Preface

We offer to our Readers Volume 2 of mathematical papers which are abstracts of Mizar articles to be found in the Main Mizar Library (MML). They are usually published in the order in which they have been approved for MML. A careful Reader may note that our publication has several peculiarities due to two facts. First, it is an endeavour to make a machine translation into English. Secondly, changes in the PC Mizar system and continually updated MML influence the quality of the texts published and the topical value of the papers. Hence, first, the standard of English is not always satisfactory. Secondly, the quality of the papers is very closely related to what is actually taking place in MML. Originally, obvious theorems (relative to the power of Checker) were not identified. As the system PC Mizar was developing, some theorems became obvious. It is likewise with repeated theorems (which accounts for the footnotes in the text of the type ”The proposition (k) was either repeated or obvious”). Those theorems can be classed in two groups. The first includes accidental repetitions: the author did not know that such a theorem was already included in MML and proved it again. There were few such cases. The other includes some 500 eliminated theorems because they were so-called definitional theorems. The authors wrote those theorems because previously it had not been possible directly to refer to definitions.

The Readers are also requested to note that in the present issue we have changed the formats of certain operations. The operation symbol of the removal of n initial terms from a sequence has been changed from ^{}into ↑ (see [2] and [1]). Likewise, the operation symbol of the multiplication of real functions ⋄ has been removed (see [1]).

Roman Matuszewski

References

[1] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

[2] Jarosław Kotowicz. Properties of real functions. Formalized Mathematics, 1(4):781–786, 1990.

2

FORMALIZED MATHEMATICS Vol.2,No.1, January–February 1991 Université Catholique de Louvain

Construction of Rings and Left-, Right-, and Bi-Modules over a Ring

Michał Muzalewski^{1} Warsaw University Białystok

Summary. Definitions of some classes of rings and left-, right-, and bi-modules over a ring and some elementary theorems on rings and skew fields.

MML Identifier: VECTSP 2.

The articles [9], [8], [11], [3], [1], [10], [7], [4], [2], [5], and [6] provide the notation and terminology for this paper. In the sequel F_{1} will denote a field structure. Let us consider F_{1}. A scalar of F_{1} is an element of the carrier of F_{1}.

In the sequel x, y will denote scalars of F_{1}. Let us consider F_{1}, x, y. The functor x − y yields a scalar of F_{1} and is defined as follows: (Def.1) x − y = x + (−y).

In the sequel F denotes a field. A field structure is called a ring if: (Def.2) Let x, y, z be scalars of it . Then

(i) x + y = y + x, (ii) (x + y) + z = x + (y + z), (iii) x + 0_{it} = x, (iv) x + (−x) = 0_{it},

(v) x · (1_{it}) = x, (vi) (1_{it}) · x = x, (vii) x · (y + z) = x · y + x · z, (viii) (y + z) · x = y · x + z · x.

The following proposition is true (1) The following conditions are equivalent:

1Supported by RPBP.III-24.C6.

(i) for all scalars x, y, z of F_{1} holds x+y = y+x and (x+y)+z = x+(y+z) and x + 0_{F_{1}} = x and x + (−x) = 0_{F_{1}} and x · (1_{F_{1}}) = x and (1_{F_{1}}) · x = x and x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x,

(ii) F_{1} is a ring. In the sequel R is a ring and x, y, z are scalars of R. Next we state several

propositions:

(2) x + y = y + x.

(3) (x + y) + z = x + (y + z).

(4) x + 0_{R} = x.

(5) x + (−x) = 0_{R}.

(6) x · (1_{R}) = x and (1_{R}) · x = x.

(7) x · (y + z) = x · y + x · z and (y + z) · x = y · x + z · x. A ring is called an associative ring if:

(Def.3) for all scalars x, y, z of it holds (x · y) · z = x · (y · z).

The following proposition is true

(8) For all scalars x, y, z of R holds (x · y) · z = x · (y · z) if and only if R is an associative ring.

In the sequel R will denote an associative ring and x, y, z will denote scalars of R. One can prove the following proposition

(9) (x · y) · z = x · (y · z). An associative ring is said to be a commutative ring if:

(Def.4) for all scalars x, y of it holds x · y = y · x.

One can prove the following proposition

(10) If for all scalars x, y of R holds x · y = y · x, then R is a commutative

ring. In the sequel R will denote a commutative ring and x, y will denote scalars

of R. The following proposition is true

(11) x · y = y · x.

A commutative ring is said to be an integral domain if: (Def.5) 0_{it} ≠ 1_{it} and for all scalars x, y of it such that x · y = 0_{it} holds x = 0_{it}

or y = 0_{it}. We now state two propositions:

(12) If 0_{R} ≠ 1_{R} and for all x, y such that x · y = 0_{R} holds x = 0_{R} or y = 0_{R}, then R is an integral domain.

(13) F is an integral domain.

In the sequel R denotes an integral domain and x, y denote scalars of R. The following propositions are true:

(14) 0_{R} ≠ 1_{R}.

(15) If x · y = 0_{R}, then x = 0_{R} or y = 0_{R}.

An associative ring is called a skew field if: (Def.6) for every scalar x of it holds if x ≠ 0_{it}, then there exists a scalar y of it

such that x · y = 1_{it} but 0_{it} ≠ 1_{it}. In the sequel R denotes an associative ring. The following proposition is true

(16) If for every scalar x of R holds if x ≠ 0_{R}, then there exists a scalar y of

R such that x · y = 1_{R} but 0_{R} ≠ 1_{R}, then R is a skew field. In the sequel S_{1} will denote a skew field and x, y will denote scalars of S_{1}.

The following propositions are true:

(17) If x ≠ 0_{S_{1}}, then there exists y such that x · y = 1_{S_{1}}.

(18) 0_{S_{1}} ≠ 1_{S_{1}}.

(19) F is a skew field.

We see that the field is a skew field.

In the sequel R is a ring and x, y, z are scalars of R. Next we state a number of propositions:

(20) x − y = x + (−y).

(21) −0_{R} = 0_{R}.

(22) x + y = z if and only if x = z − y but x + y = z if and only if y = z − x.

(23) x − 0_{R} = x and 0_{R} − x = −x.

(24) If x + y = x + z, then y = z but if x + y = z + y, then x = z.

(25) −(x + y) = (−x) + (−y).

(26) x · 0_{R} = 0_{R} and 0_{R} · x = 0_{R}.

(27) −(−x) = x.

(28) (−x) · y = −x · y.

(29) x · (−y) = −x · y.

(30) (−x) · (−y) = x · y.

(31) x · (y − z) = x · y − x · z.

(32) (x − y) · z = x · z − y · z.

(33) (x + y) − z = x + (y − z).

(34) x = 0_{R} if and only if −x = 0_{R}.

(35) x − (y + z) = (x − y) − z.

(36) x − (y − z) = (x − y) + z.

(37) x − x = 0_{R} and (−x) + x = 0_{R}.

(38) For every x, y there exists z such that x = y + z and x = z + y. In the sequel S_{1} denotes a skew field and x, y, z denote scalars of S_{1}. We

now state four propositions:

(39) If x · y = 1_{S_{1}}, then x ≠ 0_{S_{1}} and y ≠ 0_{S_{1}}.

(40) If x ≠ 0_{S_{1}}, then there exists y such that y · x = 1_{S_{1}}.

(41) If x · y = 1_{S_{1}}, then y · x = 1_{S_{1}}.

(42) If x · y = x · z and x ≠ 0_{S_{1}}, then y = z.

Let us consider S_{1}, x. Let us assume that x ≠ 0_{S_{1}}. The functor x^{−1} yielding a scalar of S_{1} is defined by: (Def.7) x · (x^{−1}) = 1_{S_{1}}.

Let us consider S_{1}, x, y. Let us assume that y ≠ 0_{S_{1}}. The functor \frac{x}{y} yielding a scalar of S_{1} is defined by: (Def.8) \frac{x}{y} = x · y^{−1}.

One can prove the following propositions:

(43) If x ≠ 0_{S_{1}}, then x · x^{−1} = 1_{S_{1}} and x^{−1} · x = 1_{S_{1}}.

(44) If y ≠ 0_{S_{1}}, then \frac{x}{y} = x · y^{−1}.

(45) If x · y = 1_{S_{1}}, then x = y^{−1} and y = x^{−1}.

(46) If x ≠ 0_{S_{1}} and y ≠ 0_{S_{1}}, then x^{−1} · y^{−1} = (y · x)^{−1}.

(47) If x · y = 0_{S_{1}}, then x = 0_{S_{1}} or y = 0_{S_{1}}.

(48) If x ≠ 0_{S_{1}}, then x^{−1} ≠ 0_{S_{1}}.

(49) If x ≠ 0_{S_{1}}, then (x^{−1})^{−1} = x.

(50) If x ≠ 0_{S_{1}}, then \frac{1_{S_{1}}}{x} = x^{−1} and \frac{1_{S_{1}}}{x^{−1}} = x.

(51) If x ≠ 0_{S_{1}}, then x · \frac{1_{S_{1}}}{x} = 1_{S_{1}} and \frac{1_{S_{1}}}{x} · x = 1_{S_{1}}.

(52) If x ≠ 0_{S_{1}}, then \frac{x}{x} = 1_{S_{1}}.

(53) If y ≠ 0_{S_{1}} and z ≠ 0_{S_{1}}, then \frac{x}{y} = \frac{x·z}{y·z}.

(54) If y ≠ 0_{S_{1}}, then −\frac{x}{y} = \frac{−x}{y} and \frac{x}{−y} = −\frac{x}{y}.

(55) If z ≠ 0_{S_{1}}, then \frac{x}{z} + \frac{y}{z} = \frac{x+y}{z} and \frac{x}{z} − \frac{y}{z} = \frac{x−y}{z}.

(56) If y ≠ 0_{S_{1}} and z ≠ 0_{S_{1}}, then \frac{x}{\frac{y}{z}} = \frac{x·z}{y}.

(57) If y ≠ 0_{S_{1}}, then \frac{x}{y} · y = x.

Let us consider F_{1}. We consider left module structures over F_{1} which are systems

⟨a carrier, a left multiplication⟩, where the carrier is an Abelian group and the left multiplication is a function from [: the carrier of F_{1}, the carrier of the carrier :] into the carrier of the carrier.

In the sequel L_{1} denotes a left module structure over F_{1}. We now define two new modes. Let us consider F_{1}, L_{1}. A scalar of L_{1} is a scalar of F_{1}.

A vector of L_{1} is an element of the carrier of L_{1}.

Let us consider F_{1}. We consider right module structures over F_{1} which are systems

⟨a carrier, a right multiplication⟩, where the carrier is an Abelian group and the right multiplication is a function from [: the carrier of the carrier, the carrier of F_{1} :] into the carrier of the carrier.

In the sequel R_{1} will denote a right module structure over F_{1}. We now define two new modes. Let us consider F_{1}, R_{1}. A scalar of R_{1} is a scalar of F_{1}.

A vector of R_{1} is an element of the carrier of R_{1}.

Let us consider F_{1}. We consider bimodule structures over F_{1} which are systems

⟨a carrier, a left multiplication, a right multiplication⟩, where the carrier is an Abelian group, the left multiplication is a function from [: the carrier of F_{1}, the carrier of the carrier :] into the carrier of the carrier, and the right multiplication is a function from [: the carrier of the carrier, the carrier of F_{1} :] into the carrier of the carrier.

In the sequel B_{1} will denote a bimodule structure over F_{1}. We now define two new modes. Let us consider F_{1}, B_{1}. A scalar of B_{1} is a scalar of F_{1}.

A vector of B_{1} is an element of the carrier of B_{1}.

In the sequel R is a ring. Let us consider R. The functor AbGr(R) yields an Abelian group and is defined by: (Def.9) AbGr(R) = ⟨ the carrier of R, the addition of R, the reverse-map of R,

the zero of R⟩.

Next we state the proposition

(58) AbGr(R) = ⟨ the carrier of R, the addition of R, the reverse-map of R,

the zero of R⟩. Let us consider R. The functor LeftModMult(R) yielding a function from [:

the carrier of R, the carrier of AbGr(R) :] into the carrier of AbGr(R) is defined as follows:

(Def.10) LeftModMult(R) = the multiplication of R.

Next we state the proposition

(59) LeftModMult(R) = the multiplication of R.

Let us consider R. The functor LeftMod(R) yielding a left module structure over R is defined as follows:

(Def.11) LeftMod(R) = ⟨AbGr(R), LeftModMult(R)⟩.

We now state the proposition

(60) LeftMod(R) = ⟨AbGr(R), LeftModMult(R)⟩.

In the sequel V will be a left module structure over R. Let us consider R, V, and let x be a scalar of R, and let v be a vector of V. The functor x · v yielding a vector of V is defined as follows:

(Def.12) for every scalar x′ of V such that x′ = x holds x · v = (the left multiplication of V)(x′, v).

The following proposition is true (62)^{2} For every V being a left module structure over R and for every scalar

x of R and for every vector v of V and for every scalar x′ of V such that x′ = x holds x · v = (the left multiplication of V)(x′, v).

Let us consider R. The functor RightModMult(R) yields a function from [: the carrier of AbGr(R), the carrier of R:] into the carrier of AbGr(R) and is defined as follows:

2The proposition (61) was either repeated or obvious.

(Def.13) RightModMult(R) = the multiplication of R.

We now state the proposition

(63) RightModMult(R) = the multiplication of R.

Let us consider R. The functor RightMod(R) yielding a right module structure over R is defined as follows:

(Def.14) RightMod(R) = ⟨AbGr(R), RightModMult(R)⟩.

We now state the proposition

(64) RightMod(R) = ⟨AbGr(R), RightModMult(R)⟩.

In the sequel V will denote a right module structure over R. Let us consider R, V, and let x be a scalar of R, and let v be a vector of V. The functor v · x yielding a vector of V is defined as follows:

(Def.15) for every scalar x′ of V such that x′ = x holds v · x = (the right multiplication of V)(v, x′).

We now state the proposition (66)^{3} For every V being a right module structure over R and for every scalar

x of R and for every vector v of V and for every scalar x′ of V such that x′ = x holds v · x = (the right multiplication of V)(v, x′).

Let us consider R. The functor BiMod(R) yielding a bimodule structure over R is defined as follows:

(Def.16) BiMod(R) = ⟨AbGr(R), LeftModMult(R), RightModMult(R)⟩.

The following proposition is true

(67) BiMod(R) = ⟨AbGr(R), LeftModMult(R), RightModMult(R)⟩. In the sequel V is a bimodule structure over R. Let us consider R, V, and

let x be a scalar of R, and let v be a vector of V. The functor x · v yields a vector of V and is defined as follows:

(Def.17) for every scalar x′ of V such that x′ = x holds x · v = (the left multiplication of V)(x′, v).

One can prove the following proposition (69)^{4} For every V being a bimodule structure over R and for every scalar x

of R and for every vector v of V and for every scalar x′ of V such that x′ = x holds x · v = (the left multiplication of V)(x′, v).

Let us consider R, V, and let x be a scalar of R, and let v be a vector of V. The functor v · x yields a vector of V and is defined by:

(Def.18) for every scalar x′ of V such that x′ = x holds v · x = (the right multiplication of V)(v, x′).

The following proposition is true (70) For every V being a bimodule structure over R and for every scalar x

3The proposition (65) was either repeated or obvious.

4The proposition (68) was either repeated or obvious.

of R and for every vector v of V and for every scalar x′ of V such that x′ = x holds v · x = (the right multiplication of V)(v, x′).

In the sequel R will denote an associative ring. Next we state the proposition (71) Let x, y be scalars of R. Let v, w be vectors of LeftMod(R). Then

x·(v +w) = x·v +x·w and (x+y)·v = x·v +y ·v and (x·y)·v = x·(y ·v) and (1_{R}) · v = v.

Let us consider R. A left module structure over R is called a left module over R if:

(Def.19) Let x, y be scalars of R. Let v, w be vectors of it . Then x · (v + w) = x · v + x · w and (x + y) · v = x · v + y · v and (x · y) · v = x · (y · v) and (1_{R}) · v = v.

We now state the proposition

(72) Let V be a left module structure over R. Then the following conditions

are equivalent:

(i) for all scalars x, y of R and for all vectors v, w of V holds x · (v + w) = x · v + x · w and (x + y) · v = x · v + y · v and (x · y) · v = x · (y · v) and (1_{R}) · v = v,

(ii) V is a left module over R. Let us consider R. Then LeftMod(R) is a left module over R. For simplicity we adopt the following rules: R is an associative ring, x, y are

scalars of R, L_{2} is a left module over R, and v, w are vectors of L_{2}. We now state several propositions:

(73) x · (v + w) = x · v + x · w.

