (h · C_{5}) · N_{1} = h · (C_{5} · N_{1}).

(17) Let h be a partial function from C_{1} to R_{1}, C_{6} be a sequence of C_{1}, and N_{1} be an increasing sequence of naturals. If rngC_{6} ⊆ domh, then (h · C_{6}) · N_{1} = h · (C_{6} · N_{1}).

(18) Let h be a partial function from R_{1} to C_{1}, R_{3} be a sequence of R_{1}, and N_{1} be an increasing sequence of naturals. If rngR_{3} ⊆ domh, then (h · R_{3}) · N_{1} = h · (R_{3} · N_{1}).

(19) Let h be a partial function from the carrier of C_{1} to ℂ, C_{6} be a sequence of C_{1}, and N_{1} be an increasing sequence of naturals. If rngC_{6} ⊆ domh, then (h · C_{6}) · N_{1} = h · (C_{6} · N_{1}).

(20) Let h be a partial function from the carrier of C_{1} to ℝ, C_{6} be a sequence of C_{1}, and N_{1} be an increasing sequence of naturals. If rngC_{6} ⊆ domh, then (h · C_{6}) · N_{1} = h · (C_{6} · N_{1}).

(21) Let h be a partial function from the carrier of R_{1} to ℂ, R_{3} be a sequence of R_{1}, and N_{1} be an increasing sequence of naturals. If rngR_{3} ⊆ domh, then (h · R_{3}) · N_{1} = h · (R_{3} · N_{1}).

(22) Let h be a partial function from C_{2} to C_{3} and C_{7}, C_{8} be sequences of C_{2}. If rngC_{7} ⊆ domh and C_{8} is a subsequence of C_{7}, then h · C_{8} is a subsequence of h · C_{7}.

(23) Let h be a partial function from C_{1} to R_{1} and C_{7}, C_{8} be sequences of C_{1}. If rngC_{7} ⊆ domh and C_{8} is a subsequence of C_{7}, then h · C_{8} is a subsequence of h · C_{7}.

(24) Let h be a partial function from R_{1} to C_{1} and R_{4}, R_{5} be sequences of R_{1}. If rngR_{4} ⊆ domh and R_{5} is a subsequence of R_{4}, then h · R_{5} is a subsequence of h · R_{4}.

(25) Let s_{1} be a complex sequence, n be a natural number, and N_{2} be an increasing sequence of naturals. Then (s_{1} · N_{2})(n) = s_{1}(N_{2}(n)).

(26) Let h be a partial function from the carrier of C_{1} to ℂ and C_{7}, C_{8} be sequences of C_{1}. If rngC_{7} ⊆ domh and C_{8} is a subsequence of C_{7}, then h · C_{8} is a subsequence of h · C_{7}.

(27) Let h be a partial function from the carrier of C_{1} to ℝ and C_{7}, C_{8} be sequences of C_{1}. If rngC_{7} ⊆ domh and C_{8} is a subsequence of C_{7}, then h · C_{8} is a subsequence of h · C_{7}.

(28) Let h be a partial function from the carrier of R_{1} to ℂ and R_{4}, R_{5} be sequences of R_{1}. If rngR_{4} ⊆ domh and R_{5} is a subsequence of R_{4}, then h · R_{5} is a subsequence of h · R_{4}.

(29) Let f be a partial function from C_{2} to C_{3} and x_{0} be a point of C_{2}. Then

f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for

every point x_{1} of C_{2} such that x_{1} ∈ domf and ∥x_{1} − x_{0}∥ < s holds ∥f_{x_{1}} − f_{x_{0}}∥ < r.

(30) Let f be a partial function from C_{1} to R_{1} and x_{0} be a point of C_{1}. Then

f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for

every point x_{1} of C_{1} such that x_{1} ∈ domf and ∥x_{1} − x_{0}∥ < s holds ∥f_{x_{1}} − f_{x_{0}}∥ < r.

(31) Let f be a partial function from R_{1} to C_{1} and x_{0} be a point of R_{1}. Then

f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for

every point x_{1} of R_{1} such that x_{1} ∈ domf and ∥x_{1} − x_{0}∥ < s holds ∥f_{x_{1}} − f_{x_{0}}∥ < r.

(32) Let f be a partial function from the carrier of C_{1} to ℝ and x_{0} be a point of C_{1}. Then f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for every

point x_{1} of C_{1} such that x_{1} ∈ domf and ∥x_{1}−x_{0}∥ < s holds |f_{x_{1}} −f_{x_{0}}| < r.

(33) Let f be a partial function from the carrier of C_{1} to ℂ and x_{0} be a point of C_{1}. Then f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for every

point x_{1} of C_{1} such that x_{1} ∈ domf and ∥x_{1}−x_{0}∥ < s holds |f_{x_{1}} −f_{x_{0}}| < r.

(34) Let f be a partial function from the carrier of R_{1} to ℂ and x_{0} be a point of R_{1}. Then f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every r such that 0 < r there exists s such that 0 < s and for every

point x_{1} of R_{1} such that x_{1} ∈ domf and ∥x_{1}−x_{0}∥ < s holds |f_{x_{1}}−f_{x_{0}}| < r.

(35) Let f be a partial function from C_{2} to C_{3} and x_{0} be a point of C_{2}. Then

f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every neighbourhood N_{3} of f_{x_{0}} there exists a neighbourhood N of

x_{0} such that for every point x_{1} of C_{2} such that x_{1} ∈ domf and x_{1} ∈ N holds f_{x_{1}} ∈ N_{3}.

(36) Let f be a partial function from C_{1} to R_{1} and x_{0} be a point of C_{1}. Then

f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every neighbourhood N_{3} of f_{x_{0}} there exists a neighbourhood N of

x_{0} such that for every point x_{1} of C_{1} such that x_{1} ∈ domf and x_{1} ∈ N holds f_{x_{1}} ∈ N_{3}.

(37) Let f be a partial function from R_{1} to C_{1} and x_{0} be a point of R_{1}. Then f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every neighbourhood N_{3} of f_{x_{0}} there exists a neighbourhood N of

x_{0} such that for every point x_{1} of R_{1} such that x_{1} ∈ domf and x_{1} ∈ N holds f_{x_{1}} ∈ N_{3}.

(38) Let f be a partial function from C_{2} to C_{3} and x_{0} be a point of C_{2}. Then

f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every neighbourhood N_{3} of f_{x_{0}} there exists a neighbourhood N of

x_{0} such that f^{∘}N ⊆ N_{3}.

(39) Let f be a partial function from C_{1} to R_{1} and x_{0} be a point of C_{1}. Then

f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every neighbourhood N_{3} of f_{x_{0}} there exists a neighbourhood N of

x_{0} such that f^{∘}N ⊆ N_{3}.

(40) Let f be a partial function from R_{1} to C_{1} and x_{0} be a point of R_{1}. Then

f is continuous in x_{0} if and only if the following conditions are satisfied:

(i) x_{0} ∈ domf, and (ii) for every neighbourhood N_{3} of f_{x_{0}} there exists a neighbourhood N of

x_{0} such that f^{∘}N ⊆ N_{3}.

(41) Let f be a partial function from C_{2} to C_{3} and x_{0} be a point of C_{2}. Suppose x_{0} ∈ domf and there exists a neighbourhood N of x_{0} such that domf ∩ N = {x_{0}}. Then f is continuous in x_{0}.

(42) Let f be a partial function from C_{1} to R_{1} and x_{0} be a point of C_{1}. Suppose x_{0} ∈ domf and there exists a neighbourhood N of x_{0} such that domf ∩ N = {x_{0}}. Then f is continuous in x_{0}.

(43) Let f be a partial function from R_{1} to C_{1} and x_{0} be a point of R_{1}. Suppose x_{0} ∈ domf and there exists a neighbourhood N of x_{0} such that domf ∩ N = {x_{0}}. Then f is continuous in x_{0}.

(44) Let h_{1}, h_{2} be partial functions from C_{2} to C_{3} and s_{1} be a sequence of C_{2}. If rngs_{1} ⊆ domh_{1} ∩ domh_{2}, then (h_{1} + h_{2}) · s_{1} = h_{1} · s_{1} + h_{2} · s_{1} and (h_{1} − h_{2}) · s_{1} = h_{1} · s_{1} − h_{2} · s_{1}.

(45) Let h_{1}, h_{2} be partial functions from C_{1} to R_{1} and s_{1} be a sequence of C_{1}. If rngs_{1} ⊆ domh_{1} ∩ domh_{2}, then (h_{1} + h_{2}) · s_{1} = h_{1} · s_{1} + h_{2} · s_{1} and (h_{1} − h_{2}) · s_{1} = h_{1} · s_{1} − h_{2} · s_{1}.

(46) Let h_{1}, h_{2} be partial functions from R_{1} to C_{1} and s_{1} be a sequence of R_{1}. If rngs_{1} ⊆ domh_{1} ∩ domh_{2}, then (h_{1} + h_{2}) · s_{1} = h_{1} · s_{1} + h_{2} · s_{1} and (h_{1} − h_{2}) · s_{1} = h_{1} · s_{1} − h_{2} · s_{1}.

(47) Let h be a partial function from C_{2} to C_{3}, s_{1} be a sequence of C_{2}, and z be a complex number. If rngs_{1} ⊆ domh, then (z h) · s_{1} = z · (h · s_{1}).

(48) Let h be a partial function from C_{1} to R_{1}, s_{1} be a sequence of C_{1}, and r be a real number. If rngs_{1} ⊆ domh, then (r h) · s_{1} = r · (h · s_{1}).

(49) Let h be a partial function from R_{1} to C_{1}, s_{1} be a sequence of R_{1}, and z be a complex number. If rngs_{1} ⊆ domh, then (z h) · s_{1} = z · (h · s_{1}).

(50) Let h be a partial function from C_{2} to C_{3} and s_{1} be a sequence of C_{2}. If rngs_{1} ⊆ domh, then ∥h · s_{1}∥ = ∥h∥ · s_{1} and − h · s_{1}= (− h) · s_{1}.

(51) Let h be a partial function from C_{1} to R_{1} and s_{1} be a sequence of C_{1}. If rngs_{1} ⊆ domh, then ∥h · s_{1}∥ = ∥h∥ · s_{1} and − h · s_{1}= (− h) · s_{1}.

(52) Let h be a partial function from R_{1} to C_{1} and s_{1} be a sequence of R_{1}. If rngs_{1} ⊆ domh, then ∥h · s_{1}∥ = ∥h∥ · s_{1} and − h · s_{1}= (− h) · s_{1}.

(53) Let f_{1}, f_{2} be partial functions from C_{2} to C_{3} and x_{0} be a point of C_{2}. Suppose f_{1} is continuous in x_{0} and f_{2} is continuous in x_{0}. Then f_{1} + f_{2} is continuous in x_{0} and f_{1} − f_{2} is continuous in x_{0}.

(54) Let f_{1}, f_{2} be partial functions from C_{1} to R_{1} and x_{0} be a point of C_{1}. Suppose f_{1} is continuous in x_{0} and f_{2} is continuous in x_{0}. Then f_{1} + f_{2} is continuous in x_{0} and f_{1} − f_{2} is continuous in x_{0}.

(55) Let f_{1}, f_{2} be partial functions from R_{1} to C_{1} and x_{0} be a point of R_{1}. Suppose f_{1} is continuous in x_{0} and f_{2} is continuous in x_{0}. Then f_{1} + f_{2} is continuous in x_{0} and f_{1} − f_{2} is continuous in x_{0}.

(56) Let f be a partial function from C_{2} to C_{3}, x_{0} be a point of C_{2}, and z be a complex number. If f is continuous in x_{0}, then z f is continuous in x_{0}.

(57) Let f be a partial function from C_{1} to R_{1}, x_{0} be a point of C_{1}, and r be a real number. If f is continuous in x_{0}, then r f is continuous in x_{0}.

(58) Let f be a partial function from R_{1} to C_{1}, x_{0} be a point of R_{1}, and z be a complex number. If f is continuous in x_{0}, then z f is continuous in x_{0}.

(59) Let f be a partial function from C_{2} to C_{3} and x_{0} be a point of C_{2}. If f is continuous in x_{0}, then ∥f∥ is continuous in x_{0} and − f is continuous in x_{0}.

(60) Let f be a partial function from C_{1} to R_{1} and x_{0} be a point of C_{1}. If f is continuous in x_{0}, then ∥f∥ is continuous in x_{0} and − f is continuous in x_{0}.

(61) Let f be a partial function from R_{1} to C_{1} and x_{0} be a point of R_{1}. If f is continuous in x_{0}, then ∥f∥ is continuous in x_{0} and − f is continuous in x_{0}.

Let C_{2}, C_{3} be complex normed spaces, let f be a partial function from C_{2} to C_{3}, and let X be a set. We say that f is continuous on X if and only if:

(Def. 21) X ⊆ domf and for every point x_{0} of C_{2} such that x_{0} ∈ X holds f↾X is

continuous in x_{0}. Let C_{1} be a complex normed space, let R_{1} be a real normed space, let f be a partial function from C_{1} to R_{1}, and let X be a set. We say that f is continuous on X if and only if:

(Def. 22) X ⊆ domf and for every point x_{0} of C_{1} such that x_{0} ∈ X holds f↾X is

continuous in x_{0}. Let R_{1} be a real normed space, let C_{1} be a complex normed space, let g be a

partial function from R_{1} to C_{1}, and let X be a set. We say that g is continuous on X if and only if:

(Def. 23) X ⊆ domg and for every point x_{0} of R_{1} such that x_{0} ∈ X holds g↾X is

continuous in x_{0}. Let C_{1} be a complex normed space, let f be a partial function from the

carrier of C_{1} to ℂ, and let X be a set. We say that f is continuous on X if and only if:

(Def. 24) X ⊆ domf and for every point x_{0} of C_{1} such that x_{0} ∈ X holds f↾X is

continuous in x_{0}. Let C_{1} be a complex normed space, let f be a partial function from the

carrier of C_{1} to ℝ, and let X be a set. We say that f is continuous on X if and only if:

(Def. 25) X ⊆ domf and for every point x_{0} of C_{1} such that x_{0} ∈ X holds f↾X is

continuous in x_{0}. Let R_{1} be a real normed space, let f be a partial function from the carrier

of R_{1} to ℂ, and let X be a set. We say that f is continuous on X if and only if: (Def. 26) X ⊆ domf and for every point x_{0} of R_{1} such that x_{0} ∈ X holds f↾X is

continuous in x_{0}. In the sequel X, X_{1} denote sets. The following propositions are true:

(62) Let f be a partial function from C_{2} to C_{3}. Then f is continuous on X if

and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every sequence s_{4} of C_{2} such that rngs_{4} ⊆ X and s_{4} is convergent

and lims_{4} ∈ X holds f · s_{4} is convergent and f_{lim s_{4}} = lim(f · s_{4}).