(74) (x + y) · v = x · v + y · v.

(75) (x · y) · v = x · (y · v).

(76) (1_{R}) · v = v.

(77) Let x, y be scalars of R. Let v, w be vectors of RightMod(R). Then (v +w)·x = v ·x+w ·x and v ·(x+y) = v ·x+v ·y and v ·(y ·x) = (v ·y)·x and v · (1_{R}) = v.

Let us consider R. A right module structure over R is said to be a right module over R if:

(Def.20) Let x, y be scalars of R. Let v, w be vectors of it . Then (v + w) · x = v · x + w · x and v · (x + y) = v · x + v · y and v · (y · x) = (v · y) · x and v · (1_{R}) = v.

The following proposition is true

(78) Let V be a right module structure over R. Then the following conditions

are equivalent:

(i) for all scalars x, y of R and for all vectors v, w of V holds (v + w) · x = v · x + w · x and v · (x + y) = v · x + v · y and v · (y · x) = (v · y) · x and v · (1_{R}) = v,

(ii) V is a right module over R.

Let us consider R. Then RightMod(R) is a right module over R.

For simplicity we follow the rules: R is an associative ring, x, y are scalars of R, R_{2} is a right module over R, and v, w are vectors of R_{2}. We now state four propositions:

(79) (v + w) · x = v · x + w · x.

(80) v · (x + y) = v · x + v · y.

(81) v · (y · x) = (v · y) · x.

(82) v · (1_{R}) = v.

Let us consider R. A bimodule structure over R is said to be a bimodule over R if:

(Def.21) Let x, y be scalars of R. Let v, w be vectors of it . Then (i) x · (v + w) = x · v + x · w, (ii) (x + y) · v = x · v + y · v, (iii) (x · y) · v = x · (y · v), (iv) (1_{R}) · v = v,

(v) (v + w) · x = v · x + w · x, (vi) v · (x + y) = v · x + v · y, (vii) v · (y · x) = (v · y) · x, (viii) v · (1_{R}) = v, (ix) x · (v · y) = (x · v) · y. Next we state two propositions:

(83) Let V be a bimodule structure over R. Then the following conditions

are equivalent:

(i) for all scalars x, y of R and for all vectors v, w of V holds x · (v + w) = x · v + x · w and (x + y) · v = x · v + y · v and (x · y) · v = x · (y · v) and (1_{R}) · v = v and (v + w) · x = v · x + w · x and v · (x + y) = v · x + v · y and v · (y · x) = (v · y) · x and v · (1_{R}) = v and x · (v · y) = (x · v) · y,

(ii) V is a bimodule over R.

(84) BiMod(R) is a bimodule over R.

Let us consider R. Then BiMod(R) is a bimodule over R.

For simplicity we follow the rules: R will be an associative ring, x, y will be scalars of R, R_{2} will be a bimodule over R, and v, w will be vectors of R_{2}. The following propositions are true:

(85) x · (v + w) = x · v + x · w.

(86) (x + y) · v = x · v + y · v.

(87) (x · y) · v = x · (y · v).

(88) (1_{R}) · v = v.

(89) (v + w) · x = v · x + w · x.

(90) v · (x + y) = v · x + v · y.

(91) v · (y · x) = (v · y) · x.

(92) v · (1_{R}) = v.

Construction of Rings and Left-, Right-, and … 11

(93) x · (v · y) = (x · v) · y.

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

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

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

[8] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

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

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

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

Received June 20, 1990 12

Desargues Theorem In Projective 3-Space

Eugeniusz Kusak^{1} Warsaw University Białystok

Summary. Proof of the Desargues theorem in Fanoian projective at least 3-dimensional space.

MML Identifier: PROJDES1.

The notation and terminology used in this paper are introduced in the following papers: [5], [1], [2], [3], and [4]. We follow a convention: F_{1} will be an at least 3-dimensional projective space defined in terms of collinearity and a, a′, b, b′, c, c′, d, d′, o, p, q, r, s, t, u, x will be elements of the points of F_{1}. One can prove the following propositions:

(1) If a, b and c are collinear, then b, c and a are collinear and c, a and b are collinear and b, a and c are collinear and a, c and b are collinear and c, b and a are collinear.

(2) If a ≠ b and a, b and c are collinear and a, b and d are collinear, then a, c and d are collinear.

(3) If p ≠ q and a, b and p are collinear and a, b and q are collinear and p, q and r are collinear, then a, b and r are collinear.

(4) If p ≠ q, then there exists r such that p, q and r are not collinear.

(5) There exist q, r such that p, q and r are not collinear.

(6) If a, b and c are not collinear and a, b and b′ are collinear and a ≠ b′, then a, b′ and c are not collinear.

(7) If a, b and c are not collinear and a, b and d are collinear and a, c and d are collinear, then a = d.

(8) If o, a and d are not collinear and o, d and d′ are collinear and a, d and s are collinear and d ≠ d′ and a′, d′ and s are collinear and o, a and a′ are collinear and o ≠ a′, then s ≠ d.

1Supported by RPBP.III-24.C6.

Let us consider F_{1}, a, b, c, d. We say that a, b, c, d are coplanar if and only if: (Def.1) there exists an element x of the points of F_{1} such that a, b and x are

collinear and c, d and x are collinear.

One can prove the following propositions: (10)^{2} If a, b and c are collinear or b, c and d are collinear or c, d and a are

collinear or d, a and b are collinear, then a, b, c, d are coplanar.

(11) Suppose a, b, c, d are coplanar. Then b, c, d, a are coplanar and c, d, a, b are coplanar and d, a, b, c are coplanar and b, a, c, d are coplanar and c, b, d, a are coplanar and d, c, a, b are coplanar and a, d, b, c are coplanar and a, c, d, b are coplanar and b, d, a, c are coplanar and c, a, b, d are coplanar and d, b, c, a are coplanar and c, a, d, b are coplanar and d, b, a, c are coplanar and a, c, b, d are coplanar and b, d, c, a are coplanar and a, b, d, c are coplanar and a, d, c, b are coplanar and b, c, a, d are coplanar and b, a, d, c are coplanar and c, b, a, d are coplanar and c, d, b, a are coplanar and d, a, c, b are coplanar and d, c, b, a are coplanar.

(12) If a, b and c are not collinear and a, b, c, p are coplanar and a, b, c, q are coplanar and a, b, c, r are coplanar and a, b, c, s are coplanar, then p, q, r, s are coplanar.

(13) If p, q and r are not collinear and a, b, c, p are coplanar and a, b, c, r are coplanar and a, b, c, q are coplanar and p, q, r, s are coplanar, then a, b, c, s are coplanar.

(14) If p ≠ q and p, q and r are collinear and a, b, c, p are coplanar and a, b, c, q are coplanar, then a, b, c, r are coplanar.

(15) If a, b and c are not collinear and a, b, c, p are coplanar and a, b, c, q are coplanar and a, b, c, r are coplanar and a, b, c, s are coplanar, then there exists x such that p, q and x are collinear and r, s and x are collinear.

(16) There exist a, b, c, d such that a, b, c, d are not coplanar.

(17) If p, q and r are not collinear, then there exists s such that p, q, r, s are not coplanar.

(18) If a = b or a = c or b = c or a = d or b = d or d = c, then a, b, c, d are coplanar.

(19) If a, b, c, o are not coplanar and o, a and a′ are collinear and a ≠ a′, then a, b, c, a′ are not coplanar.

(20) Suppose that (i) a, b and c are not collinear, (ii) a′, b′ and c′ are not collinear, (iii) a, b, c, p are coplanar, (iv) a, b, c, q are coplanar,

(v) a, b, c, r are coplanar, (vi) a′, b′, c′, p are coplanar, (vii) a′, b′, c′, q are coplanar, (viii) a′, b′, c′, r are coplanar, (ix) a, b, c, a′ are not coplanar.

2The proposition (9) was either repeated or obvious.

Then p, q and r are collinear.

(21) Suppose that (i) a ≠ a′, (ii) o, a and a′ are collinear, (iii) a, b, c, o are not coplanar, (iv) a′, b′ and c′ are not collinear,

(v) a, b and p are collinear, (vi) a′, b′ and p are collinear, (vii) b, c and q are collinear, (viii) b′, c′ and q are collinear, (ix) a, c and r are collinear,

(x) a′, c′ and r are collinear.

Then p, q and r are collinear.

(22) If a, b, c, d are not coplanar and a, b, c, o are coplanar and a, b and o are not collinear, then a, b, d, o are not coplanar.

(23) If a, b, c, o are not coplanar and o, a and a′ are collinear and o, b and b′ are collinear and o, c and c′ are collinear and o ≠ a′ and o ≠ b′ and o ≠ c′, then a′, b′ and c′ are not collinear and a′, b′, c′, o are not coplanar.

(24) Suppose that (i) a, b, c, o are coplanar, (ii) a, b, c, d are not coplanar, (iii) a, b, d, o are not coplanar, (iv) b, c, d, o are not coplanar,

(v) a, c, d, o are not coplanar, (vi) o, d and d′ are collinear, (vii) o, a and a′ are collinear, (viii) o, b and b′ are collinear, (ix) o, c and c′ are collinear,

(x) a, d and s are collinear, (xi) a′, d′ and s are collinear, (xii) b, d and t are collinear, (xiii) b′, d′ and t are collinear, (xiv) c, d and u are collinear, (xv) o ≠ a′, (xvi) o ≠ d′, (xvii) d ≠ d′, (xviii) o ≠ b′.

Then s, t and u are not collinear. Let us consider F_{1}, o, a, b, c. We say that o, a, b, and c constitute a

quadrangle if and only if:

(Def.2) a, b and c are not collinear and o, a and b are not collinear and o, b and c are not collinear and o, c and a are not collinear.

The following propositions are true: (26)^{3} Suppose that

(i) o, a and b are not collinear, (ii) o, b and c are not collinear, (iii) o, a and c are not collinear, (iv) o, a and a′ are collinear,

(v) o, b and b′ are collinear, (vi) o, c and c′ are collinear, (vii) a, b and p are collinear, (viii) a′, b′ and p are collinear, (ix) a ≠ a′,

(x) b, c and r are collinear, (xi) b′, c′ and r are collinear, (xii) a, c and q are collinear, (xiii) b ≠ b′, (xiv) a′, c′ and q are collinear, (xv) o ≠ a′, (xvi) o ≠ b′, (xvii) o ≠ c′.

Then r, q and p are collinear.

(27) For every at least 3-dimensional projective space C_{1} defined in terms of collinearity holds C_{1} is a Desarguesian at least 3-dimensional projective space defined in terms of collinearity.

We see that the at least 3-dimensional projective space defined in terms of collinearity is a Desarguesian at least 3-dimensional projective space defined in terms of collinearity.

References

[1] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part I. Formalized Mathematics, 1(4):767–776, 1990.

[2] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part II. Formalized Mathematics, 1(5):901–907, 1990.

[3] Wojciech Leoṅczuk and Krzysztof Prażmowski. Projective spaces - part III. Formalized Mathematics, 1(5):909–918, 1990.

[4] Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces - part IV. Formalized Mathematics, 1(5):919–927, 1990.

[5] Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657–659, 1990.

Received August 13, 1990

3The proposition (25) was either repeated or obvious.

The Limit of a Real Function at Infinity

Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. We introduce the halflines (open and closed), real sequences divergent to infinity (plus and minus) and the proper and improper limit of a real function at infinity. We prove basic properties of halflines, sequeces divergent to infinity and the limit of function at infinity.

MML Identifier: LIMFUNC1.

The articles [14], [4], [1], [2], [12], [10], [5], [6], [11], [15], [3], [7], [8], [13], and [9] provide the terminology and notation for this paper. For simplicity we follow a convention: r, r_{1}, r_{2}, g, g_{1}, g_{2} are real numbers, X is a subset of , n, m, k are natural numbers, s_{1}, s_{2}, s_{3} are sequences of real numbers, and f, f_{1}, f_{2} are partial functions from to . Let us consider n, m. Then max(n, m) is a natural number.

We now state four propositions:

(1) If 0 ≤ r_{1} and r_{1} < r_{2} and 0 < g_{1} and g_{1} ≤ g_{2}, then r_{1} · g_{1} < r_{2} · g_{2}.

(2) If r ≠ 0, then (−r)^{−1} = −r^{−1}.

(3) If r_{1} < r_{2} and r_{2} < 0 and 0 < g, then \frac{g}{r_{2}} < \frac{g}{r_{1}} .

(4) If r < 0, then r^{−1} < 0.

Let us consider r. We introduce the functor ]−∞, r[ as a synonym of HL(r). We now define three new functors. Let us consider r. The functor ]−∞, r]

yielding a subset of is defined as follows: (Def.1) ]−∞, r] = {g : g ≤ r}. The functor [r, +∞[ yields a subset of and is defined as follows: (Def.2) [r, +∞[ = {g : r ≤ g}. The functor ]r, +∞[ yielding a subset of is defined by: (Def.3) ]r, +∞[ = {g : r < g}.

1Supported by RPBP.III-24.C8

One can prove the following propositions:

(5) X = ]−∞, r] if and only if X = {g : g ≤ r}.

(6) X = [r, +∞[ if and only if X = {g : r ≤ g}.

(7) X = ]r, +∞[ if and only if X = {g : r < g}.

(8) If r_{1} ≤ r_{2}, then ]r_{2}, +∞[ ⊆ ]r_{1}, +∞[.

(9) If r_{1} ≤ r_{2}, then [r_{2}, +∞[ ⊆ [r_{1}, +∞[.

(10) ]r, +∞[ ⊆ [r, +∞[.

(11) ]r, g[ ⊆ ]r, +∞[.

(12) [r, g] ⊆ [r, +∞[.

(13) If r_{1} ≤ r_{2}, then ]−∞, r_{1}[ ⊆ ]−∞, r_{2}[.

(14) If r_{1} ≤ r_{2}, then ]−∞, r_{1}] ⊆ ]−∞, r_{2}].

(15) ]−∞, r[ ⊆ ]−∞, r].

(16) ]g, r[ ⊆ ]−∞, r[.

(17) [g, r] ⊆ ]−∞, r].

(18) ]−∞, r[ ∩]g, +∞[ = ]g, r[.

(19) ]−∞, r] ∩ [g, +∞[ = [g, r].

(20) If r ≤ r_{1}, then ]r_{1}, r_{2}[ ⊆ ]r, +∞[ and [r_{1}, r_{2}] ⊆ [r, +∞[.

(21) If r < r_{1}, then [r_{1}, r_{2}] ⊆ ]r, +∞[.

(22) If r_{2} ≤ r, then ]r_{1}, r_{2}[ ⊆ ]−∞, r[ and [r_{1}, r_{2}] ⊆ ]−∞, r].

(23) If r_{2} < r, then [r_{1}, r_{2}] ⊆ ]−∞, r[.

(24) \]r, +∞[ = ]−∞, r] and \[r, +∞[ = ]−∞, r[ and \]−∞, r[ = [r, +∞[ and \]−∞, r] = ]r, +∞[.

(25) \]r_{1}, r_{2}[ = ]−∞, r_{1}] ∪ [r_{2}, +∞[ and \ [r_{1}, r_{2}] = ]−∞, r_{1}[ ∪]r_{2}, +∞[.

(26) If s_{1} is non-decreasing, then s_{1} is lower bounded but if s_{1} is nonincreasing, then s_{1} is upper bounded.

(27) If s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0 and s_{1} is nondecreasing, then for every n holds s_{1}(n) < 0.

(28) If s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0 and s_{1} is nonincreasing, then for every n holds 0 < s_{1}(n).

(29) If s_{1} is convergent and 0 < lim s_{1}, then there exists n such that for every m such that n ≤ m holds 0 < s_{1}(m).

(30) If s_{1} is convergent and 0 < lim s_{1}, then there exists n such that for

every m such that n ≤ m holds \frac{lim s_{1}}{2} < s_{1}(m). We now define two new predicates. Let us consider s_{1}. We say that s_{1} is

divergent to +∞ if and only if: (Def.4) for every r there exists n such that for every m such that n ≤ m holds

r < s_{1}(m). We say that s_{1} is divergent to −∞ if and only if: (Def.5) for every r there exists n such that for every m such that n ≤ m holds

s_{1}(m) < r.