(63) Let f be a partial function from C_{1} to R_{1}. Then f is continuous on X if

and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every sequence s_{4} of C_{1} such that rngs_{4} ⊆ X and s_{4} is convergent

and lims_{4} ∈ X holds f · s_{4} is convergent and f_{lim s_{4}} = lim(f · s_{4}).

(64) Let f be a partial function from R_{1} to C_{1}. Then f is continuous on X if

and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every sequence s_{4} of R_{1} such that rngs_{4} ⊆ X and s_{4} is convergent

and lims_{4} ∈ X holds f · s_{4} is convergent and f_{lim s_{4}} = lim(f · s_{4}).

(65) Let f be a partial function from C_{2} to C_{3}. Then f is continuous on X if

and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every point x_{0} of C_{2} and for every r such that x_{0} ∈ X and 0 < r

there exists s such that 0 < s and for every point x_{1} of C_{2} such that x_{1} ∈ X and ∥x_{1} − x_{0}∥ < s holds ∥f_{x_{1}} − f_{x_{0}}∥ < r.

(66) Let f be a partial function from C_{1} to R_{1}. Then f is continuous on X if

and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every point x_{0} of C_{1} and for every r such that x_{0} ∈ X and 0 < r

there exists s such that 0 < s and for every point x_{1} of C_{1} such that x_{1} ∈ X and ∥x_{1} − x_{0}∥ < s holds ∥f_{x_{1}} − f_{x_{0}}∥ < r.

(67) Let f be a partial function from R_{1} to C_{1}. Then f is continuous on X if

and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every point x_{0} of R_{1} and for every r such that x_{0} ∈ X and 0 < r

there exists s such that 0 < s and for every point x_{1} of R_{1} such that x_{1} ∈ X and ∥x_{1} − x_{0}∥ < s holds ∥f_{x_{1}} − f_{x_{0}}∥ < r.

(68) Let f be a partial function from the carrier of C_{1} to ℂ. Then f is conti-

nuous on X if and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every point x_{0} of C_{1} and for every r such that x_{0} ∈ X and 0 < r

there exists s such that 0 < s and for every point x_{1} of C_{1} such that x_{1} ∈ X and ∥x_{1} − x_{0}∥ < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(69) Let f be a partial function from the carrier of C_{1} to ℝ. Then f is conti-

nuous on X if and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every point x_{0} of C_{1} and for every r such that x_{0} ∈ X and 0 < r

there exists s such that 0 < s and for every point x_{1} of C_{1} such that x_{1} ∈ X and ∥x_{1} − x_{0}∥ < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(70) Let f be a partial function from the carrier of R_{1} to ℂ. Then f is conti-

nuous on X if and only if the following conditions are satisfied:

(i) X ⊆ domf, and (ii) for every point x_{0} of R_{1} and for every r such that x_{0} ∈ X and 0 < r

there exists s such that 0 < s and for every point x_{1} of R_{1} such that x_{1} ∈ X and ∥x_{1} − x_{0}∥ < s holds |f_{x_{1}} − f_{x_{0}}| < r.

(71) For every partial function f from C_{2} to C_{3} holds f is continuous on X iff f↾X is continuous on X.

(72) For every partial function f from C_{1} to R_{1} holds f is continuous on X iff f↾X is continuous on X.

(73) For every partial function f from R_{1} to C_{1} holds f is continuous on X iff f↾X is continuous on X.

(74) Let f be a partial function from the carrier of C_{1} to ℂ. Then f is continuous on X if and only if f↾X is continuous on X.

(75) Let f be a partial function from the carrier of C_{1} to ℝ. Then f is continuous on X if and only if f↾X is continuous on X.

(76) Let f be a partial function from the carrier of R_{1} to ℂ. Then f is continuous on X if and only if f↾X is continuous on X.

(77) For every partial function f from C_{2} to C_{3} such that f is continuous on X and X_{1} ⊆ X holds f is continuous on X_{1}.

(78) For every partial function f from C_{1} to R_{1} such that f is continuous on X and X_{1} ⊆ X holds f is continuous on X_{1}.

(79) For every partial function f from R_{1} to C_{1} such that f is continuous on X and X_{1} ⊆ X holds f is continuous on X_{1}.

(80) For every partial function f from C_{2} to C_{3} and for every point x_{0} of C_{2} such that x_{0} ∈ domf holds f is continuous on {x_{0}}.

(81) For every partial function f from C_{1} to R_{1} and for every point x_{0} of C_{1} such that x_{0} ∈ domf holds f is continuous on {x_{0}}.

(82) For every partial function f from R_{1} to C_{1} and for every point x_{0} of R_{1} such that x_{0} ∈ domf holds f is continuous on {x_{0}}.

(83) Let f_{1}, f_{2} be partial functions from C_{2} to C_{3}. Suppose f_{1} is continuous on X and f_{2} is continuous on X. Then f_{1} + f_{2} is continuous on X and f_{1} − f_{2} is continuous on X.

(84) Let f_{1}, f_{2} be partial functions from C_{1} to R_{1}. Suppose f_{1} is continuous on X and f_{2} is continuous on X. Then f_{1} + f_{2} is continuous on X and f_{1} − f_{2} is continuous on X.

(85) Let f_{1}, f_{2} be partial functions from R_{1} to C_{1}. Suppose f_{1} is continuous on X and f_{2} is continuous on X. Then f_{1} + f_{2} is continuous on X and f_{1} − f_{2} is continuous on X.

(86) Let f_{1}, f_{2} be partial functions from C_{2} to C_{3}. Suppose f_{1} is continuous on X and f_{2} is continuous on X_{1}. Then f_{1} + f_{2} is continuous on X ∩ X_{1} and f_{1} − f_{2} is continuous on X ∩ X_{1}.

(87) Let f_{1}, f_{2} be partial functions from C_{1} to R_{1}. Suppose f_{1} is continuous on X and f_{2} is continuous on X_{1}. Then f_{1} + f_{2} is continuous on X ∩ X_{1} and f_{1} − f_{2} is continuous on X ∩ X_{1}.

(88) Let f_{1}, f_{2} be partial functions from R_{1} to C_{1}. Suppose f_{1} is continuous on X and f_{2} is continuous on X_{1}. Then f_{1} + f_{2} is continuous on X ∩ X_{1} and f_{1} − f_{2} is continuous on X ∩ X_{1}.

(89) For every partial function f from C_{2} to C_{3} such that f is continuous on

X holds z f is continuous on X.

(90) For every partial function f from C_{1} to R_{1} such that f is continuous on X holds r f is continuous on X.

(91) For every partial function f from R_{1} to C_{1} such that f is continuous on X holds z f is continuous on X.

(92) Let f be a partial function from C_{2} to C_{3}. If f is continuous on X, then ∥f∥ is continuous on X and − f is continuous on X.

(93) Let f be a partial function from C_{1} to R_{1}. If f is continuous on X, then ∥f∥ is continuous on X and − f is continuous on X.

(94) Let f be a partial function from R_{1} to C_{1}. If f is continuous on X, then ∥f∥ is continuous on X and − f is continuous on X.

(95) Let f be a partial function from C_{2} to C_{3}. Suppose f is total and for all points x_{1}, x_{2} of C_{2} holds f_{x_{1}+x_{2}} = f_{x_{1}} + f_{x_{2}} and there exists a point x_{0} of C_{2} such that f is continuous in x_{0}. Then f is continuous on the carrier of C_{2}.

(96) Let f be a partial function from C_{1} to R_{1}. Suppose f is total and for all points x_{1}, x_{2} of C_{1} holds f_{x_{1}+x_{2}} = f_{x_{1}} + f_{x_{2}} and there exists a point x_{0} of C_{1} such that f is continuous in x_{0}. Then f is continuous on the carrier of C_{1}.

(97) Let f be a partial function from R_{1} to C_{1}. Suppose f is total and for all points x_{1}, x_{2} of R_{1} holds f_{x_{1}+x_{2}} = f_{x_{1}} + f_{x_{2}} and there exists a point x_{0} of R_{1} such that f is continuous in x_{0}. Then f is continuous on the carrier of R_{1}.

(98) For every partial function f from C_{2} to C_{3} such that domf is compact and f is continuous on domf holds rngf is compact.

(99) For every partial function f from C_{1} to R_{1} such that domf is compact and f is continuous on domf holds rngf is compact.

(100) For every partial function f from R_{1} to C_{1} such that domf is compact and f is continuous on domf holds rngf is compact.

(101) Let f be a partial function from the carrier of C_{1} to ℂ. If domf is compact and f is continuous on domf, then rngf is compact.

(102) Let f be a partial function from the carrier of C_{1} to ℝ. If domf is compact and f is continuous on domf, then rngf is compact.

(103) Let f be a partial function from the carrier of R_{1} to ℂ. If domf is compact and f is continuous on domf, then rngf is compact.

(104) Let Y be a subset of C_{2} and f be a partial function from C_{2} to C_{3}. If Y ⊆ domf and Y is compact and f is continuous on Y, then f^{∘}Y is compact.

(105) Let Y be a subset of C_{1} and f be a partial function from C_{1} to R_{1}.

If Y ⊆ domf and Y is compact and f is continuous on Y, then f^{∘}Y is compact.

(106) Let Y be a subset of R_{1} and f be a partial function from R_{1} to C_{1}. If Y ⊆ domf and Y is compact and f is continuous on Y, then f^{∘}Y is compact.

(107) Let f be a partial function from the carrier of C_{1} to ℝ. Suppose domf ≠ ∅ and domf is compact and f is continuous on domf. Then there exist points x_{1}, x_{2} of C_{1} such that x_{1} ∈ domf and x_{2} ∈ domf and f_{x_{1}} = suprngf and f_{x_{2}} = inf rngf.

(108) Let f be a partial function from C_{2} to C_{3}. Suppose domf ≠ ∅ and domf is compact and f is continuous on domf. Then there exist points x_{1}, x_{2} of C_{2} such that x_{1} ∈ domf and x_{2} ∈ domf and ∥f∥_{x_{1}} = suprng∥f∥ and ∥f∥_{x_{2}} = inf rng∥f∥.

(109) Let f be a partial function from C_{1} to R_{1}. Suppose domf ≠ ∅ and domf is compact and f is continuous on domf. Then there exist points x_{1}, x_{2} of C_{1} such that x_{1} ∈ domf and x_{2} ∈ domf and ∥f∥_{x_{1}} = suprng∥f∥ and ∥f∥_{x_{2}} = inf rng∥f∥.

(110) Let f be a partial function from R_{1} to C_{1}. Suppose domf ≠ ∅ and domf is compact and f is continuous on domf. Then there exist points x_{1}, x_{2} of R_{1} such that x_{1} ∈ domf and x_{2} ∈ domf and ∥f∥_{x_{1}} = suprng∥f∥ and ∥f∥_{x_{2}} = inf rng∥f∥.

(111) For every partial function f from C_{2} to C_{3} holds ∥f∥↾X = ∥f↾X∥.

(112) For every partial function f from C_{1} to R_{1} holds ∥f∥↾X = ∥f↾X∥.

(113) For every partial function f from R_{1} to C_{1} holds ∥f∥↾X = ∥f↾X∥.

(114) Let f be a partial function from C_{2} to C_{3} and Y be a subset of C_{2}. Suppose Y ≠ ∅ and Y ⊆ domf and Y is compact and f is continuous on Y. Then there exist points x_{1}, x_{2} of C_{2} such that x_{1} ∈ Y and x_{2} ∈ Y and ∥f∥_{x_{1}} = sup(∥f∥^{∘}Y) and ∥f∥_{x_{2}} = inf(∥f∥^{∘}Y).

(115) Let f be a partial function from C_{1} to R_{1} and Y be a subset of C_{1}. Suppose Y ≠ ∅ and Y ⊆ domf and Y is compact and f is continuous on Y. Then there exist points x_{1}, x_{2} of C_{1} such that x_{1} ∈ Y and x_{2} ∈ Y and ∥f∥_{x_{1}} = sup(∥f∥^{∘}Y) and ∥f∥_{x_{2}} = inf(∥f∥^{∘}Y).

(116) Let f be a partial function from R_{1} to C_{1} and Y be a subset of R_{1}. Suppose Y ≠ ∅ and Y ⊆ domf and Y is compact and f is continuous on Y. Then there exist points x_{1}, x_{2} of R_{1} such that x_{1} ∈ Y and x_{2} ∈ Y and ∥f∥_{x_{1}} = sup(∥f∥^{∘}Y) and ∥f∥_{x_{2}} = inf(∥f∥^{∘}Y).

(117) Let f be a partial function from the carrier of C_{1} to ℝ and Y be a subset of C_{1}. Suppose Y ≠ ∅ and Y ⊆ domf and Y is compact and f is continuous on Y. Then there exist points x_{1}, x_{2} of C_{1} such that x_{1} ∈ Y and x_{2} ∈ Y and f_{x_{1}} = sup(f^{∘}Y) and f_{x_{2}} = inf(f^{∘}Y).