Next we state a number of propositions: (33)^{2} If s_{1} is divergent to +∞ or s_{1} is divergent to −∞, then there exists n

such that for every m such that n ≤ m holds s_{1} ↑ m is non-zero.

(34) If s_{1} ↑ k is divergent to +∞, then s_{1} is divergent to +∞ but if s_{1} ↑ k is divergent to −∞, then s_{1} is divergent to −∞.

(35) If s_{2} is divergent to +∞ and s_{3} is divergent to +∞, then s_{2} + s_{3} is divergent to +∞.

(36) If s_{2} is divergent to +∞ and s_{3} is lower bounded, then s_{2}+s_{3} is divergent to +∞.

(37) If s_{2} is divergent to +∞ and s_{3} is divergent to +∞, then s_{2}s_{3} is divergent to +∞.

(38) If s_{2} is divergent to −∞ and s_{3} is divergent to −∞, then s_{2} + s_{3} is divergent to −∞.

(39) If s_{2} is divergent to −∞ and s_{3} is upper bounded, then s_{2} + s_{3} is divergent to −∞.

(40) If s_{1} is divergent to +∞ and r > 0, then rs_{1} is divergent to +∞ but if s_{1} is divergent to +∞ and r < 0, then rs_{1} is divergent to −∞ but if s_{1} is divergent to +∞ and r = 0, then rng(rs_{1}) = {0} and rs_{1} is constant.

(41) If s_{1} is divergent to −∞ and r > 0, then rs_{1} is divergent to −∞ but if s_{1} is divergent to −∞ and r < 0, then rs_{1} is divergent to +∞ but if s_{1} is divergent to −∞ and r = 0, then rng(rs_{1}) = {0} and rs_{1} is constant.

(42) If s_{1} is divergent to +∞, then −s_{1} is divergent to −∞ but if s_{1} is divergent to −∞, then −s_{1} is divergent to +∞.

(43) If s_{1} is lower bounded and s_{2} is divergent to −∞, then s_{1}−s_{2} is divergent to +∞.

(44) If s_{1} is upper bounded and s_{2} is divergent to +∞, then s_{1} − s_{2} is divergent to −∞.

(45) If s_{1} is divergent to +∞ and s_{2} is convergent, then s_{1} + s_{2} is divergent to +∞.

(46) If s_{1} is divergent to −∞ and s_{2} is convergent, then s_{1} + s_{2} is divergent to −∞.

(47) If for every n holds s_{1}(n) = n, then s_{1} is divergent to +∞.

(48) If for every n holds s_{1}(n) = −n, then s_{1} is divergent to −∞.

(49) If s_{2} is divergent to +∞ and there exists r such that r > 0 and for every n holds s_{3}(n) ≥ r, then s_{2}s_{3} is divergent to +∞.

(50) If s_{2} is divergent to −∞ and there exists r such that 0 < r and for every n holds s_{3}(n) ≥ r, then s_{2}s_{3} is divergent to −∞.

(51) If s_{2} is divergent to −∞ and s_{3} is divergent to −∞, then s_{2}s_{3} is divergent to +∞.

2The propositions (31)–(32) were either repeated or obvious.

(52) If s_{1} is divergent to +∞ or s_{1} is divergent to −∞, then |s_{1}| is divergent to +∞.

(53) If s_{1} is divergent to +∞ and s_{2} is a subsequence of s_{1}, then s_{2} is divergent to +∞.

(54) If s_{1} is divergent to −∞ and s_{2} is a subsequence of s_{1}, then s_{2} is divergent to −∞.

(55) If s_{2} is divergent to +∞ and s_{3} is convergent and 0 < lim s_{3}, then s_{2}s_{3} is divergent to +∞.

(56) If s_{1} is non-decreasing and s_{1} is not upper bounded, then s_{1} is divergent to +∞.

(57) If s_{1} is non-increasing and s_{1} is not lower bounded, then s_{1} is divergent to −∞.

(58) If s_{1} is increasing and s_{1} is not upper bounded, then s_{1} is divergent to +∞.

(59) If s_{1} is decreasing and s_{1} is not lower bounded, then s_{1} is divergent to −∞.

(60) If s_{1} is monotone, then s_{1} is convergent or s_{1} is divergent to +∞ or s_{1} is divergent to −∞.

(61) If s_{1} is divergent to +∞ or s_{1} is divergent to −∞ but s_{1} is non-zero,

then s_{1}^{−1} is convergent and lim s_{1}^{−1} = 0. Next we state several propositions:

(62) If s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0 and there exists k such that for every n such that k ≤ n holds 0 < s_{1}(n), then s_{1}^{−1} is divergent to +∞.

(63) If s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0 and there exists k such that for every n such that k ≤ n holds s_{1}(n) < 0, then s_{1}^{−1} is divergent to −∞.

(64) If s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0 and s_{1} is nondecreasing, then s_{1}^{−1} is divergent to −∞.

(65) If s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0 and s_{1} is nonincreasing, then s_{1}^{−1} is divergent to +∞.

(66) If s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0 and s_{1} is increasing, then s_{1}^{−1} is divergent to −∞.

(67) If s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0 and s_{1} is decreasing, then s_{1}^{−1} is divergent to +∞.

(68) If s_{2} is bounded but s_{3} is divergent to +∞ or s_{3} is divergent to −∞ and s_{3} is non-zero, then \frac{s_{2}}{s_{3}} is convergent and lim \frac{s_{2}}{s_{3}} = 0.

(69) If s_{1} is divergent to +∞ and for every n holds s_{1}(n) ≤ s_{2}(n), then s_{2} is divergent to +∞.

(70) If s_{1} is divergent to −∞ and for every n holds s_{2}(n) ≤ s_{1}(n), then s_{2} is divergent to −∞.

We now define several new predicates. Let us consider f. We say that f is convergent in +∞ if and only if: (Def.6) for every r there exists g such that r < g and g ∈ dom f and there exists

g such that for every s_{1} such that s_{1} is divergent to +∞ and rng s_{1} ⊆ dom f holds f · s_{1} is convergent and lim(f · s_{1}) = g.

We say that f is divergent in +∞ to +∞ if and only if: (Def.7) for every r there exists g such that r < g and g ∈ dom f and for every

s_{1} such that s_{1} is divergent to +∞ and rng s_{1} ⊆ dom f holds f · s_{1} is divergent to +∞.

We say that f is divergent in +∞ to −∞ if and only if: (Def.8) for every r there exists g such that r < g and g ∈ dom f and for every

s_{1} such that s_{1} is divergent to +∞ and rng s_{1} ⊆ dom f holds f · s_{1} is divergent to −∞.

We say that f is convergent in −∞ if and only if: (Def.9) for every r there exists g such that g < r and g ∈ dom f and there exists

g such that for every s_{1} such that s_{1} is divergent to −∞ and rng s_{1} ⊆ dom f holds f · s_{1} is convergent and lim(f · s_{1}) = g.

We say that f is divergent in −∞ to +∞ if and only if: (Def.10) for every r there exists g such that g < r and g ∈ dom f and for every

s_{1} such that s_{1} is divergent to −∞ and rng s_{1} ⊆ dom f holds f · s_{1} is divergent to +∞.

We say that f is divergent in −∞ to −∞ if and only if: (Def.11) for every r there exists g such that g < r and g ∈ dom f and for every

s_{1} such that s_{1} is divergent to −∞ and rng s_{1} ⊆ dom f holds f · s_{1} is divergent to −∞.

We now state a number of propositions: (77)^{3} f is convergent in +∞ if and only if for every r there exists g such

that r < g and g ∈ dom f and there exists g such that for every g_{1} such that 0 < g_{1} there exists r such that for every r_{1} such that r < r_{1} and r_{1} ∈ dom f holds |f(r_{1}) − g| < g_{1}.

(78) f is convergent in −∞ if and only if for every r there exists g such that g < r and g ∈ dom f and there exists g such that for every g_{1} such that 0 < g_{1} there exists r such that for every r_{1} such that r_{1} < r and r_{1} ∈ dom f holds |f(r_{1}) − g| < g_{1}.

(79) f is divergent in +∞ to +∞ if and only if for every r there exists g such that r < g and g ∈ dom f and for every g there exists r such that for every r_{1} such that r < r_{1} and r_{1} ∈ dom f holds g < f(r_{1}).

(80) f is divergent in +∞ to −∞ if and only if for every r there exists g such that r < g and g ∈ dom f and for every g there exists r such that for every r_{1} such that r < r_{1} and r_{1} ∈ dom f holds f(r_{1}) < g.

3The propositions (71)–(76) were either repeated or obvious.

(81) f is divergent in −∞ to +∞ if and only if for every r there exists g such that g < r and g ∈ dom f and for every g there exists r such that for every r_{1} such that r_{1} < r and r_{1} ∈ dom f holds g < f(r_{1}).

(82) f is divergent in −∞ to −∞ if and only if for every r there exists g such that g < r and g ∈ dom f and for every g there exists r such that for every r_{1} such that r_{1} < r and r_{1} ∈ dom f holds f(r_{1}) < g.

(83) If f_{1} is divergent in +∞ to +∞ and f_{2} is divergent in +∞ to +∞ and for every r there exists g such that r < g and g ∈ dom f_{1} ∩ dom f_{2}, then f_{1} + f_{2} is divergent in +∞ to +∞ and f_{1}f_{2} is divergent in +∞ to +∞.

(84) If f_{1} is divergent in +∞ to −∞ and f_{2} is divergent in +∞ to −∞ and for every r there exists g such that r < g and g ∈ dom f_{1} ∩ dom f_{2}, then f_{1} + f_{2} is divergent in +∞ to −∞ and f_{1}f_{2} is divergent in +∞ to +∞.

(85) If f_{1} is divergent in −∞ to +∞ and f_{2} is divergent in −∞ to +∞ and for every r there exists g such that g < r and g ∈ dom f_{1} ∩ dom f_{2}, then f_{1} + f_{2} is divergent in −∞ to +∞ and f_{1}f_{2} is divergent in −∞ to +∞.

(86) If f_{1} is divergent in −∞ to −∞ and f_{2} is divergent in −∞ to −∞ and for every r there exists g such that g < r and g ∈ dom f_{1} ∩ dom f_{2}, then f_{1} + f_{2} is divergent in −∞ to −∞ and f_{1}f_{2} is divergent in −∞ to +∞.

(87) If f_{1} is divergent in +∞ to +∞ and for every r there exists g such that r < g and g ∈ dom(f_{1} + f_{2}) and there exists r such that f_{2} is lower bounded on ]r, +∞[, then f_{1} + f_{2} is divergent in +∞ to +∞.

(88) If f_{1} is divergent in +∞ to +∞ and for every r there exists g such that r < g and g ∈ dom(f_{1}f_{2}) and there exist r, r_{1} such that 0 < r and for every g such that g ∈ dom f_{2} ∩ ]r_{1}, +∞[ holds r ≤ f_{2}(g), then f_{1}f_{2} is divergent in +∞ to +∞.

(89) If f_{1} is divergent in −∞ to +∞ and for every r there exists g such that g < r and g ∈ dom(f_{1} + f_{2}) and there exists r such that f_{2} is lower bounded on ]−∞, r[, then f_{1} + f_{2} is divergent in −∞ to +∞.

(90) If f_{1} is divergent in −∞ to +∞ and for every r there exists g such that g < r and g ∈ dom(f_{1}f_{2}) and there exist r, r_{1} such that 0 < r and for every g such that g ∈ dom f_{2} ∩ ]−∞, r_{1}[ holds r ≤ f_{2}(g), then f_{1}f_{2} is divergent in −∞ to +∞.

(91) If f is divergent in +∞ to +∞ and r > 0, then rf is divergent in +∞ to +∞ but if f is divergent in +∞ to +∞ and r < 0, then rf is divergent in +∞ to −∞ but if f is divergent in +∞ to −∞ and r > 0, then rf is divergent in +∞ to −∞ but if f is divergent in +∞ to −∞ and r < 0, then rf is divergent in +∞ to +∞.

(92) If f is divergent in −∞ to +∞ and r > 0, then rf is divergent in −∞ to +∞ but if f is divergent in −∞ to +∞ and r < 0, then rf is divergent in −∞ to −∞ but if f is divergent in −∞ to −∞ and r > 0, then rf is divergent in −∞ to −∞ but if f is divergent in −∞ to −∞ and r < 0, then rf is divergent in −∞ to +∞.

(93) If f is divergent in +∞ to +∞ or f is divergent in +∞ to −∞, then

|f| is divergent in +∞ to +∞.

(94) If f is divergent in −∞ to +∞ or f is divergent in −∞ to −∞, then |f| is divergent in −∞ to +∞.

(95) If there exists r such that f is non-decreasing on ]r, +∞[ and f is not upper bounded on ]r, +∞[ and for every r there exists g such that r < g and g ∈ dom f, then f is divergent in +∞ to +∞.

(96) If there exists r such that f is increasing on ]r, +∞[ and f is not upper bounded on ]r, +∞[ and for every r there exists g such that r < g and g ∈ dom f, then f is divergent in +∞ to +∞.

(97) If there exists r such that f is non-increasing on ]r, +∞[ and f is not lower bounded on ]r, +∞[ and for every r there exists g such that r < g and g ∈ dom f, then f is divergent in +∞ to −∞.

(98) If there exists r such that f is decreasing on ]r, +∞[ and f is not lower bounded on ]r, +∞[ and for every r there exists g such that r < g and g ∈ dom f, then f is divergent in +∞ to −∞.

(99) If there exists r such that f is non-increasing on ]−∞, r[ and f is not upper bounded on ]−∞, r[ and for every r there exists g such that g < r and g ∈ dom f, then f is divergent in −∞ to +∞.

(100) If there exists r such that f is decreasing on ]−∞, r[ and f is not upper bounded on ]−∞, r[ and for every r there exists g such that g < r and g ∈ dom f, then f is divergent in −∞ to +∞.

(101) If there exists r such that f is non-decreasing on ]−∞, r[ and f is not lower bounded on ]−∞, r[ and for every r there exists g such that g < r and g ∈ dom f, then f is divergent in −∞ to −∞.

The following propositions are true:

(102) If there exists r such that f is increasing on ]−∞, r[ and f is not lower bounded on ]−∞, r[ and for every r there exists g such that g < r and g ∈ dom f, then f is divergent in −∞ to −∞.

(103) Suppose f_{1} is divergent in +∞ to +∞ and for every r there exists g such that r < g and g ∈ dom f and there exists r such that dom f ∩]r, +∞[ ⊆ dom f_{1} ∩ ]r, +∞[ and for every g such that g ∈ dom f ∩ ]r, +∞[ holds f_{1}(g) ≤ f(g). Then f is divergent in +∞ to +∞.

(104) Suppose f_{1} is divergent in +∞ to −∞ and for every r there exists g such that r < g and g ∈ dom f and there exists r such that dom f ∩]r, +∞[ ⊆ dom f_{1} ∩ ]r, +∞[ and for every g such that g ∈ dom f ∩ ]r, +∞[ holds f(g) ≤ f_{1}(g). Then f is divergent in +∞ to −∞.

(105) Suppose f_{1} is divergent in −∞ to +∞ and for every r there exists g such that g < r and g ∈ dom f and there exists r such that dom f ∩]−∞, r[ ⊆ dom f_{1} ∩ ]−∞, r[ and for every g such that g ∈ dom f ∩ ]−∞, r[ holds f_{1}(g) ≤ f(g). Then f is divergent in −∞ to +∞.

(106) Suppose f_{1} is divergent in −∞ to −∞ and for every r there exists g such that g < r and g ∈ dom f and there exists r such that dom f ∩]−∞, r[ ⊆ dom f_{1} ∩ ]−∞, r[ and for every g such that g ∈ dom f ∩ ]−∞, r[ holds f(g) ≤ f_{1}(g). Then f is divergent in −∞ to −∞.

(107) If f_{1} is divergent in +∞ to +∞ and there exists r such that ]r, +∞[ ⊆ dom f ∩ dom f_{1} and for every g such that g ∈ ]r, +∞[ holds f_{1}(g) ≤ f(g), then f is divergent in +∞ to +∞.

(108) If f_{1} is divergent in +∞ to −∞ and there exists r such that ]r, +∞[ ⊆ dom f ∩ dom f_{1} and for every g such that g ∈ ]r, +∞[ holds f(g) ≤ f_{1}(g), then f is divergent in +∞ to −∞.