Let C_{2}, C_{3} be complex normed spaces, let X be a set, and let f be a partial function from C_{2} to C_{3}. We say that f is Lipschitzian on X if and only if:

(Def. 27) X ⊆ domf and there exists r such that 0 < r and for all points x_{1}, x_{2}

of C_{2} such that x_{1} ∈ X and x_{2} ∈ X holds ∥f_{x_{1}} − f_{x_{2}}∥ r · ∥x_{1} − x_{2}∥. Let C_{1} be a complex normed space, let R_{1} be a real normed space, let X be a

set, and let f be a partial function from C_{1} to R_{1}. We say that f is Lipschitzian on X if and only if:

(Def. 28) X ⊆ domf and there exists r such that 0 < r and for all points x_{1}, x_{2}

of C_{1} such that x_{1} ∈ X and x_{2} ∈ X holds ∥f_{x_{1}} − f_{x_{2}}∥ r · ∥x_{1} − x_{2}∥. Let R_{1} be a real normed space, let C_{1} be a complex normed space, let X be a

set, and let f be a partial function from R_{1} to C_{1}. We say that f is Lipschitzian on X if and only if:

(Def. 29) X ⊆ domf and there exists r such that 0 < r and for all points x_{1}, x_{2}

of R_{1} such that x_{1} ∈ X and x_{2} ∈ X holds ∥f_{x_{1}} − f_{x_{2}}∥ r · ∥x_{1} − x_{2}∥. Let C_{1} be a complex normed space, let X be a set, and let f be a partial

function from the carrier of C_{1} to ℂ. We say that f is Lipschitzian on X if and only if:

(Def. 30) X ⊆ domf and there exists r such that 0 < r and for all points x_{1}, x_{2}

of C_{1} such that x_{1} ∈ X and x_{2} ∈ X holds |f_{x_{1}} − f_{x_{2}}| r · ∥x_{1} − x_{2}∥. Let C_{1} be a complex normed space, let X be a set, and let f be a partial

function from the carrier of C_{1} to ℝ. We say that f is Lipschitzian on X if and only if:

(Def. 31) X ⊆ domf and there exists r such that 0 < r and for all points x_{1}, x_{2}

of C_{1} such that x_{1} ∈ X and x_{2} ∈ X holds |f_{x_{1}} − f_{x_{2}}| r · ∥x_{1} − x_{2}∥. Let R_{1} be a real normed space, let X be a set, and let f be a partial function

from the carrier of R_{1} to ℂ. We say that f is Lipschitzian on X if and only if: (Def. 32) X ⊆ domf and there exists r such that 0 < r and for all points x_{1}, x_{2}

of R_{1} such that x_{1} ∈ X and x_{2} ∈ X holds |f_{x_{1}} − f_{x_{2}}| r · ∥x_{1} − x_{2}∥. Next we state a number of propositions:

(118) For every partial function f from C_{2} to C_{3} such that f is Lipschitzian on X and X_{1} ⊆ X holds f is Lipschitzian on X_{1}.

(119) For every partial function f from C_{1} to R_{1} such that f is Lipschitzian on X and X_{1} ⊆ X holds f is Lipschitzian on X_{1}.

(120) For every partial function f from R_{1} to C_{1} such that f is Lipschitzian on X and X_{1} ⊆ X holds f is Lipschitzian on X_{1}.

(121) Let f_{1}, f_{2} be partial functions from C_{2} to C_{3}. Suppose f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1}. Then f_{1} +f_{2} is Lipschitzian on X ∩X_{1}.

(122) Let f_{1}, f_{2} be partial functions from C_{1} to R_{1}. Suppose f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1}. Then f_{1} +f_{2} is Lipschitzian on X ∩X_{1}.

(123) Let f_{1}, f_{2} be partial functions from R_{1} to C_{1}. Suppose f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1}. Then f_{1} +f_{2} is Lipschitzian on X ∩X_{1}.

(124) Let f_{1}, f_{2} be partial functions from C_{2} to C_{3}. Suppose f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1}. Then f_{1} −f_{2} is Lipschitzian on X ∩X_{1}.

(125) Let f_{1}, f_{2} be partial functions from C_{1} to R_{1}. Suppose f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1}. Then f_{1} −f_{2} is Lipschitzian on X ∩X_{1}.

(126) Let f_{1}, f_{2} be partial functions from R_{1} to C_{1}. Suppose f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1}. Then f_{1} −f_{2} is Lipschitzian on X ∩X_{1}.

(127) For every partial function f from C_{2} to C_{3} such that f is Lipschitzian on X holds z f is Lipschitzian on X.

(128) For every partial function f from C_{1} to R_{1} such that f is Lipschitzian on X holds r f is Lipschitzian on X.

(129) For every partial function f from R_{1} to C_{1} such that f is Lipschitzian on X holds z f is Lipschitzian on X.

(130) Let f be a partial function from C_{2} to C_{3}. Suppose f is Lipschitzian on X. Then − f is Lipschitzian on X and ∥f∥ is Lipschitzian on X.

(131) Let f be a partial function from C_{1} to R_{1}. Suppose f is Lipschitzian on X. Then − f is Lipschitzian on X and ∥f∥ is Lipschitzian on X.

(132) Let f be a partial function from R_{1} to C_{1}. Suppose f is Lipschitzian on X. Then − f is Lipschitzian on X and ∥f∥ is Lipschitzian on X.

(133) Let X be a set and f be a partial function from C_{2} to C_{3}. If X ⊆ domf and f is a constant on X, then f is Lipschitzian on X.

(134) Let X be a set and f be a partial function from C_{1} to R_{1}. If X ⊆ domf and f is a constant on X, then f is Lipschitzian on X.

(135) Let X be a set and f be a partial function from R_{1} to C_{1}. If X ⊆ domf and f is a constant on X, then f is Lipschitzian on X.

(136) For every subset Y of C_{1} holds id_{Y} is Lipschitzian on Y.

(137) For every partial function f from C_{2} to C_{3} such that f is Lipschitzian on X holds f is continuous on X.

(138) For every partial function f from C_{1} to R_{1} such that f is Lipschitzian on X holds f is continuous on X.

(139) For every partial function f from R_{1} to C_{1} such that f is Lipschitzian on X holds f is continuous on X.

(140) Let f be a partial function from the carrier of C_{1} to ℂ. If f is Lipschitzian on X, then f is continuous on X.

(141) Let f be a partial function from the carrier of C_{1} to ℝ. If f is Lipschitzian on X, then f is continuous on X.

(142) Let f be a partial function from the carrier of R_{1} to ℂ. If f is Lipschitzian on X, then f is continuous on X.

(143) For every partial function f from C_{2} to C_{3} such that there exists a point r of C_{3} such that rngf = {r} holds f is continuous on domf.

(144) For every partial function f from C_{1} to R_{1} such that there exists a point r of R_{1} such that rngf = {r} holds f is continuous on domf.

(145) For every partial function f from R_{1} to C_{1} such that there exists a point r of C_{1} such that rngf = {r} holds f is continuous on domf.

(146) For every partial function f from C_{2} to C_{3} such that X ⊆ domf and f is a constant on X holds f is continuous on X.

(147) For every partial function f from C_{1} to R_{1} such that X ⊆ domf and f is a constant on X holds f is continuous on X.

(148) For every partial function f from R_{1} to C_{1} such that X ⊆ domf and f is a constant on X holds f is continuous on X.

(149) Let f be a partial function from C_{1} to C_{1}. Suppose that for every point x_{0} of C_{1} such that x_{0} ∈ domf holds f_{x_{0}} = x_{0}. Then f is continuous on domf.

(150) For every partial function f from C_{1} to C_{1} such that f = id_{dom f} holds f is continuous on domf.