(109) If f_{1} is divergent in −∞ to +∞ and there exists r such that ]−∞, r[ ⊆ dom f ∩ dom f_{1} and for every g such that g ∈ ]−∞, r[ holds f_{1}(g) ≤ f(g), then f is divergent in −∞ to +∞.

(110) If f_{1} is divergent in −∞ to −∞ and there exists r such that ]−∞, r[ ⊆ dom f ∩ dom f_{1} and for every g such that g ∈ ]−∞, r[ holds f(g) ≤ f_{1}(g), then f is divergent in −∞ to −∞.

Let us consider f. Let us assume that f is convergent in +∞. The functor lim_{+∞} f yielding a real number is defined by:

(Def.12) for every s_{1} such that s_{1} is divergent to +∞ and rng s_{1} ⊆ dom f holds f · s_{1} is convergent and lim(f · s_{1}) = lim_{+∞} f.

Let us consider f. Let us assume that f is convergent in −∞. The functor lim_{−∞} f yields a real number and is defined by:

(Def.13) for every s_{1} such that s_{1} is divergent to −∞ and rng s_{1} ⊆ dom f holds f · s_{1} is convergent and lim(f · s_{1}) = lim_{−∞} f.

Next we state a number of propositions:

(111) If f is convergent in +∞, then lim_{+∞} f = g if and only if for every s_{1} such that s_{1} is divergent to +∞ and rng s_{1} ⊆ dom f holds f · s_{1} is convergent and lim(f · s_{1}) = g.

(112) If f is convergent in −∞, then lim_{−∞} f = g if and only if for every s_{1} such that s_{1} is divergent to −∞ and rng s_{1} ⊆ dom f holds f · s_{1} is convergent and lim(f · s_{1}) = g.

(113) If f is convergent in −∞, then lim_{−∞} f = g if and only if for every g_{1} such that 0 < g_{1} there exists r such that for every r_{1} such that r_{1} < r and r_{1} ∈ dom f holds |f(r_{1}) − g| < g_{1}.

(114) If f is convergent in +∞, then lim_{+∞} f = g if and only if for every g_{1} such that 0 < g_{1} there exists r such that for every r_{1} such that r < r_{1} and r_{1} ∈ dom f holds |f(r_{1}) − g| < g_{1}.

(115) If f is convergent in +∞, then rf is convergent in +∞ and lim_{+∞}(rf) = r · (lim_{+∞} f).

(116) If f is convergent in +∞, then −f is convergent in +∞ and lim_{+∞}(−f) = − lim_{+∞} f.

(117) If f_{1} is convergent in +∞ and f_{2} is convergent in +∞ and for every r there exists g such that r < g and g ∈ dom(f_{1} + f_{2}), then f_{1} + f_{2} is convergent in +∞ and lim_{+∞}(f_{1} + f_{2}) = lim_{+∞} f_{1} + lim_{+∞} f_{2}.

The Limit of a Real Function at Infinity 25

(118) If f_{1} is convergent in +∞ and f_{2} is convergent in +∞ and for every r there exists g such that r < g and g ∈ dom(f_{1} − f_{2}), then f_{1} − f_{2} is convergent in +∞ and lim_{+∞}(f_{1} − f_{2}) = lim_{+∞} f_{1} − lim_{+∞} f_{2}.

(119) If f is convergent in +∞ and f ^{−1} \{0\} = ∅ and lim_{+∞} f ≠ 0, then \frac{1}{f} is convergent in +∞ and lim_{+∞} \frac{1}{f} = (lim_{+∞} f)^{−1}.

(120) If f is convergent in +∞, then |f| is convergent in +∞ and lim_{+∞} |f| = | lim_{+∞} f|.

(121) If f is convergent in +∞ and lim_{+∞} f ≠ 0 and for every r there exists g such that r < g and g ∈ dom f and f(g) ≠ 0, then \frac{1}{f} is convergent in +∞ and lim_{+∞} \frac{1}{f} = (lim_{+∞} f)^{−1}.

(122) If f_{1} is convergent in +∞ and f_{2} is convergent in +∞ and for every r there exists g such that r < g and g ∈ dom(f_{1}f_{2}), then f_{1}f_{2} is convergent in +∞ and lim_{+∞}(f_{1}f_{2}) = (lim_{+∞} f_{1}) · (lim_{+∞} f_{2}).

(123) If f_{1} is convergent in +∞ and f_{2} is convergent in +∞ and lim_{+∞} f_{2} ≠ 0 and for every r there exists g such that r < g and g ∈ dom \frac{f_{1}}{f_{2}} , then \frac{f_{1}}{f_{2}} is convergent in +∞ and lim_{+∞} \frac{f_{1}}{f_{2}} = \frac{lim_{+∞} f_{1}}{lim_{+∞} f_{2}} .

(124) If f is convergent in −∞, then rf is convergent in −∞ and lim_{−∞}(rf) = r · (lim_{−∞} f).

(125) If f is convergent in −∞, then −f is convergent in −∞ and lim_{−∞}(−f) = − lim_{−∞} f.

(126) If f_{1} is convergent in −∞ and f_{2} is convergent in −∞ and for every r there exists g such that g < r and g ∈ dom(f_{1} + f_{2}), then f_{1} + f_{2} is convergent in −∞ and lim_{−∞}(f_{1} + f_{2}) = lim_{−∞} f_{1} + lim_{−∞} f_{2}.

(127) If f_{1} is convergent in −∞ and f_{2} is convergent in −∞ and for every r there exists g such that g < r and g ∈ dom(f_{1} − f_{2}), then f_{1} − f_{2} is convergent in −∞ and lim_{−∞}(f_{1} − f_{2}) = lim_{−∞} f_{1} − lim_{−∞} f_{2}.

(128) If f is convergent in −∞ and f ^{−1} \{0\} = ∅ and lim_{−∞} f ≠ 0, then \frac{1}{f} is convergent in −∞ and lim_{−∞} \frac{1}{f} = (lim_{−∞} f)^{−1}.

(129) If f is convergent in −∞, then |f| is convergent in −∞ and lim_{−∞} |f| = | lim_{−∞} f|.

(130) If f is convergent in −∞ and lim_{−∞} f ≠ 0 and for every r there exists g such that g < r and g ∈ dom f and f(g) ≠ 0, then \frac{1}{f} is convergent in −∞ and lim_{−∞} \frac{1}{f} = (lim_{−∞} f)^{−1}.

(131) If f_{1} is convergent in −∞ and f_{2} is convergent in −∞ and for every r there exists g such that g < r and g ∈ dom(f_{1}f_{2}), then f_{1}f_{2} is convergent in −∞ and lim_{−∞}(f_{1}f_{2}) = (lim_{−∞} f_{1}) · (lim_{−∞} f_{2}).

(132) If f_{1} is convergent in −∞ and f_{2} is convergent in −∞ and lim_{−∞} f_{2} ≠ 0 and for every r there exists g such that g < r and g ∈ dom \frac{f_{1}}{f_{2}} , then \frac{f_{1}}{f_{2}} is convergent in −∞ and lim_{−∞} \frac{f_{1}}{f_{2}} = \frac{lim_{−∞} f_{1}}{lim_{−∞} f_{2}} .

(133) If f_{1} is convergent in +∞ and lim_{+∞} f_{1} = 0 and for every r there exists g such that r < g and g ∈ dom(f_{1}f_{2}) and there exists r such that f_{2} is bounded on ]r, +∞[, then f_{1}f_{2} is convergent in +∞ and lim_{+∞}(f_{1}f_{2}) = 0.

(134) If f_{1} is convergent in −∞ and lim_{−∞} f_{1} = 0 and for every r there exists g such that g < r and g ∈ dom(f_{1}f_{2}) and there exists r such that f_{2} is bounded on ]−∞, r[, then f_{1}f_{2} is convergent in −∞ and lim_{−∞}(f_{1}f_{2}) = 0.

(135) Suppose that (i) f_{1} is convergent in +∞, (ii) f_{2} is convergent in +∞, (iii) lim_{+∞} f_{1} = lim_{+∞} f_{2}, (iv) for every r there exists g such that r < g and g ∈ dom f,

(v) there exists r such that dom f_{1} ∩ ]r, +∞[ ⊆ dom f_{2} ∩ ]r, +∞[ and dom f ∩]r, +∞[ ⊆ dom f_{1}∩]r, +∞[ or dom f_{2}∩]r, +∞[ ⊆ dom f_{1}∩]r, +∞[ and dom f ∩ ]r, +∞[ ⊆ dom f_{2} ∩ ]r, +∞[ but for every g such that g ∈ dom f ∩]r, +∞[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g). Then f is convergent in +∞ and lim_{+∞} f = lim_{+∞} f_{1}.

(136) Suppose f_{1} is convergent in +∞ and f_{2} is convergent in +∞ and lim_{+∞} f_{1} = lim_{+∞} f_{2} and there exists r such that ]r, +∞[ ⊆ (dom f_{1} ∩ dom f_{2}) ∩ dom f and for every g such that g ∈ ]r, +∞[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g). Then f is convergent in +∞ and lim_{+∞} f = lim_{+∞} f_{1}.

(137) Suppose that (i) f_{1} is convergent in −∞, (ii) f_{2} is convergent in −∞, (iii) lim_{−∞} f_{1} = lim_{−∞} f_{2}, (iv) for every r there exists g such that g < r and g ∈ dom f,

(v) there exists r such that dom f_{1} ∩ ]−∞, r[ ⊆ dom f_{2} ∩ ]−∞, r[ and dom f ∩]−∞, r[ ⊆ dom f_{1}∩]−∞, r[ or dom f_{2}∩]−∞, r[ ⊆ dom f_{1}∩]−∞, r[ and dom f ∩ ]−∞, r[ ⊆ dom f_{2} ∩ ]−∞, r[ but for every g such that g ∈ dom f ∩]−∞, r[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g). Then f is convergent in −∞ and lim_{−∞} f = lim_{−∞} f_{1}.

(138) Suppose f_{1} is convergent in −∞ and f_{2} is convergent in −∞ and lim_{−∞} f_{1} = lim_{−∞} f_{2} and there exists r such that ]−∞, r[ ⊆ (dom f_{1} ∩ dom f_{2}) ∩ dom f and for every g such that g ∈ ]−∞, r[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g). Then f is convergent in −∞ and lim_{−∞} f = lim_{−∞} f_{1}.

(139) Suppose that (i) f_{1} is convergent in +∞, (ii) f_{2} is convergent in +∞, (iii) there exists r such that dom f_{1} ∩]r, +∞[ ⊆ dom f_{2} ∩]r, +∞[ and for

every g such that g ∈ dom f_{1} ∩ ]r, +∞[ holds f_{1}(g) ≤ f_{2}(g) or dom f_{2} ∩ ]r, +∞[ ⊆ dom f_{1} ∩]r, +∞[ and for every g such that g ∈ dom f_{2} ∩]r, +∞[ holds f_{1}(g) ≤ f_{2}(g). Then lim_{+∞} f_{1} ≤ lim_{+∞} f_{2}.

(140) Suppose that (i) f_{1} is convergent in −∞, (ii) f_{2} is convergent in −∞, (iii) there exists r such that dom f_{1} ∩]−∞, r[ ⊆ dom f_{2} ∩]−∞, r[ and for

every g such that g ∈ dom f_{1} ∩]−∞, r[ holds f_{1}(g) ≤ f_{2}(g) or dom f_{2} ∩ ]−∞, r[ ⊆ dom f_{1} ∩]−∞, r[ and for every g such that g ∈ dom f_{2} ∩]−∞, r[ holds f_{1}(g) ≤ f_{2}(g). Then lim_{−∞} f_{1} ≤ lim_{−∞} f_{2}.

(141) If f is divergent in +∞ to +∞ or f is divergent in +∞ to −∞ but for every r there exists g such that r < g and g ∈ dom f and f(g) ≠ 0, then \frac{1}{f} is convergent in +∞ and lim_{+∞} \frac{1}{f} = 0.

We now state several propositions:

(142) If f is divergent in −∞ to +∞ or f is divergent in −∞ to −∞ but for every r there exists g such that g < r and g ∈ dom f and f(g) ≠ 0, then \frac{1}{f} is convergent in −∞ and lim_{−∞} \frac{1}{f} = 0.

(143) If f is convergent in +∞ and lim_{+∞} f = 0 and for every r there exists g such that r < g and g ∈ dom f and f(g) ≠ 0 and there exists r such that for every g such that g ∈ dom f ∩]r, +∞[ holds 0 ≤ f(g), then \frac{1}{f} is divergent in +∞ to +∞.

(144) If f is convergent in +∞ and lim_{+∞} f = 0 and for every r there exists g such that r < g and g ∈ dom f and f(g) ≠ 0 and there exists r such that for every g such that g ∈ dom f ∩]r, +∞[ holds f(g) ≤ 0, then \frac{1}{f} is divergent in +∞ to −∞.

(145) If f is convergent in −∞ and lim_{−∞} f = 0 and for every r there exists g such that g < r and g ∈ dom f and f(g) ≠ 0 and there exists r such that for every g such that g ∈ dom f ∩]−∞, r[ holds 0 ≤ f(g), then \frac{1}{f} is divergent in −∞ to +∞.

(146) If f is convergent in −∞ and lim_{−∞} f = 0 and for every r there exists g such that g < r and g ∈ dom f and f(g) ≠ 0 and there exists r such that for every g such that g ∈ dom f ∩]−∞, r[ holds f(g) ≤ 0, then \frac{1}{f} is divergent in −∞ to −∞.

(147) If f is convergent in +∞ and lim_{+∞} f = 0 and there exists r such that for every g such that g ∈ dom f ∩]r, +∞[ holds 0 < f(g), then \frac{1}{f} is divergent in +∞ to +∞.

(148) If f is convergent in +∞ and lim_{+∞} f = 0 and there exists r such that for every g such that g ∈ dom f ∩]r, +∞[ holds f(g) < 0, then \frac{1}{f} is divergent in +∞ to −∞.

(149) If f is convergent in −∞ and lim_{−∞} f = 0 and there exists r such that for every g such that g ∈ dom f ∩]−∞, r[ holds 0 < f(g), then \frac{1}{f} is divergent in −∞ to +∞.

(150) If f is convergent in −∞ and lim_{−∞} f = 0 and there exists r such that for every g such that g ∈ dom f ∩]−∞, r[ holds f(g) < 0, then \frac{1}{f} is divergent in −∞ to −∞.

28 Jarosław Kotowicz

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 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 sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 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] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

[9] Jarosław Kotowicz. Properties of real functions. Formalized Mathematics, 1(4):781–786, 1990.

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

[11] Andrzej Ne‘dzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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

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

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

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

One-Side Limits of a Real Function at a Point

Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. We introduce the left-side and the right-side limit of a real function at a point. We prove a few properties of the operations on the proper and improper one-side limits and show that Cauchy and Heine characterizations of one-side limit are equivalent.

MML Identifier: LIMFUNC2.

The articles [15], [4], [1], [2], [13], [11], [5], [7], [12], [14], [3], [8], [9], [10], and [6] provide the terminology and notation for this paper. For simplicity we adopt the following convention: r, r_{1}, r_{2}, g, g_{1}, x_{0} will be real numbers, n, k will be natural numbers, s_{1} will be a sequence of real numbers, and f, f_{1}, f_{2} will be partial functions from to . We now state several propositions:

(1) If s_{1} is convergent and r < lim s_{1}, then there exists n such that for every k such that n ≤ k holds r < s_{1}(k).

(2) If s_{1} is convergent and lim s_{1} < r, then there exists n such that for every k such that n ≤ k holds s_{1}(k) < r.

(3) If 0 < r_{2} and ]r_{1} − r_{2}, r_{1}[ ⊆ dom f, then for every r such that r < r_{1} there exists g such that r < g and g < r_{1} and g ∈ dom f.

(4) If 0 < r_{2} and ]r_{1}, r_{1} + r_{2}[ ⊆ dom f, then for every r such that r_{1} < r there exists g such that g < r and r_{1} < g and g ∈ dom f.

(5) If for every n holds x_{0}− \frac{1}{n+1} < s_{1}(n) and s_{1}(n) < x_{0} and s_{1}(n) ∈ dom f, then s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f and rng s_{1} ⊆ dom f ∩]−∞, x_{0}[.

(6) If for every n holds x_{0} < s_{1}(n) and s_{1}(n) < x_{0}+ \frac{1}{n+1} and s_{1}(n) ∈ dom f, then s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f and rng s_{1} ⊆ dom f ∩]x_{0}, +∞[.

1Supported by RPBP.III-24.C8

We now define several new predicates. Let us consider f, x_{0}. We say that f is left convergent in x_{0} if and only if: (Def.1) (i) for every r such that r < x_{0} there exists g such that r < g and

g < x_{0} and g ∈ dom f, (ii) there exists g such that for every s_{1} such that s_{1} is convergent and

lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]−∞, x_{0}[ holds f · s_{1} is convergent and lim(f · s_{1}) = g.

We say that f is left divergent to +∞ in x_{0} if and only if: (Def.2) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom f and for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]−∞, x_{0}[ holds f · s_{1} is divergent to +∞.

We say that f is left divergent to −∞ in x_{0} if and only if: (Def.3) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom f and for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]−∞, x_{0}[ holds f · s_{1} is divergent to −∞.

We say that f is right convergent in x_{0} if and only if: (Def.4) (i) for every r such that x_{0} < r there exists g such that g < r and

x_{0} < g and g ∈ dom f, (ii) there exists g such that for every s_{1} such that s_{1} is convergent and

lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]x_{0}, +∞[ holds f · s_{1} is convergent and lim(f · s_{1}) = g.

We say that f is right divergent to +∞ in x_{0} if and only if: (Def.5) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom f and for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]x_{0}, +∞[ holds f · s_{1} is divergent to +∞.

We say that f is right divergent to −∞ in x_{0} if and only if: (Def.6) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom f and for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]x_{0}, +∞[ holds f · s_{1} is divergent to −∞.

We now state a number of propositions:

(7) f is left convergent in x_{0} if and only if the following conditions are satisfied:

(i) for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f,

(ii) there exists g such that for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]−∞, x_{0}[ holds f · s_{1} is convergent and lim(f · s_{1}) = g.

(8) f is left divergent to +∞ in x_{0} if and only if for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f and for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]−∞, x_{0}[ holds f · s_{1} is divergent to +∞.

(9) f is left divergent to −∞ in x_{0} if and only if for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f and for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]−∞, x_{0}[ holds f · s_{1} is divergent to −∞.

(10) f is right convergent in x_{0} if and only if the following conditions are

satisfied:

(i) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f,

(ii) there exists g such that for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]x_{0}, +∞[ holds f · s_{1} is convergent and lim(f · s_{1}) = g.

(11) f is right divergent to +∞ in x_{0} if and only if for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f and for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]x_{0}, +∞[ holds f · s_{1} is divergent to +∞.

(12) f is right divergent to −∞ in x_{0} if and only if for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f and for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]x_{0}, +∞[ holds f · s_{1} is divergent to −∞.

(13) f is left convergent in x_{0} if and only if the following conditions are

satisfied:

(i) for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f,

(ii) there exists g such that for every g_{1} such that 0 < g_{1} there exists r such that r < x_{0} and for every r_{1} such that r < r_{1} and r_{1} < x_{0} and r_{1} ∈ dom f holds |f(r_{1}) − g| < g_{1}.

(14) f is left divergent to +∞ in x_{0} if and only if the following conditions

are satisfied:

(i) for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f,

(ii) for every g_{1} there exists r such that r < x_{0} and for every r_{1} such that

r < r_{1} and r_{1} < x_{0} and r_{1} ∈ dom f holds g_{1} < f(r_{1}).

(15) f is left divergent to −∞ in x_{0} if and only if the following conditions

are satisfied:

(i) for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f,

(ii) for every g_{1} there exists r such that r < x_{0} and for every r_{1} such that

r < r_{1} and r_{1} < x_{0} and r_{1} ∈ dom f holds f(r_{1}) < g_{1}.

(16) f is right convergent in x_{0} if and only if the following conditions are

satisfied:

(i) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f,

(ii) there exists g such that for every g_{1} such that 0 < g_{1} there exists r such that x_{0} < r and for every r_{1} such that r_{1} < r and x_{0} < r_{1} and r_{1} ∈ dom f

holds |f(r_{1}) − g| < g_{1}.

(17) f is right divergent to +∞ in x_{0} if and only if the following conditions

are satisfied:

(i) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f,

(ii) for every g_{1} there exists r such that x_{0} < r and for every r_{1} such that

r_{1} < r and x_{0} < r_{1} and r_{1} ∈ dom f holds g_{1} < f(r_{1}).

(18) f is right divergent to −∞ in x_{0} if and only if the following conditions

are satisfied:

(i) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f,

(ii) for every g_{1} there exists r such that x_{0} < r and for every r_{1} such that

r_{1} < r and x_{0} < r_{1} and r_{1} ∈ dom f holds f(r_{1}) < g_{1}.

(19) If f_{1} is left divergent to +∞ in x_{0} and f_{2} is left divergent to +∞ in x_{0} and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f_{1} ∩ dom f_{2}, then f_{1} + f_{2} is left divergent to +∞ in x_{0} and f_{1}f_{2} is left divergent to +∞ in x_{0}.

(20) If f_{1} is left divergent to −∞ in x_{0} and f_{2} is left divergent to −∞ in x_{0} and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f_{1} ∩ dom f_{2}, then f_{1} + f_{2} is left divergent to −∞ in x_{0} and f_{1}f_{2} is left divergent to +∞ in x_{0}.

(21) If f_{1} is right divergent to +∞ in x_{0} and f_{2} is right divergent to +∞ in x_{0} and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f_{1} ∩ dom f_{2}, then f_{1} + f_{2} is right divergent to +∞ in x_{0} and f_{1}f_{2} is right divergent to +∞ in x_{0}.

(22) If f_{1} is right divergent to −∞ in x_{0} and f_{2} is right divergent to −∞ in x_{0} and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f_{1} ∩ dom f_{2}, then f_{1} + f_{2} is right divergent to −∞ in x_{0} and f_{1}f_{2} is right divergent to +∞ in x_{0}.

(23) If f_{1} is left divergent to +∞ in x_{0} and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{1} + f_{2}) and there exists r such that 0 < r and f_{2} is lower bounded on ]x_{0} − r, x_{0}[, then f_{1} + f_{2} is left divergent to +∞ in x_{0}.

(24) Suppose that (i) f_{1} is left divergent to +∞ in x_{0}, (ii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{1}f_{2}), (iii) there exist r, r_{1} such that 0 < r and 0 < r_{1} and for every g such that

g ∈ dom f_{2} ∩]x_{0} − r, x_{0}[ holds r_{1} ≤ f_{2}(g). Then f_{1}f_{2} is left divergent to +∞ in x_{0}.

(25) If f_{1} is right divergent to +∞ in x_{0} and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{1} + f_{2}) and there exists r such that 0 < r and f_{2} is lower bounded on ]x_{0}, x_{0} + r[, then

f_{1} + f_{2} is right divergent to +∞ in x_{0}.

(26) Suppose that (i) f_{1} is right divergent to +∞ in x_{0}, (ii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{1}f_{2}), (iii) there exist r, r_{1} such that 0 < r and 0 < r_{1} and for every g such that

g ∈ dom f_{2} ∩]x_{0}, x_{0} + r[ holds r_{1} ≤ f_{2}(g). Then f_{1}f_{2} is right divergent to +∞ in x_{0}.

(27) (i) If f is left divergent to +∞ in x_{0} and r > 0, then rf is left divergent

to +∞ in x_{0}, (ii) if f is left divergent to +∞ in x_{0} and r < 0, then rf is left divergent

to −∞ in x_{0}, (iii) if f is left divergent to −∞ in x_{0} and r > 0, then rf is left divergent

to −∞ in x_{0}, (iv) if f is left divergent to −∞ in x_{0} and r < 0, then rf is left divergent

to +∞ in x_{0}.

(28) (i) If f is right divergent to +∞ in x_{0} and r > 0, then rf is right

divergent to +∞ in x_{0}, (ii) if f is right divergent to +∞ in x_{0} and r < 0, then rf is right divergent

to −∞ in x_{0}, (iii) if f is right divergent to −∞ in x_{0} and r > 0, then rf is right divergent

to −∞ in x_{0}, (iv) if f is right divergent to −∞ in x_{0} and r < 0, then rf is right divergent

to +∞ in x_{0}.

(29) If f is left divergent to +∞ in x_{0} or f is left divergent to −∞ in x_{0}, then |f| is left divergent to +∞ in x_{0}.

(30) If f is right divergent to +∞ in x_{0} or f is right divergent to −∞ in x_{0}, then |f| is right divergent to +∞ in x_{0}.

(31) If there exists r such that 0 < r and f is non-decreasing on ]x_{0} − r, x_{0}[ and f is not upper bounded on ]x_{0}−r, x_{0}[ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f, then f is left divergent to +∞ in x_{0}.

(32) If there exists r such that 0 < r and f is increasing on ]x_{0} − r, x_{0}[ and f is not upper bounded on ]x_{0} − r, x_{0}[ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f, then f is left divergent to +∞ in x_{0}.

(33) If there exists r such that 0 < r and f is non-increasing on ]x_{0} − r, x_{0}[ and f is not lower bounded on ]x_{0} −r, x_{0}[ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f, then f is left divergent to −∞ in x_{0}.

(34) If there exists r such that 0 < r and f is decreasing on ]x_{0} − r, x_{0}[ and f is not lower bounded on ]x_{0} − r, x_{0}[ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f, then f is left

divergent to −∞ in x_{0}.

(35) If there exists r such that 0 < r and f is non-increasing on ]x_{0}, x_{0} + r[ and f is not upper bounded on ]x_{0}, x_{0}+r[ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f, then f is right divergent to +∞ in x_{0}.

(36) If there exists r such that 0 < r and f is decreasing on ]x_{0}, x_{0} + r[ and f is not upper bounded on ]x_{0}, x_{0} + r[ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f, then f is right divergent to +∞ in x_{0}.

(37) If there exists r such that 0 < r and f is non-decreasing on ]x_{0}, x_{0} + r[ and f is not lower bounded on ]x_{0}, x_{0} +r[ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f, then f is right divergent to −∞ in x_{0}.

Next we state several propositions:

(38) If there exists r such that 0 < r and f is increasing on ]x_{0}, x_{0} + r[ and f is not lower bounded on ]x_{0}, x_{0} + r[ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f, then f is right divergent to −∞ in x_{0}.

(39) Suppose that (i) f_{1} is left divergent to +∞ in x_{0}, (ii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom f, (iii) there exists r such that 0 < r and dom f ∩]x_{0} − r, x_{0}[ ⊆ dom f_{1} ∩]x_{0} −

r, x_{0}[ and for every g such that g ∈ dom f ∩]x_{0} −r, x_{0}[ holds f_{1}(g) ≤ f(g). Then f is left divergent to +∞ in x_{0}.

(40) Suppose that (i) f_{1} is left divergent to −∞ in x_{0}, (ii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom f, (iii) there exists r such that 0 < r and dom f ∩]x_{0} − r, x_{0}[ ⊆ dom f_{1} ∩]x_{0} −

r, x_{0}[ and for every g such that g ∈ dom f ∩]x_{0} −r, x_{0}[ holds f(g) ≤ f_{1}(g). Then f is left divergent to −∞ in x_{0}.

(41) Suppose that (i) f_{1} is right divergent to +∞ in x_{0}, (ii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom f, (iii) there exists r such that 0 < r and dom f∩]x_{0}, x_{0}+r[ ⊆ dom f_{1}∩]x_{0}, x_{0}+

r[ and for every g such that g ∈ dom f ∩]x_{0}, x_{0} + r[ holds f_{1}(g) ≤ f(g). Then f is right divergent to +∞ in x_{0}.

(42) Suppose that (i) f_{1} is right divergent to −∞ in x_{0}, (ii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom f, (iii) there exists r such that 0 < r and dom f∩]x_{0}, x_{0}+r[ ⊆ dom f_{1}∩]x_{0}, x_{0}+

r[ and for every g such that g ∈ dom f ∩]x_{0}, x_{0} + r[ holds f(g) ≤ f_{1}(g). Then f is right divergent to −∞ in x_{0}.

(43) If f_{1} is left divergent to +∞ in x_{0} and there exists r such that 0 < r and ]x_{0} −r, x_{0}[ ⊆ dom f ∩dom f_{1} and for every g such that g ∈ ]x_{0} −r, x_{0}[ holds f_{1}(g) ≤ f(g), then f is left divergent to +∞ in x_{0}.

(44) If f_{1} is left divergent to −∞ in x_{0} and there exists r such that 0 < r and ]x_{0} −r, x_{0}[ ⊆ dom f ∩dom f_{1} and for every g such that g ∈ ]x_{0} −r, x_{0}[ holds f(g) ≤ f_{1}(g), then f is left divergent to −∞ in x_{0}.

(45) If f_{1} is right divergent to +∞ in x_{0} and there exists r such that 0 < r and ]x_{0}, x_{0} +r[ ⊆ dom f ∩dom f_{1} and for every g such that g ∈ ]x_{0}, x_{0} +r[ holds f_{1}(g) ≤ f(g), then f is right divergent to +∞ in x_{0}.

(46) If f_{1} is right divergent to −∞ in x_{0} and there exists r such that 0 < r and ]x_{0}, x_{0} +r[ ⊆ dom f ∩dom f_{1} and for every g such that g ∈ ]x_{0}, x_{0} +r[ holds f(g) ≤ f_{1}(g), then f is right divergent to −∞ in x_{0}.

Let us consider f, x_{0}. Let us assume that f is left convergent in x_{0}. The functor lim_{x_{0}}_{−} f yields a real number and is defined by: (Def.7) for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆

dom f ∩]−∞, x_{0}[ holds f · s_{1} is convergent and lim(f · s_{1}) = lim_{x_{0}}_{−} f.

Let us consider f, x_{0}. Let us assume that f is right convergent in x_{0}. The functor lim_{x_{0}}_{+} f yields a real number and is defined by: (Def.8) for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆

dom f ∩]x_{0}, +∞[ holds f · s_{1} is convergent and lim(f · s_{1}) = lim_{x_{0}}_{+} f.

One can prove the following propositions:

(47) If f is left convergent in x_{0}, then lim_{x_{0}}_{−} f = g if and only if for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]−∞, x_{0}[ holds f · s_{1} is convergent and lim(f · s_{1}) = g.

(48) If f is right convergent in x_{0}, then lim_{x_{0}}_{+} f = g if and only if for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f ∩]x_{0}, +∞[ holds f · s_{1} is convergent and lim(f · s_{1}) = g.

(49) If f is left convergent in x_{0}, then lim_{x_{0}}_{−} f = g if and only if for every g_{1} such that 0 < g_{1} there exists r such that r < x_{0} and for every r_{1} such that r < r_{1} and r_{1} < x_{0} and r_{1} ∈ dom f holds |f(r_{1}) − g| < g_{1}.

(50) If f is right convergent in x_{0}, then lim_{x_{0}}_{+} f = g if and only if for every g_{1} such that 0 < g_{1} there exists r such that x_{0} < r and for every r_{1} such that r_{1} < r and x_{0} < r_{1} and r_{1} ∈ dom f holds |f(r_{1}) − g| < g_{1}.

(51) If f is left convergent in x_{0}, then rf is left convergent in x_{0} and lim_{x_{0}}_{−}(rf) = r · (lim_{x_{0}}_{−} f).

(52) If f is left convergent in x_{0}, then −f is left convergent in x_{0} and lim_{x_{0}}_{−}(−f) = − lim_{x_{0}}_{−} f.

(53) Suppose f_{1} is left convergent in x_{0} and f_{2} is left convergent in x_{0} and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{1} + f_{2}). Then f_{1} + f_{2} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{1} + f_{2}) = lim_{x_{0}}_{−} f_{1} + lim_{x_{0}}_{−} f_{2}.

(54) Suppose f_{1} is left convergent in x_{0} and f_{2} is left convergent in x_{0} and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{1} − f_{2}). Then f_{1} − f_{2} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{1} − f_{2}) = lim_{x_{0}}_{−} f_{1} − lim_{x_{0}}_{−} f_{2}.

(55) If f is left convergent in x_{0} and f ^{−1} \{0\} = ∅ and lim_{x_{0}}_{−} f ≠ 0, then \frac{1}{f} is left convergent in x_{0} and lim_{x_{0}}_{−} \frac{1}{f} = (lim_{x_{0}}_{−} f)^{−1}.

(56) If f is left convergent in x_{0}, then |f| is left convergent in x_{0} and lim_{x_{0}}_{−} |f| = | lim_{x_{0}}_{−} f|.