(151) Let f be a partial function from C_{1} to C_{1} and Y be a subset of C_{1}. If Y ⊆ domf and f↾Y = id_{Y} , then f is continuous on Y.

(152) Let f be a partial function from C_{1} to C_{1}, z be a complex number, and p be a point of C_{1}. Suppose X ⊆ domf and for every point x_{0} of C_{1} such that x_{0} ∈ X holds f_{x_{0}} = z · x_{0} + p. Then f is continuous on X.

(153) Let f be a partial function from the carrier of C_{1} to ℝ. Suppose that for every point x_{0} of C_{1} such that x_{0} ∈ domf holds f_{x_{0}} = ∥x_{0}∥. Then f is continuous on domf.

(154) Let f be a partial function from the carrier of C_{1} to ℝ. Suppose X ⊆ domf and for every point x_{0} of C_{1} such that x_{0} ∈ X holds f_{x_{0}} = ∥x_{0}∥. Then f is continuous on X.

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121–124, 1993.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[3] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[7] Noboru Endou. Algebra of complex vector valued functions. Formalized Mathematics, 12(3):397–401, 2004.

[8] Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93–102, 2004.

continuous functions on real and complex … 419

[9] Noboru Endou. Series on complex Banach algebra. Formalized Mathematics, 12(3):281– 288, 2004.

[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[11] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[12] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[13] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[14] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[15] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[16] Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex sequence and continuity of complex function. Formalized Mathematics, 9(1):185–190, 2001.

[17] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265–268, 1997.

[18] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269–275, 2004.

[19] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[20] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111–115, 1991.

[21] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[22] Yasunari Shidama. The series on Banach algebra. Formalized Mathematics, 12(2):131– 138, 2004.

[23] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403–410, 1997.

[24] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.

[25] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[26] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.

[27] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.

[28] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[29] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[30] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

[31] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171–175, 1992.

On the Fundamental Groups of Products of Topological Spaces

Artur Korniłowicz^{1} University of Białystok

Summary. In the paper we show that fundamental group of the product of two topological spaces is isomorphic to the product of fundamental groups of the spaces.

MML Identifier: TOPALG 4.

The articles [15], [7], [14], [19], [5], [20], [6], [3], [4], [1], [2], [12], [17], [18], [10], [13], [16], [8], [9], and [11] provide the terminology and notation for this paper.

1. On the Product of Groups

The following proposition is true

(1) Let G, H be non empty groupoids and x be an element of ∏⟨G, H⟩. Then there exists an element g of G and there exists an element h of H such that x = ⟨g, h⟩.

Let G_{1}, G_{2}, H_{1}, H_{2} be non empty groupoids, let f be a map from G_{1} into H_{1}, and let g be a map from G_{2} into H_{2}. The functor Gr2Iso(f, g) yields a map from ∏⟨G_{1}, G_{2}⟩ into ∏⟨H_{1}, H_{2}⟩ and is defined by the condition (Def. 1).

(Def. 1) Let x be an element of ∏⟨G_{1}, G_{2}⟩. Then there exists an element x_{1} of G_{1} and there exists an element x_{2} of G_{2} such that x = ⟨x_{1}, x_{2}⟩ and (Gr2Iso(f, g))(x) = ⟨f(x_{1}), g(x_{2})⟩.

The following proposition is true (2) Let G_{1}, G_{2}, H_{1}, H_{2} be non empty groupoids, f be a map from G_{1} into

1The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan. This work has been partially supported by KBN grant 4 T11C 039 24.

H_{1}, g be a map from G_{2} into H_{2}, x_{1} be an element of G_{1}, and x_{2} be an element of G_{2}. Then (Gr2Iso(f, g))(⟨x_{1}, x_{2}⟩) = ⟨f(x_{1}), g(x_{2})⟩.

Let G_{1}, G_{2}, H_{1}, H_{2} be groups, let f be a homomorphism from G_{1} to H_{1}, and let g be a homomorphism from G_{2} to H_{2}. Then Gr2Iso(f, g) is a homomorphism from ∏⟨G_{1}, G_{2}⟩ to ∏⟨H_{1}, H_{2}⟩.

One can prove the following four propositions:

(3) Let G_{1}, G_{2}, H_{1}, H_{2} be non empty groupoids, f be a map from G_{1} into H_{1}, and g be a map from G_{2} into H_{2}. If f is one-to-one and g is one-to-one, then Gr2Iso(f, g) is one-to-one.

(4) Let G_{1}, G_{2}, H_{1}, H_{2} be non empty groupoids, f be a map from G_{1} into H_{1}, and g be a map from G_{2} into H_{2}. If f is onto and g is onto, then Gr2Iso(f, g) is onto.

(5) Let G_{1}, G_{2}, H_{1}, H_{2} be groups, f be a homomorphism from G_{1} to H_{1}, and g be a homomorphism from G_{2} to H_{2}. If f is an isomorphism and g is an isomorphism, then Gr2Iso(f, g) is an isomorphism.

(6) Let G_{1}, G_{2}, H_{1}, H_{2} be groups. Suppose G_{1} and H_{1} are isomorphic and G_{2} and H_{2} are isomorphic. Then ∏⟨G_{1}, G_{2}⟩ and ∏⟨H_{1}, H_{2}⟩ are isomorphic.

2. On the Fundamental Groups of Products of Topological Spaces

For simplicity, we adopt the following rules: S, T, Y denote non empty topological spaces, s, s_{1}, s_{2}, s_{3} denote points of S, t, t_{1}, t_{2}, t_{3} denote points of T, l_{1}, l_{2} denote paths from ⟨⟨s_{1}, t_{1}⟩⟩ to ⟨⟨s_{2}, t_{2}⟩⟩, and H denotes a homotopy between l_{1} and l_{2}.

We now state two propositions:

(7) For all functions f, g such that domf = domg holds pr1(⟨f, g⟩) = f.

(8) For all functions f, g such that domf = domg holds pr2(⟨f, g⟩) = g. Let us consider S, T, Y, let f be a map from Y into S, and let g be a map

from Y into T. Then ⟨f, g⟩ is a map from Y into [: S, T :].

Let us consider S, T, Y and let f be a map from Y into [: S, T :]. Then pr1(f) is a map from Y into S. Then pr2(f) is a map from Y into T.

The following propositions are true:

(9) For every continuous map f from Y into [: S, T :] holds pr1(f) is continuous.

(10) For every continuous map f from Y into [: S, T :] holds pr2(f) is continuous.

(11) If ⟨⟨s_{1}, t_{1}⟩⟩, ⟨⟨s_{2}, t_{2}⟩⟩ are connected, then s_{1}, s_{2} are connected.

(12) If ⟨⟨s_{1}, t_{1}⟩⟩, ⟨⟨s_{2}, t_{2}⟩⟩ are connected, then t_{1}, t_{2} are connected.

(13) If ⟨⟨s_{1}, t_{1}⟩⟩, ⟨⟨s_{2}, t_{2}⟩⟩ are connected, then for every path L from ⟨⟨s_{1}, t_{1}⟩⟩ to ⟨⟨s_{2}, t_{2}⟩⟩ holds pr1(L) is a path from s_{1} to s_{2}.

(14) If ⟨⟨s_{1}, t_{1}⟩⟩, ⟨⟨s_{2}, t_{2}⟩⟩ are connected, then for every path L from ⟨⟨s_{1}, t_{1}⟩⟩ to ⟨⟨s_{2}, t_{2}⟩⟩ holds pr2(L) is a path from t_{1} to t_{2}.

(15) If s_{1}, s_{2} are connected and t_{1}, t_{2} are connected, then ⟨⟨s_{1}, t_{1}⟩⟩, ⟨⟨s_{2}, t_{2}⟩⟩ are connected.

(16) Suppose s_{1}, s_{2} are connected and t_{1}, t_{2} are connected. Let L_{1} be a path from s_{1} to s_{2} and L_{2} be a path from t_{1} to t_{2}. Then ⟨L_{1}, L_{2}⟩ is a path from ⟨⟨s_{1}, t_{1}⟩⟩ to ⟨⟨s_{2}, t_{2}⟩⟩.

Let S, T be non empty arcwise connected topological spaces, let s_{1}, s_{2} be points of S, let t_{1}, t_{2} be points of T, let L_{1} be a path from s_{1} to s_{2}, and let L_{2} be a path from t_{1} to t_{2}. Then ⟨L_{1}, L_{2}⟩ is a path from ⟨⟨s_{1}, t_{1}⟩⟩ to ⟨⟨s_{2}, t_{2}⟩⟩.

Let S, T be non empty topological spaces, let s be a point of S, let t be a point of T, let L_{1} be a loop of s, and let L_{2} be a loop of t. Then ⟨L_{1}, L_{2}⟩ is a loop of ⟨⟨s, t⟩⟩.

Let S, T be non empty arcwise connected topological spaces. One can verify that [: S, T :] is arcwise connected.

Let S, T be non empty arcwise connected topological spaces, let s_{1}, s_{2} be points of S, let t_{1}, t_{2} be points of T, and let L be a path from ⟨⟨s_{1}, t_{1}⟩⟩ to ⟨⟨s_{2}, t_{2}⟩⟩. Then pr1(L) is a path from s_{1} to s_{2}. Then pr2(L) is a path from t_{1} to t_{2}.

Let S, T be non empty topological spaces, let s be a point of S, let t be a point of T, and let L be a loop of ⟨⟨s, t⟩⟩. Then pr1(L) is a loop of s. Then pr2(L) is a loop of t.

Next we state a number of propositions:

(17) Let p, q be paths from s_{1} to s_{2}. Suppose p = pr1(l_{1}) and q = pr1(l_{2}) and l_{1}, l_{2} are homotopic. Then pr1(H) is a homotopy between p and q.

(18) Let p, q be paths from t_{1} to t_{2}. Suppose p = pr2(l_{1}) and q = pr2(l_{2}) and l_{1}, l_{2} are homotopic. Then pr2(H) is a homotopy between p and q.

(19) For all paths p, q from s_{1} to s_{2} such that p = pr1(l_{1}) and q = pr1(l_{2}) and l_{1}, l_{2} are homotopic holds p, q are homotopic.

(20) For all paths p, q from t_{1} to t_{2} such that p = pr2(l_{1}) and q = pr2(l_{2}) and l_{1}, l_{2} are homotopic holds p, q are homotopic.

(21) Let p, q be paths from s_{1} to s_{2}, x, y be paths from t_{1} to t_{2}, f be a homotopy between p and q, and g be a homotopy between x and y. Suppose p = pr1(l_{1}) and q = pr1(l_{2}) and x = pr2(l_{1}) and y = pr2(l_{2}) and p, q are homotopic and x, y are homotopic. Then ⟨f, g⟩ is a homotopy between l_{1} and l_{2}.

(22) Let p, q be paths from s_{1} to s_{2} and x, y be paths from t_{1} to t_{2}. Suppose p = pr1(l_{1}) and q = pr1(l_{2}) and x = pr2(l_{1}) and y = pr2(l_{2}) and p, q are homotopic and x, y are homotopic. Then l_{1}, l_{2} are homotopic.

(23) Let l_{1} be a path from ⟨⟨s_{1}, t_{1}⟩⟩to ⟨⟨s_{2}, t_{2}⟩⟩, l_{2} be a path from ⟨⟨s_{2}, t_{2}⟩⟩to ⟨⟨s_{3}, t_{3}⟩⟩, p_{1} be a path from s_{1} to s_{2}, and p_{2} be a path from s_{2} to s_{3}. Suppose ⟨⟨s_{1}, t_{1}⟩⟩, ⟨⟨s_{2}, t_{2}⟩⟩ are connected and ⟨⟨s_{2}, t_{2}⟩⟩, ⟨⟨s_{3}, t_{3}⟩⟩ are connected and p_{1} = pr1(l_{1}) and p_{2} = pr1(l_{2}). Then pr1(l_{1} + l_{2}) = p_{1} + p_{2}.

(24) Let S, T be non empty arcwise connected topological spaces, s_{1}, s_{2}, s_{3} be points of S, t_{1}, t_{2}, t_{3} be points of T, l_{1} be a path from ⟨⟨s_{1}, t_{1}⟩⟩ to ⟨⟨s_{2}, t_{2}⟩⟩, and l_{2} be a path from ⟨⟨s_{2}, t_{2}⟩⟩ to ⟨⟨s_{3}, t_{3}⟩⟩. Then pr1(l_{1} + l_{2}) = pr1(l_{1}) + pr1(l_{2}).

(25) Let l_{1} be a path from ⟨⟨s_{1}, t_{1}⟩⟩to ⟨⟨s_{2}, t_{2}⟩⟩, l_{2} be a path from ⟨⟨s_{2}, t_{2}⟩⟩to ⟨⟨s_{3}, t_{3}⟩⟩, p_{1} be a path from t_{1} to t_{2}, and p_{2} be a path from t_{2} to t_{3}. Suppose ⟨⟨s_{1}, t_{1}⟩⟩, ⟨⟨s_{2}, t_{2}⟩⟩ are connected and ⟨⟨s_{2}, t_{2}⟩⟩, ⟨⟨s_{3}, t_{3}⟩⟩ are connected and p_{1} = pr2(l_{1}) and p_{2} = pr2(l_{2}). Then pr2(l_{1} + l_{2}) = p_{1} + p_{2}.

(26) Let S, T be non empty arcwise connected topological spaces, s_{1}, s_{2}, s_{3} be points of S, t_{1}, t_{2}, t_{3} be points of T, l_{1} be a path from ⟨⟨s_{1}, t_{1}⟩⟩ to ⟨⟨s_{2}, t_{2}⟩⟩, and l_{2} be a path from ⟨⟨s_{2}, t_{2}⟩⟩ to ⟨⟨s_{3}, t_{3}⟩⟩. Then pr2(l_{1} + l_{2}) = pr2(l_{1}) + pr2(l_{2}).