(57) Suppose f is left convergent in x_{0} and lim_{x_{0}}_{−} f ≠ 0 and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f and f(g) ≠ 0. Then \frac{1}{f} is left convergent in x_{0} and lim_{x_{0}}_{−} \frac{1}{f} = (lim_{x_{0}}_{−} f)^{−1}.

(58) Suppose f_{1} is left convergent in x_{0} and f_{2} is left convergent in x_{0} and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{1}f_{2}). Then f_{1}f_{2} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{1}f_{2}) = (lim_{x_{0}}_{−} f_{1}) · (lim_{x_{0}}_{−} f_{2}).

(59) Suppose f_{1} is left convergent in x_{0} and f_{2} is left convergent in x_{0} and lim_{x_{0}}_{−} f_{2} ≠ 0 and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom \frac{f_{1}}{f_{2}} . Then \frac{f_{1}}{f_{2}} is left convergent in x_{0} and

lim_{x_{0}}_{−} \frac{f_{1}}{f_{2}} = \frac{lim_{_{x_{0}}−} f_{1}}{lim_{_{x_{0}}−} f_{2}} .

(60) If f is right convergent in x_{0}, then rf is right convergent in x_{0} and lim_{x_{0}}_{+}(rf) = r · (lim_{x_{0}}_{+} f).

(61) If f is right convergent in x_{0}, then −f is right convergent in x_{0} and lim_{x_{0}}_{+}(−f) = − lim_{x_{0}}_{+} f.

(62) Suppose f_{1} is right convergent in x_{0} and f_{2} is right convergent in x_{0} and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{1} + f_{2}). Then f_{1} + f_{2} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{1} + f_{2}) = lim_{x_{0}}_{+} f_{1} + lim_{x_{0}}_{+} f_{2}.

(63) Suppose f_{1} is right convergent in x_{0} and f_{2} is right convergent in x_{0} and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{1} − f_{2}). Then f_{1} − f_{2} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{1} − f_{2}) = lim_{x_{0}}_{+} f_{1} − lim_{x_{0}}_{+} f_{2}.

(64) If f is right convergent in x_{0} and f ^{−1} {0} = ∅ and lim_{x_{0}}_{+} f ≠ 0, then \frac{1}{f} is right convergent in x_{0} and lim_{x_{0}}_{+} \frac{1}{f} = (lim_{x_{0}}_{+} f)^{−1}.

(65) If f is right convergent in x_{0}, then |f| is right convergent in x_{0} and lim_{x_{0}}_{+} |f| = | lim_{x_{0}}_{+} f|.

(66) Suppose f is right convergent in x_{0} and lim_{x_{0}}_{+} f ≠ 0 and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f and f(g) ≠ 0. Then \frac{1}{f} is right convergent in x_{0} and lim_{x_{0}}_{+} \frac{1}{f} = (lim_{x_{0}}_{+} f)^{−1}.

(67) Suppose f_{1} is right convergent in x_{0} and f_{2} is right convergent in x_{0} and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{1}f_{2}). Then f_{1}f_{2} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{1}f_{2}) = (lim_{x_{0}}_{+} f_{1}) · (lim_{x_{0}}_{+} f_{2}).

(68) Suppose f_{1} is right convergent in x_{0} and f_{2} is right convergent in x_{0} and lim_{x_{0}}_{+} f_{2} ≠ 0 and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom \frac{f_{1}}{f_{2}} . Then \frac{f_{1}}{f_{2}} is right convergent in x_{0}

and lim_{x_{0}}_{+} \frac{f_{1}}{f_{2}} = \frac{lim_{_{x_{0}}+} f_{1}}{lim_{_{x_{0}}+} f_{2}} .

(69) Suppose f_{1} is left convergent in x_{0} and lim_{x_{0}}_{−} f_{1} = 0 and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{1}f_{2}) and there exists r such that 0 < r and f_{2} is bounded on ]x_{0} − r, x_{0}[. Then f_{1}f_{2} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{1}f_{2}) = 0.

(70) Suppose f_{1} is right convergent in x_{0} and lim_{x_{0}}_{+} f_{1} = 0 and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{1}f_{2}) and there exists r such that 0 < r and f_{2} is bounded on ]x_{0}, x_{0} + r[. Then f_{1}f_{2} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{1}f_{2}) = 0.

(71) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is left convergent in x_{0}, (iii) lim_{x_{0}}_{−} f_{1} = lim_{x_{0}}_{−} f_{2}, (iv) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom f,

(v) there exists r such that 0 < r and for every g such that g ∈ dom f ∩ ]x_{0} −r, x_{0}[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g) but dom f_{1} ∩]x_{0} −r, x_{0}[ ⊆ dom f_{2} ∩ ]x_{0} − r, x_{0}[ and dom f ∩ ]x_{0} − r, x_{0}[ ⊆ dom f_{1} ∩ ]x_{0} − r, x_{0}[ or dom f_{2} ∩ ]x_{0} − r, x_{0}[ ⊆ dom f_{1} ∩ ]x_{0} − r, x_{0}[ and dom f ∩ ]x_{0} − r, x_{0}[ ⊆ dom f_{2} ∩]x_{0} − r, x_{0}[. Then f is left convergent in x_{0} and lim_{x_{0}}_{−} f = lim_{x_{0}}_{−} f_{1}.

(72) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is left convergent in x_{0}, (iii) lim_{x_{0}}_{−} f_{1} = lim_{x_{0}}_{−} f_{2}, (iv) there exists r such that 0 < r and ]x_{0} − r, x_{0}[ ⊆ (dom f_{1} ∩ dom f_{2}) ∩

dom f and for every g such that g ∈ ]x_{0} − r, x_{0}[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g). Then f is left convergent in x_{0} and lim_{x_{0}}_{−} f = lim_{x_{0}}_{−} f_{1}.

(73) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is right convergent in x_{0}, (iii) lim_{x_{0}}_{+} f_{1} = lim_{x_{0}}_{+} f_{2}, (iv) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom f,

(v) there exists r such that 0 < r and for every g such that g ∈ dom f ∩ ]x_{0}, x_{0} +r[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g) but dom f_{1} ∩]x_{0}, x_{0} +r[ ⊆ dom f_{2} ∩ ]x_{0}, x_{0} + r[ and dom f ∩ ]x_{0}, x_{0} + r[ ⊆ dom f_{1} ∩ ]x_{0}, x_{0} + r[ or dom f_{2} ∩ ]x_{0}, x_{0} + r[ ⊆ dom f_{1} ∩ ]x_{0}, x_{0} + r[ and dom f ∩ ]x_{0}, x_{0} + r[ ⊆ dom f_{2} ∩]x_{0}, x_{0} + r[. Then f is right convergent in x_{0} and lim_{x_{0}}_{+} f = lim_{x_{0}}_{+} f_{1}.

(74) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is right convergent in x_{0}, (iii) lim_{x_{0}}_{+} f_{1} = lim_{x_{0}}_{+} f_{2}, (iv) there exists r such that 0 < r and ]x_{0}, x_{0} + r[ ⊆ (dom f_{1} ∩ dom f_{2}) ∩

dom f and for every g such that g ∈ ]x_{0}, x_{0} + r[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g). Then f is right convergent in x_{0} and lim_{x_{0}}_{+} f = lim_{x_{0}}_{+} f_{1}.

(75) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is left convergent in x_{0}, (iii) there exists r such that 0 < r but dom f_{1} ∩]x_{0} − r, x_{0}[ ⊆ dom f_{2} ∩]x_{0} −

r, x_{0}[ and for every g such that g ∈ dom f_{1}∩]x_{0}−r, x_{0}[ holds f_{1}(g) ≤ f_{2}(g) or dom f_{2} ∩]x_{0} − r, x_{0}[ ⊆ dom f_{1} ∩]x_{0} − r, x_{0}[ and for every g such that g ∈ dom f_{2} ∩]x_{0} − r, x_{0}[ holds f_{1}(g) ≤ f_{2}(g). Then lim_{x_{0}}_{−} f_{1} ≤ lim_{x_{0}}_{−} f_{2}.

(76) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is right convergent in x_{0}, (iii) there exists r such that 0 < r but dom f_{1} ∩ ]x_{0}, x_{0} + r[ ⊆ dom f_{2} ∩

]x_{0}, x_{0} + r[ and for every g such that g ∈ dom f_{1} ∩ ]x_{0}, x_{0} + r[ holds f_{1}(g) ≤ f_{2}(g) or dom f_{2} ∩]x_{0}, x_{0} + r[ ⊆ dom f_{1} ∩]x_{0}, x_{0} + r[ and for every g such that g ∈ dom f_{2} ∩]x_{0}, x_{0} + r[ holds f_{1}(g) ≤ f_{2}(g). Then lim_{x_{0}}_{+} f_{1} ≤ lim_{x_{0}}_{+} f_{2}.

(77) If f is left divergent to +∞ in x_{0} or f is left divergent to −∞ in x_{0} but for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f and f(g) ≠ 0, then \frac{1}{f} is left convergent in x_{0} and lim_{x_{0}}_{−} \frac{1}{f} = 0.

One can prove the following propositions:

(78) If f is right divergent to +∞ in x_{0} or f is right divergent to −∞ in x_{0} but for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f and f(g) ≠ 0, then \frac{1}{f} is right convergent in x_{0} and lim_{x_{0}}_{+} \frac{1}{f} = 0.

(79) If f is left convergent in x_{0} and lim_{x_{0}}_{−} f = 0 and there exists r such that 0 < r and for every g such that g ∈ dom f ∩]x_{0} − r, x_{0}[ holds 0 < f(g), then \frac{1}{f} is left divergent to +∞ in x_{0}.

(80) If f is left convergent in x_{0} and lim_{x_{0}}_{−} f = 0 and there exists r such that 0 < r and for every g such that g ∈ dom f ∩]x_{0} − r, x_{0}[ holds f(g) < 0, then \frac{1}{f} is left divergent to −∞ in x_{0}.

(81) If f is right convergent in x_{0} and lim_{x_{0}}_{+} f = 0 and there exists r such that 0 < r and for every g such that g ∈ dom f ∩]x_{0}, x_{0}+r[ holds 0 < f(g), then \frac{1}{f} is right divergent to +∞ in x_{0}.

(82) If f is right convergent in x_{0} and lim_{x_{0}}_{+} f = 0 and there exists r such that 0 < r and for every g such that g ∈ dom f ∩]x_{0}, x_{0}+r[ holds f(g) < 0, then \frac{1}{f} is right divergent to −∞ in x_{0}.

(83) Suppose that (i) f is left convergent in x_{0}, (ii) lim_{x_{0}}_{−} f = 0, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom f and f(g) ≠ 0, (iv) there exists r such that 0 < r and for every g such that g ∈ dom f ∩

]x_{0} − r, x_{0}[ holds 0 ≤ f(g). Then \frac{1}{f} is left divergent to +∞ in x_{0}.

(84) Suppose that (i) f is left convergent in x_{0}, (ii) lim_{x_{0}}_{−} f = 0, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom f and f(g) ≠ 0, (iv) there exists r such that 0 < r and for every g such that g ∈ dom f ∩

]x_{0} − r, x_{0}[ holds f(g) ≤ 0. Then \frac{1}{f} is left divergent to −∞ in x_{0}.

(85) Suppose that (i) f is right convergent in x_{0}, (ii) lim_{x_{0}}_{+} f = 0, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom f and f(g) ≠ 0, (iv) there exists r such that 0 < r and for every g such that g ∈ dom f ∩

]x_{0}, x_{0} + r[ holds 0 ≤ f(g). Then \frac{1}{f} is right divergent to +∞ in x_{0}.

(86) Suppose that (i) f is right convergent in x_{0}, (ii) lim_{x_{0}}_{+} f = 0, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom f and f(g) ≠ 0, (iv) there exists r such that 0 < r and for every g such that g ∈ dom f ∩

]x_{0}, x_{0} + r[ holds f(g) ≤ 0. Then \frac{1}{f} is right divergent to −∞ in x_{0}.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

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

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

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[9] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

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

Lattice of Subgroups of a Group. Frattini Subgroup

Wojciech A. Trybulec^{1} Warsaw University

Summary. We define the notion of a subgroup generated by a set of elements of a group and two closely connected notions, namely lattice of subgroups and the Frattini subgroup. The operations on the lattice are the intersection of subgroups (introduced in [18]) and multiplication of subgroups, which result is defined as a subgroup generated by a sum of carriers of the two subgroups. In order to define the Frattini subgroup and to prove theorems concerning it we introduce notion of maximal subgroup and non-generating element of the group (see page 30 in [6]). The Frattini subgroup is defined as in [6] as an intersection of all maximal subgroups. We show that an element of the group belongs to the Frattini subgroup of the group if and only if it is a non-generating element. We also prove theorems that should be proved in [1] but are not.

MML Identifier: GROUP 4.

The notation and terminology used here are introduced in the following articles: [3], [13], [4], [11], [20], [10], [19], [8], [16], [5], [17], [2], [15], [18], [14], [12], [21], [7], [9], and [1]. Let D be a non-empty set, and let F be a finite sequence of elements of D, and let X be a set. Then F − X is a finite sequence of elements of D.

In this article we present several logical schemes. The scheme SubsetD deals with a non-empty set A, and a unary predicate P, and states that:

{d : P[d]}, where d is an element of A, is a subset of A for all values of the parameters.

The scheme MeetSbgEx deals with a group A, and a unary predicate P, and states that:

there exists a subgroup H of A such that the carrier of H = ⋂{A : ⋁_{K}[A = the carrier of K ∧ P[K]]}, where A is a subset of A, and K is a subgroup of A provided the parameters have the following property:

1Supported by RPBP.III-24.C1

• there exists a subgroup H of A such that P[H].

For simplicity we adopt the following rules: X denotes a set, k, l, m, n denote natural numbers, i, i_{1}, i_{2}, i_{3}, j denote integers, G denotes a group, a, b, c denote elements of G, A, B denote subsets of G, H, H_{1}, H_{2}, H_{3}, K denote subgroups of G, N_{1}, N_{2} denote normal subgroups of G, h denotes an element of H, F, F_{1}, F_{2} denote finite sequences of elements of the carrier of G, and I, I_{1}, I_{2} denote finite sequences of elements of . The scheme SubgrSep deals with a group A, and a unary predicate P, and states that:

there exists X such that X ⊆ SubGr A and for every subgroup H of A holds H ∈ X if and only if P[H] for all values of the parameters.

Let i be an element of . The functor @i yields an integer and is defined by: (Def.1) @i = i.

We now state the proposition

(1) For every element i of holds @i = i.

Let us consider i. The functor @i yielding an element of is defined as follows: (Def.2) @i = i.

Next we state several propositions:

(2) @i = i.

(3) If a = h, then a^{n} = h^{n}.

(4) If a = h, then a^{i} = h^{i}.

(5) If a ∈ H, then a^{n} ∈ H.

(6) If a ∈ H, then a^{i} ∈ H.

Let us consider G, F. The functor ∏ F yielding an element of G is defined as follows: (Def.3) ∏ F = the operation of G ⊙ F.

Next we state a number of propositions:

(7) ∏ F = the operation of G ⊙ F.

(8) ∏(F_{1} ^{}F_{2}) = ∏ F_{1} · ∏ F_{2}.

(9) ∏(F ^{}⟨a⟩) = ∏ F · a.

(10) ∏(⟨a⟩^{}F) = a · ∏ F.

(11) ∏ ε_{the carrier of G} = 1_{G}.

(12) ∏⟨a⟩ = a.

(13) ∏⟨a, b⟩ = a · b.

(14) ∏⟨a, b, c⟩ = (a · b) · c and ∏⟨a, b, c⟩ = a · (b · c).

(15) ∏(n ⟼ a) = a^{n}.

(16) ∏(F − {1_{G}}) = ∏ F.

(17) If len F_{1} = len F_{2} and for every k such that k ∈ Seg(len F_{1}) holds F_{2}((len F_{1} − k) + 1) = (π_{k}F_{1})^{−1}, then ∏ F_{1} = (∏ F_{2})^{−1}.

(18) If G is an Abelian group, then for every permutation P of Seg(len F_{1}) such that F_{2} = F_{1} · P holds ∏ F_{1} = ∏ F_{2}.

(19) If G is an Abelian group and F_{1} is one-to-one and F_{2} is one-to-one and rng F_{1} = rng F_{2}, then ∏ F_{1} = ∏ F_{2}.

(20) If G is an Abelian group and len F = len F_{1} and len F = len F_{2} and for every k such that k ∈ Seg(len F) holds F (k) = π_{k}F_{1} · π_{k}F_{2}, then ∏ F = ∏ F_{1} · ∏ F_{2}.