Let S, T be non empty topological spaces, let s be a point of S, and let t be a point of T. The functor FGPrIso(s, t) yielding a map from π_{1}([: S, T :], ⟨⟨s, t⟩⟩) into ∏⟨π_{1}(S, s), π_{1}(T, t)⟩ is defined as follows:

(Def. 2) For every point x of π_{1}([: S, T :], ⟨⟨s, t⟩⟩) there exists a loop l of ⟨⟨s, t⟩⟩such that x = [l]_{EqRel([: S, T :],⟨⟨s, t⟩⟩)} and (FGPrIso(s, t))(x) = ⟨[pr1(l)]_{EqRel(S,s)}, [pr2(l)]_{EqRel(T,t)}⟩.

The following propositions are true:

(27) For every point x of π_{1}([: S, T :], ⟨⟨s, t⟩⟩) and for every loop l of ⟨⟨s, t⟩⟩such that x = [l]_{EqRel([: S, T :],⟨⟨s, t⟩⟩)} holds (FGPrIso(s, t))(x) = ⟨[pr1(l)]_{EqRel(S,s)}, [pr2(l)]_{EqRel(T,t)}⟩.

(28) For every loop l of ⟨⟨s, t⟩⟩ holds (FGPrIso(s, t))([l]_{EqRel([: S, T :],⟨⟨s, t⟩⟩)}) =

⟨[pr1(l)]_{EqRel(S,s)}, [pr2(l)]_{EqRel(T,t)}⟩. Let S, T be non empty topological spaces, let s be a point of S, and let t be

a point of T. Observe that FGPrIso(s, t) is one-to-one and onto.

Let S, T be non empty topological spaces, let s be a point of S, and let t be a point of T. Then FGPrIso(s, t) is a homomorphism from π_{1}([: S, T :], ⟨⟨s, t⟩⟩) to ∏⟨π_{1}(S, s), π_{1}(T, t)⟩.

The following propositions are true:

(29) FGPrIso(s, t) is an isomorphism.

(30) π_{1}([: S, T :], ⟨⟨s, t⟩⟩) and ∏⟨π_{1}(S, s), π_{1}(T, t)⟩ are isomorphic.

(31) Let f be a homomorphism from π_{1}(S, s_{1}) to π_{1}(S, s_{2}) and g be a homomorphism from π_{1}(T, t_{1}) to π_{1}(T, t_{2}). Suppose f is an isomorphism and g is an isomorphism. Then Gr2Iso(f, g) · FGPrIso(s_{1}, t_{1}) is an isomorphism.

on the fundamental groups of products of … 425

(32) Let S, T be non empty arcwise connected topological spaces, s_{1}, s_{2} be points of S, and t_{1}, t_{2} be points of T. Then π_{1}([: S, T :], ⟨⟨s_{1}, t_{1}⟩⟩) and ∏⟨π_{1}(S, s_{2}), π_{1}(T, t_{2})⟩ are isomorphic.

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[2] Grzegorz Bancerek and Piotr Rudnicki. On defining functions on trees. Formalized Mathematics, 4(1):91–101, 1993.

[3] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.

[4] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[7] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.

[8] Adam Grabowski. Introduction to the homotopy theory. Formalized Mathematics, 6(4):449–454, 1997.

[9] Adam Grabowski and Artur Korniłowicz. Algebraic properties of homotopies. Formalized Mathematics, 12(3):251–260, 2004.

[10] Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127–134, 1998.

[11] Artur Korniłowicz, Yasunari Shidama, and Adam Grabowski. The fundamental group. Formalized Mathematics, 12(3):261–268, 2004.

[12] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[13] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223–230, 1990.

[14] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[15] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[16] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535–545, 1991.

[17] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.

[18] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573–578, 1991.

[19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[20] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

Index of MML Identifiers

BORSUK 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251 CATALAN1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 CFUNCDOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 CLOPBAN2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 CLOPBAN3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 CLOPBAN4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 FIB NUM2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307 FIB NUM3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .329 FINTOPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 GROUP 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 HALLMAR1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 LATSUM 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .335 NAGATA 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .341 NAGATA 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .385 NCFCONT1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 NDIFF 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 NDIFF 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 NFCONT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .269 NFCONT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .277 POLYEQ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247 PRGCOR 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375 SHEFFER1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 SHEFFER2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 SIN COS5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243 TOPALG 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261 TOPALG 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295 TOPALG 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .391 TOPALG 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .421 TOPREAL9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 VFUNCT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .397

Contents Formaliz. Math. 12 (3)

Complex Valued Functions Space

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Banach Algebra of Bounded Complex Linear Operators

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Formulas and Identities of Trigonometric Functions

By Yuzhong Ding and Xiquan Liang  . . . . . . . . . . . . . . . . . . . . . . 243

Solving Roots of the Special Polynomial Equation with Real Co-

efficients

By Yuzhong Ding and Xiquan Liang  . . . . . . . . . . . . . . . . . . . . . . 247

Algebraic Properties of Homotopies

By Adam Grabowski and Artur Korniłowicz  . . . . . . . . . . . . 251

The Fundamental Group

By Artur Korniłowicz et al.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

The Continuous Functions on Normed Linear Spaces

By Takaya Nishiyama et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

The Uniform Continuity of Functions on Normed Linear Spaces

By Takaya Nishiyama et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Series on Complex Banach Algebra

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Exponential Function on Complex Banach Algebra

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

The Fundamental Group of Convex Subspaces of E_{T}^{n}

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Intersections of Intervals and Balls in E_{T}^{n}

By Artur Korniłowicz and Yasunari Shidama  . . . . . . . . . . . 301

Continued on inside back cover

Some Properties of Fibonacci Numbers

By Magdalena Jastrzȩbska and Adam Grabowski  . . . . . . . 307

The Hall Marriage Theorem

By Ewa Romanowicz and Adam Grabowski  . . . . . . . . . . . . . . . 315

The Differentiable Functions on Normed Linear Spaces

By Hiroshi Imura et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Lucas Numbers and Generalized Fibonacci Numbers

By Piotr Wojtecki and Adam Grabowski  . . . . . . . . . . . . . . . . 329

The Operation of Addition of Relational Structures

By Katarzyna Romanowicz and Adam Grabowski  . . . . . . . 335

The Nagata-Smirnov Theorem. Part I

By Karol Pa̧k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Properties of Groups

By Gijs Geleijnse and Grzegorz Bancerek  . . . . . . . . . . . . . . . 347

Catalan Numbers

By Dorota Czȩstochowska and Adam Grabowski  . . . . . . . 351

Axiomatization of Boolean Algebras Based on Sheffer Stroke

By Violetta Kozarkiewicz and Adam Grabowski  . . . . . . . . 355

Short Sheffer Stroke-Based Single Axiom for Boolean Algebras

By Aneta Łukaszuk and Adam Grabowski  . . . . . . . . . . . . . . . . 363

Differentiable Functions on Normed Linear Spaces. Part II

By Hiroshi Imura et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Logical Correctness of Vector Calculation Programs

By Takaya Nishiyama et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

Continuous Mappings between Finite and One-Dimensional Finite

Topological Spaces

By Hiroshi Imura et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

The Nagata-Smirnov Theorem. Part II

By Karol Pa̧k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

On the Isomorphism of Fundamental Groups

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

Algebra of Complex Vector Valued Functions

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Continuous Functions on Real and Complex Normed Linear Spa-

ces

By Noboru Endou  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

On the Fundamental Groups of Products of Topological Spaces

By Artur Korniłowicz  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Continued on inside back cover