(21) If rng F ⊆ \overline{H}, then ∏ F ∈ H.

Let us consider G, I, F. Let us assume that len F = len I. The functor F^{I} yields a finite sequence of elements of the carrier of G and is defined as follows: (Def.4) len(F^{I}) = len F and for every k such that k ∈ Seg(len F) holds (F^{I})(k) =

π_{k}F^{@(π_{k}I)}. One can prove the following propositions:

(22) If len F = len I and len F_{1} = len F and for every k such that k ∈ Seg(len F) holds F_{1}(k) = π_{k}F^{@(π_{k}I)}, then F_{1} = F^{I}.

(23) If len F = len I, then for every k such that k ∈ Seg(len F) holds (F^{I})(k) = π_{k}F^{@(π_{k}I)}.

(24) If len F = len I, then len(F^{I}) = len F.

(25) If len F_{1} = len I_{1} and len F_{2} = len I_{2}, then (F_{1} ^{}F_{2})^{I_{1}^{}I_{2}} = F_{1}^{I_{1}} ^{}F_{2}^{I_{2}}.

(26) If len F = len I and rng F ⊆ \overline{H}, then ∏(F^{I}) ∈ H.

(27) ε_{the carrier of G}^{ε_{}}= ε.

(28) ⟨a⟩^{⟨@i⟩} = ⟨a^{i}⟩.

(29) ⟨a, b⟩^{⟨@i,@j⟩} = ⟨a^{i}, b^{j}⟩.

(30) ⟨a, b, c⟩^{⟨@i_{1},@i_{2},@i_{3}⟩} = ⟨a^{i_{1}}, b^{i_{2}}, c^{i_{3}}⟩.

(31) F^{len F ⟼@(+1)} = F.

(32) F^{len F ⟼@(+0)} = len F ⟼ 1_{G}.

(33) If len I = n, then (n ⟼ 1_{G})^{I} = n ⟼ 1_{G}.

Let us consider G, A. The functor gr(A) yielding a subgroup of G is defined as follows: (Def.5) A ⊆ the carrier of gr(A) and for every H such that A ⊆ the carrier of

H holds gr(A) is a subgroup of H. We now state a number of propositions:

(34) If A ⊆ the carrier of H_{1} and for every H_{2} such that A ⊆ the carrier of H_{2} holds H_{1} is a subgroup of H_{2}, then H_{1} = gr(A).

(35) A ⊆ the carrier of gr(A).

(36) If A ⊆ the carrier of H, then gr(A) is a subgroup of H.

(37) a ∈ gr(A) if and only if there exist F, I such that len F = len I and rng F ⊆ A and ∏(F^{I}) = a.

(38) If a ∈ A, then a ∈ gr(A).

(39) gr(∅_{the carrier of G}) = {1}_{G}.

(40) gr(\overline{H}) = H.

(41) If A ⊆ B, then gr(A) is a subgroup of gr(B).

(42) gr(A ∩ B) is a subgroup of gr(A) ∩ gr(B).

(43) The carrier of gr(A) = ⋂\{B : ⋁_{H}[B = the carrier of H ∧ A ⊆ \overline{H}]\}.

(44) gr(A) = gr(A \ {1_{G}}).

We now define two new predicates. Let us consider G, a. We say that a is non-generating if and only if: (Def.6) for every A such that gr(A) = G holds gr(A \ {a}) = G. a is generating stands for a is not non-generating.

We now state the proposition (46)^{2} 1_{G} is non-generating.

Let us consider G, H. We say that H is maximal if and only if: (Def.7) H ≠ G and for every K such that H ≠ K and H is a subgroup of K

holds K = G. Next we state the proposition

(48)^{3} If H is maximal and a ∉ H, then gr(\overline{H} ∪ \{a\}) = G.

Let us consider G. The functor Φ(G) yields a subgroup of G and is defined as follows: (Def.8) the carrier of Φ(G) = ⋂{A : ⋁_{H}[A = the carrier of H∧ H is maximal

]} if there exists H such that H is maximal, Φ(G) = G, otherwise. We now state several propositions:

(49) If there exists H such that H is maximal and the carrier of H = ⋂{A : ⋁_{K}[A = the carrier of K∧ K is maximal ]}, then H = Φ(G).

(50) If for every H holds H is not maximal, then Φ(G) = G.

(51) If there exists H such that H is maximal, then the carrier of Φ(G) = ⋂{A : ⋁_{K}[A = the carrier of K∧ K is maximal ]}.

(52) If there exists H such that H is maximal, then a ∈ Φ(G) if and only if for every H such that H is maximal holds a ∈ H.

(53) If for every H holds H is not maximal, then a ∈ Φ(G).

(54) If H is maximal, then Φ(G) is a subgroup of H.

(55) The carrier of Φ(G) = {a : a is non-generating }.

(56) a ∈ Φ(G) if and only if a is non-generating.

Let us consider G, H_{1}, H_{2}. The functor H_{1} · H_{2} yielding a subset of G is defined as follows: (Def.9) H_{1} · H_{2} = \overline{H_{1}} · \overline{H_{2}}.

The following propositions are true:

(57) H_{1} · H_{2} = \overline{H_{1}} · \overline{H_{2}} and H_{1} · H_{2} = H_{1} · \overline{H_{2}} and H_{1} · H_{2} = \overline{H_{1}} · H_{2}.

2The proposition (45) was either repeated or obvious.

3The proposition (47) was either repeated or obvious.

(58) H · H = \overline{H}.

(59) (H_{1} · H_{2}) · H_{3} = H_{1} · (H_{2} · H_{3}).

(60) (a · H_{1}) · H_{2} = a · (H_{1} · H_{2}).

(61) (H_{1} · H_{2}) · a = H_{1} · (H_{2} · a).

(62) (A · H_{1}) · H_{2} = A · (H_{1} · H_{2}).

(63) (H_{1} · H_{2}) · A = H_{1} · (H_{2} · A).

(64) N_{1} · N_{2} = N_{2} · N_{1}.

(65) If G is an Abelian group, then H_{1} · H_{2} = H_{2} · H_{1}.

Let us consider G, H_{1}, H_{2}. The functor H_{1} ⊔ H_{2} yielding a subgroup of G is defined as follows:

(Def.10) H_{1} ⊔ H_{2} = gr(\overline{H_{1}} ∪ \overline{H_{2}}).

One can prove the following propositions:

(66) H_{1} ⊔ H_{2} = gr(\overline{H_{1}} ∪ \overline{H_{2}}).

(67) a ∈ H_{1} ⊔ H_{2} if and only if there exist F, I such that len F = len I and rng F ⊆ \overline{H_{1}} ∪ \overline{H_{2}} and a = ∏(F^{I}).

(68) H_{1} ⊔ H_{2} = gr(H_{1} · H_{2}).

(69) If H_{1} · H_{2} = H_{2} · H_{1}, then the carrier of H_{1} ⊔ H_{2} = H_{1} · H_{2}.

(70) If G is an Abelian group, then the carrier of H_{1} ⊔ H_{2} = H_{1} · H_{2}.

(71) The carrier of N_{1} ⊔ N_{2} = N_{1} · N_{2}.

(72) N_{1} ⊔ N_{2} is a normal subgroup of G.

(73) H ⊔ H = H.

(74) H_{1} ⊔ H_{2} = H_{2} ⊔ H_{1}.

(75) (H_{1} ⊔ H_{2}) ⊔ H_{3} = H_{1} ⊔ (H_{2} ⊔ H_{3}).

(76) {1}_{G} ⊔ H = H and H ⊔ {1}_{G} = H.

(77) Ω_{G} ⊔ H = G and H ⊔ Ω_{G} = G.

(78) H_{1} is a subgroup of H_{1} ⊔ H_{2} and H_{2} is a subgroup of H_{1} ⊔ H_{2}.

(79) H_{1} is a subgroup of H_{2} if and only if H_{1} ⊔ H_{2} = H_{2}.

(80) If H_{1} is a subgroup of H_{2}, then H_{1} is a subgroup of H_{2} ⊔ H_{3}.

(81) If H_{1} is a subgroup of H_{3} and H_{2} is a subgroup of H_{3}, then H_{1} ⊔ H_{2} is a subgroup of H_{3}.

(82) If H_{1} is a subgroup of H_{2}, then H_{1} ⊔ H_{3} is a subgroup of H_{2} ⊔ H_{3}.

(83) H_{1} ∩ H_{2} is a subgroup of H_{1} ⊔ H_{2}.

(84) (H_{1} ∩ H_{2}) ⊔ H_{2} = H_{2}.

(85) H_{1} ∩ (H_{1} ⊔ H_{2}) = H_{1}.

(86) H_{1} ⊔ H_{2} = H_{2} if and only if H_{1} ∩ H_{2} = H_{1}.

In the sequel S_{1}, S_{2} are elements of SubGr G and o is a binary operation on SubGr G. Let us consider G. The functor SubJoin G yields a binary operation on SubGr G and is defined by:

(Def.11) for all S_{1}, S_{2}, H_{1}, H_{2} such that S_{1} = H_{1} and S_{2} = H_{2} holds

(SubJoin G)(S_{1}, S_{2}) = H_{1} ⊔ H_{2}. Next we state two propositions:

(87) If for all S_{1}, S_{2}, H_{1}, H_{2} such that S_{1} = H_{1} and S_{2} = H_{2} holds o(S_{1}, S_{2}) = H_{1} ⊔ H_{2}, then o = SubJoin G.

(88) If H_{1} = S_{1} and H_{2} = S_{2}, then SubJoin G(S_{1}, S_{2}) = H_{1} ⊔ H_{2}. Let us consider G. The functor SubMeet G yields a binary operation on

SubGr G and is defined as follows: (Def.12) for all S_{1}, S_{2}, H_{1}, H_{2} such that S_{1} = H_{1} and S_{2} = H_{2} holds

(SubMeet G)(S_{1}, S_{2}) = H_{1} ∩ H_{2}. One can prove the following two propositions:

(89) If for all S_{1}, S_{2}, H_{1}, H_{2} such that S_{1} = H_{1} and S_{2} = H_{2} holds o(S_{1}, S_{2}) = H_{1} ∩ H_{2}, then o = SubMeet G.

(90) If H_{1} = S_{1} and H_{2} = S_{2}, then SubMeet G(S_{1}, S_{2}) = H_{1} ∩ H_{2}.

Let us consider G. The functor _{G} yielding a lattice is defined as follows: (Def.13) _{G} = ⟨SubGr G, SubJoin G, SubMeet G⟩.

One can prove the following propositions:

(91) _{G} = ⟨SubGr G, SubJoin G, SubMeet G⟩.

(92) The carrier of _{G} = SubGr G.

(93) The join operation of _{G} = SubJoin G.

(94) The meet operation of _{G} = SubMeet G.

(95) _{G} is a lower bound lattice.

(96) _{G} is an upper bound lattice.

(97) _{G} is a bound lattice.

(98) ⊥_{_{G}} = {1}_{G}.

(99) ⊤_{_{G}} = Ω_{G}.

(100) n mod 2 = 0 or n mod 2 = 1.

(101) k · n mod k = 0 and k · n mod n = 0.

(102) If k > 1, then 1 mod k = 1.

(103) If k mod n = 0 and l = k − m · n, then l mod n = 0.

(104) If n ≠ 0 and k mod n = 0 and l < n, then (k + l) mod n = l.

(105) If k mod n = 0 and l mod n = 0, then (k + l) mod n = 0.

(106) If n ≠ 0 and k mod n = 0 and l mod n = 0, then (k + l) ÷ n = (k ÷ n) + (l ÷ n).

(107) If k ≠ 0, then k · n ÷ k = n.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

Lattice of Subgroups of a Group. Frattini … 47

[2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[6] M. I. Kargapołow and J. I. Mierzlakow. Podstawy teorii grup. PWN, Warszawa, 1989.

[7] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829–832, 1990.

[8] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

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

[10] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

[12] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

[13] Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1(5):979–981, 1990.

[14] Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955–962, 1990.

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

[16] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.

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

[18] Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855–864, 1990.

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

[20] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

[21] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

48

Equalities and Inequalities in Real Numbers

Andrzej Kondracki Warsaw University

Summary. The article is to give a number of useful theorems concerning equalities and inequalities in real numbers. Some of the theorems are extentions of [1] theorems, others were found to be needed in practice.

MML Identifier: REAL 2.

The terminology and notation used here are introduced in the following articles: [1], [3], [2], and [4]. In the sequel a, b, d, e will be real numbers. One can prove the following propositions:

(1) If b + a = b or a + b = b or b − a = b, then a = 0.

(2) Suppose that

(i) a − b = 0 or a + (−b) = 0 or (−b) + a = 0 or −a = −b or a − e = b − e or a − e = b + (−e) or a − e = (−e) + b or e − a = e − b or e − a = e + (−b) or e − a = (−b) + e. Then a = b.

(3) If a = −b, then a + b = 0 and b + a = 0 and −a = b.

(4) If a + b = 0 or b + a = 0, then a = −b.

(5) (−a) − b = (−b) − a.

(6) −(a + b) = (−a) + (−b) and −(a + b) = (−b) + (−a) and −(a + b) = (−b) − a and −(a + b) = (−a) − b.

(7) a − b = (−b) + a.

(8) −(a − b) = (−a) + b and −(a − b) = b − a and −(a − b) = b + (−a).

(9) −((−a) + b) = a − b and −((−a) + b) = a + (−b) and −((−a) + b) = (−b) + a.

(10) (i) a + b = −((−a) − b), (ii) a + b = −((−b) − a), (iii) a + b = −((−b) + (−a)), (iv) a + b = −((−a) + (−b)),

(v) a + b = a − (−b), (vi) a + b = b − (−a).

(11) If a+b = e or b+a = e, then a = e−b and a = e+(−b) and a = (−b)+e.

(12) If a = e−b or a = e+(−b) or a = (−b)+e, then a+b = e and b+a = e and b = e − a.

(13) If a + b = e + d, then a − e = d − b and a − d = e − b and b − e = d − a and b − d = e − a.

(14) If a − e = d − b, then a + b = e + d and a + b = d + e and b + a = d + e and b + a = e + d.

(15) If a − b = e − d, then a − e = b − d.

(16) If a + b = e − d or b + a = e − d, then a + d = e − b and d + a = e − b.

(17) (i) a = a + (b − b), (ii) a = (a + b) − b, (iii) a = a + (b + (−b)), (iv) a = (a + b) + (−b),

(v) a = a − (b − b), (vi) a = (a − b) + b, (vii) a = a − (b + (−b)), (viii) a = a + ((−b) + b), (ix) a = (a + (−b)) + b,

(x) a = b + (a − b), (xi) a = (b + a) − b, (xii) a = b + (a + (−b)), (xiii) a = (b + a) + (−b), (xiv) a = b − (b − a), (xv) a = (b − b) + a, (xvi) a = (−b) + (a + b), (xvii) a = ((−b) + a) + b, (xviii) a = (−b) + (b + a), (xix) a = ((−b) + b) + a, (xx) a = (−b) − ((−a) − b), (xxi) a = (−b) − ((−b) − a).

(18) a − (b − e) = a + (e − b) and a + (b − e) = (a + b) − e. (20)^{1} a + ((−b) − e) = (a − b) − e and a − ((−b) − e) = (a + b) + e.

(21) (a+b)+e = (a+e)+b and (a+b)+e = (b+e)+a and (a+b)+e = (e+a)+b and (a + b) + e = (e + b) + a.

(22) (a + b) − e = (a − e) + b and (a + b) − e = (b − e) + a and (a + b) − e = ((−e) + a) + b and (a + b) − e = ((−e) + b) + a.

(23) (a−b)+e = (e−b)+a and (a−b)+e = ((−b)+a)+e and (a−b)+e = ((−b) + e) + a.

1The proposition (19) was either repeated or obvious.

(24) (i) (a − b) − e = (a − e) − b, (ii) (a − b) − e = ((−b) + a) − e, (iii) (a − b) − e = ((−b) − e) + a, (iv) (a − b) − e = ((−e) + a) − b,

(v) (a − b) − e = ((−e) − b) + a.

(25) ((−a) + b) − e = ((−e) + b) − a and ((−a) + b) − e = ((−e) − a) + b.

(26) (i) ((−a) − b) − e = ((−a) − e) − b, (ii) ((−a) − b) − e = ((−b) − e) − a, (iii) ((−a) − b) − e = ((−e) − a) − b, (iv) ((−a) − b) − e = ((−e) − b) − a.

(27) (i) −((a + b) + e) = ((−a) − b) − e, (ii) −((a + b) − e) = ((−a) − b) + e, (iii) −((a − b) + e) = ((−a) + b) − e, (iv) −(((−a) + b) + e) = (a − b) − e,

(v) −((a − b) − e) = ((−a) + b) + e, (vi) −(((−a) + b) − e) = (a − b) + e, (vii) −(((−a) − b) + e) = (a + b) − e, (viii) −(((−a) − b) − e) = (a + b) + e.

(28) (i) a + e = (a + b) + (e − b), (ii) a + e = (b + a) + (e − b), (iii) a + e = (a − b) + (e + b), (iv) a + e = (a − b) + (b + e),

(v) e + a = (a + b) + (e − b), (vi) e + a = (b + a) + (e − b), (vii) e + a = (a − b) + (e + b), (viii) e + a = (a − b) + (b + e), (ix) a + e = (a + b) − (b − e),

(x) a + e = (b + a) − (b − e), (xi) e + a = (b + a) − (b − e), (xii) e + a = (a + b) − (b − e).

(29) (i) a − e = (a − b) − (e − b), (ii) a − e = (a − b) + (b − e), (iii) a − e = (a + b) − (e + b), (iv) a − e = (b + a) − (e + b),

(v) a − e = (b + a) − (b + e).

(30) If b ≠ 0, then if \frac{a}{b} = 1 or a · b^{−1} = 1 or b^{−1} · a = 1, then a = b.

(31) If e ≠ 0 and \frac{a}{e} = \frac{b}{e}, then a = b.

Next we state a number of propositions:

(32) If a · 1 = b · 1 or a · 1 = 1 · b or 1 · a = 1 · b or 1 · a = b · 1, then a = b.

(33) If a ≠ 0 and b ≠ 0, then if a^{−1} = b^{−1} or \frac{1}{a} = \frac{1}{b} or \frac{1}{a} = b^{−1}, then a = b.

(34) If b ≠ 0 and \frac{a}{b} = −1, then a = −b and b = −a.

(35) If a · b = 1 or b · a = 1, then a = \frac{1}{b} and a = b^{−1}.

(36) If b ≠ 0, then if a = \frac{1}{b} or a = b^{−1}, then a · b = 1 and b · a = 1 and a^{−1} = b and b = \frac{1}{a}.

(37) If b ≠ 0 but a · b = b or b · a = b, then a = 1.

(38) If b ≠ 0 but a · b = −b or b · a = −b, then a = −1.

(39) If a ≠ 0 and b ≠ 0 and \frac{b}{a} = b, then a = 1.

(40) If a ≠ 0 and b ≠ 0 and \frac{b}{a} = −b, then a = −1.

(41) If a ≠ 0, then \frac{1}{a} ≠ 0.

(42) If a ≠ 0 and b ≠ 0, then a · b^{−1} ≠ 0 and b^{−1} · a ≠ 0 and \frac{a}{b} ≠ 0 and a^{−1} · b^{−1} ≠ 0 and \frac{1}{a·b} ≠ 0.

(43) \frac{1}{1} = 1 and 1^{−1} = 1 and \frac{1}{−1} = −1 and (−1)^{−1} = −1 and (−1) · (−1) = 1.

(44) \frac{a}{1} = a and a · 1^{−1} = a and 1^{−1} · a = a.

(45) If a ≠ 0, then \frac{−a}{a} = −1 and \frac{a}{−a} = −1 and (−a)^{−1} = −a^{−1}.

(46) If a ≠ 0, then if a = a^{−1} or a = \frac{1}{a}, then a = 1 or a = −1.

(47) Suppose a ≠ 0 and b ≠ 0. Then (i) (a · b^{−1})^{−1} = a^{−1} · b, (ii) (a · b^{−1})^{−1} = b · a^{−1}, (iii) (b^{−1} · a)^{−1} = b · a^{−1}, (iv) (b^{−1} · a)^{−1} = a^{−1} · b,

(v) (a^{−1} · b^{−1})^{−1} = a · b.

(48) If a ≠ 0 and b ≠ 0, then \frac{1}{\frac{a}{b}} = \frac{b}{a} and \frac{a}{b}^{−1} = \frac{b}{a}.

(49) (i) (−a) · b = −b · a, (ii) a · (−b) = −b · a, (iii) (−a) · b = (−b) · a, (iv) (−a) · (−b) = a · b,

(v) (−a) · (−b) = b · a, (vi) −a · (−b) = a · b, (vii) −a · (−b) = b · a, (viii) −(−a) · b = a · b, (ix) −(−a) · b = b · a.

(50) If b ≠ 0, then \frac{a}{b} = 0 if and only if a = 0.

(51) If a ≠ 0 and b ≠ 0, then \frac{1}{a} · \frac{1}{b} = \frac{1}{a·b}.

(52) If a ≠ 0, then _{\frac{1}{\frac{1}{a}}} = a.

(53) Suppose e ≠ 0 and d ≠ 0. Then (i) \frac{a}{e} · \frac{b}{d} = \frac{b·a}{e·d}, (ii) \frac{a}{e} · \frac{b}{d} = \frac{b·a}{d·e}, (iii) \frac{a}{e} · \frac{b}{d} = \frac{a·b}{d·e}, (iv) \frac{a}{e} · \frac{b}{d} = \frac{a}{d} · \frac{b}{e}.

(54) If a ≠ 0, then a · \frac{1}{a} = 1.

(55) Suppose b ≠ 0 and e ≠ 0. Then (i) \frac{a}{b} = \frac{a·e}{e·b}, (ii) \frac{a}{b} = \frac{e·a}{b·e}, (iii) \frac{a}{b} = \frac{e·a}{e·b}, (iv) \frac{a}{b} = \frac{\frac{a}{e}}{\frac{b}{e}},

(v) \frac{a}{b} = e · \frac{a}{b·e}, (vi) \frac{a}{b} = e · \frac{a}{e·b}, (vii) \frac{a}{b} = \frac{a}{e·b} · e, (viii) \frac{a}{b} = \frac{a}{b·e} · e, (ix) \frac{a}{b} = e · \frac{\frac{a}{e}}{b},

(x) \frac{a}{b} = \frac{\frac{a}{e}}{b} · e, (xi) \frac{a}{b} = \frac{a}{e} · \frac{e}{b}.

(56) If b ≠ 0, then a · \frac{1}{b} = \frac{a}{b} and \frac{1}{b} · a = \frac{a}{b}.

(57) If b ≠ 0, then _{\frac{a}{\frac{1}{b}}} = a · b and _{\frac{a}{\frac{1}{b}}} = b · a.

(58) If b ≠ 0, then −\frac{a}{−b} = \frac{a}{b} and −\frac{−a}{b} = \frac{a}{b} and \frac{−a}{−b} = \frac{a}{b} and \frac{−a}{b} = \frac{a}{−b}.

(59) If e ≠ 0, then \frac{a+b}{e} = \frac{b}{e} + \frac{a}{e}.

(60) Suppose e ≠ 0 and d ≠ 0. Then (i) \frac{a}{e} + \frac{b}{d} = \frac{d·a+b·e}{e·d}, (ii) \frac{a}{e} + \frac{b}{d} = \frac{d·a+e·b}{e·d}, (iii) \frac{a}{e} + \frac{b}{d} = \frac{a·d+e·b}{e·d}, (iv) \frac{a}{e} + \frac{b}{d} = \frac{d·a+b·e}{d·e},

(v) \frac{a}{e} + \frac{b}{d} = \frac{d·a+e·b}{d·e}, (vi) \frac{a}{e} + \frac{b}{d} = \frac{a·d+e·b}{d·e}, (vii) \frac{a}{e} − \frac{b}{d} = \frac{d·a−b·e}{e·d}, (viii) \frac{a}{e} − \frac{b}{d} = \frac{d·a−e·b}{e·d}, (ix) \frac{a}{e} − \frac{b}{d} = \frac{a·d−e·b}{e·d},

(x) \frac{a}{e} − \frac{b}{d} = \frac{d·a−b·e}{d·e}, (xi) \frac{a}{e} − \frac{b}{d} = \frac{d·a−e·b}{d·e}, (xii) \frac{a}{e} − \frac{b}{d} = \frac{a·d−e·b}{d·e}.

(61) Suppose b ≠ 0 and e ≠ 0. Then (i) _{\frac{a}{\frac{b}{e}}} = \frac{e·a}{b}, (ii) _{\frac{a}{\frac{b}{e}}} = a · \frac{e}{b}, (iii) _{\frac{a}{\frac{b}{e}}} = \frac{e}{b} · a, (iv) _{\frac{a}{\frac{b}{e}}} = e · \frac{a}{b},

(v) _{\frac{a}{\frac{b}{e}}} = \frac{a}{b} · e.

(62) Suppose b ≠ 0. Then (i) a = a · \frac{b}{b}, (ii) a = \frac{a·b}{b}, (iii) a = a · (b · \frac{1}{b}), (iv) a = (a · b) · \frac{1}{b}, (v) a = _{\frac{a}{\frac{b}{b}}}, (vi) a = \frac{a}{b} · b, (vii) a = _{\frac{a}{b· \frac{1}{b}}}, (viii) a = a · (\frac{1}{b} · b), (ix) a = (a · \frac{1}{b}) · b,

(x) a = b · \frac{a}{b}, (xi) a = \frac{b·a}{b}, (xii) a = b · (a · \frac{1}{b}), (xiii) a = (b · a) · \frac{1}{b}, (xiv) a = \frac{b}{b} · a, (xv) a = (\frac{1}{b} · b) · a, (xvi) a = \frac{1}{b} · (b · a), (xvii) a = \frac{1}{b} · (a · b), (xviii) a = (\frac{1}{b} · a) · b.

The following propositions are true:

(63) For every a, b there exists e such that a = b − e.

(64) For all a, b such that a ≠ 0 and b ≠ 0 there exists e such that a = \frac{b}{e}.

(65) Suppose b ≠ 0. Then (i) \frac{a}{b} + e = \frac{a+e·b}{b}, (ii) \frac{a}{b} + e = \frac{a+b·e}{b}, (iii) \frac{a}{b} + e = \frac{b·e+a}{b}, (iv) \frac{a}{b} + e = \frac{e·b+a}{b},

(v) e + \frac{a}{b} = \frac{e·b+a}{b}, (vi) e + \frac{a}{b} = \frac{a+e·b}{b}, (vii) e + \frac{a}{b} = \frac{a+b·e}{b}, (viii) e + \frac{a}{b} = \frac{b·e+a}{b}.

(66) Suppose b ≠ 0. Then (i) \frac{a}{b} − e = \frac{a−e·b}{b}, (ii) \frac{a}{b} − e = \frac{a−b·e}{b}, (iii) e − \frac{a}{b} = \frac{e·b−a}{b}, (iv) e − \frac{a}{b} = \frac{b·e−a}{b}.

(67) Suppose b ≠ 0 and e ≠ 0. Then (i) \frac{\frac{a}{b}}{e} = \frac{a}{b·e}, (ii) \frac{\frac{a}{b}}{e} = \frac{a}{e·b}, (iii) \frac{\frac{a}{b}}{e} = \frac{\frac{a}{e}}{b}, (iv) \frac{\frac{a}{b}}{e} = \frac{1}{b} · \frac{a}{e},

(v) \frac{\frac{a}{b}}{e} = \frac{a}{e} · \frac{1}{b}, (vi) \frac{\frac{a}{b}}{e} = \frac{1}{e} · \frac{a}{b}, (vii) \frac{\frac{a}{b}}{e} = \frac{a}{b} · \frac{1}{e}, (viii) \frac{1}{e} · \frac{a}{b} = \frac{a}{b·e}, (ix) \frac{1}{e} · \frac{a}{b} = \frac{a}{e·b},

(x) \frac{a}{b} · \frac{1}{e} = \frac{a}{e·b}, (xi) \frac{a}{b} · \frac{1}{e} = \frac{a}{b·e}.

(68) Suppose b ≠ 0. Then e · \frac{a}{b} = \frac{e·a}{b} and e · \frac{a}{b} = \frac{a·e}{b} and \frac{a}{b} · e = \frac{a·e}{b} and \frac{a}{b} · e = \frac{e·a}{b}.

(69) (a · b) · e = (a · e) · b and (a · b) · e = (b · e) · a and (a · b) · e = (e · a) · b and (a · b) · e = (e · b) · a.

(70) Suppose e ≠ 0 and d ≠ 0. Then (i) \frac{a·b}{e·d} = \frac{\frac{a}{e} ·b}{d}, (ii) \frac{a·b}{e·d} = \frac{b· \frac{a}{e}}{d},

(iii) \frac{a·b}{e·d} = \frac{\frac{b}{e} ·a}{d},

(iv) \frac{a·b}{e·d} = \frac{a· \frac{b}{e}}{d},

(v) \frac{a·b}{d·e} = \frac{\frac{a}{e} ·b}{d}, (vi) \frac{a·b}{d·e} = \frac{b· \frac{a}{e}}{d},

(vii) \frac{a·b}{d·e} = \frac{a· \frac{b}{e}}{d},

(viii) \frac{a·b}{d·e} = \frac{\frac{b}{e} ·a}{d}.

(71) (−1)·a = −a and a·(−1) = −a and (−a)·(−1) = a and (−1)·(−a) = a and −a = \frac{a}{−1} and a = \frac{−a}{−1}.

(72) If e ≠ 0, then if a · e = b or e · a = b, then a = \frac{b}{e}.

(73) If e ≠ 0 and a = \frac{b}{e}, then a · e = b and e · a = b.

(74) If a ≠ 0 and e ≠ 0 and a = \frac{b}{e}, then e = \frac{b}{a}.

(75) If e ≠ 0 and d ≠ 0, then if a · e = b · d or e · a = b · d or e · a = d · b or a · e = d · b, then \frac{a}{d} = \frac{b}{e}.

(76) If e ≠ 0 and d ≠ 0 and \frac{a}{d} = \frac{b}{e}, then a · e = b · d and e · a = b · d and e · a = d · b and a · e = d · b.

(77) If e ≠ 0 and d ≠ 0, then if a · e = \frac{b}{d} or e · a = \frac{b}{d}, then a · d = \frac{b}{e} and d · a = \frac{b}{e}.

(78) Suppose b ≠ 0. Then (i) a · e = (a · b) · \frac{e}{b}, (ii) a · e = (b · a) · \frac{e}{b}, (iii) a · e = \frac{a}{b} · (e · b), (iv) a · e = \frac{a}{b} · (b · e),

(v) e · a = (a · b) · \frac{e}{b}, (vi) e · a = (b · a) · \frac{e}{b}, (vii) e · a = \frac{a}{b} · (e · b), (viii) e · a = \frac{a}{b} · (b · e).

(79) Suppose b ≠ 0 and e ≠ 0. Then a · e = _{\frac{a·b}{\frac{b}{e}}} and a · e = _{\frac{b·a}{\frac{b}{e}}} and e · a = _{\frac{b·a}{\frac{b}{e}}} and e · a = _{\frac{a·b}{\frac{b}{e}}}.

(80) If b ≠ 0, then \frac{a}{b} · e = \frac{e}{b} · a and \frac{a}{b} · e = (\frac{1}{b} · a) · e and \frac{a}{b} · e = (\frac{1}{b} · e) · a.

(81) (−a) · (−b) = a · b and (−a) · (−b) = b · a.

(82) If b ≠ 0 and d ≠ 0 and b ≠ d and \frac{a}{b} = \frac{e}{d}, then \frac{a}{b} = \frac{a−e}{b−d}.

(83) Suppose b ≠ 0 and d ≠ 0 and b ≠ −d and \frac{a}{b} = \frac{e}{d}. Then \frac{a}{b} = \frac{a+e}{b+d} and \frac{a}{b} = \frac{e+a}{b+d} and \frac{a}{b} = \frac{e+a}{d+b} and \frac{a}{b} = \frac{a+e}{d+b}.

(84) (i) e · (a + b) = a · e + e · b, (ii) e · (a + b) = e · a + b · e, (iii) e · (a + b) = a · e + b · e, (iv) (a + b) · e = e · a + b · e,

(v) (a + b) · e = a · e + e · b, (vi) (a + b) · e = e · a + e · b, (vii) e · (b + a) = a · e + e · b, (viii) e · (b + a) = e · a + b · e, (ix) e · (b + a) = a · e + b · e,

(x) (b + a) · e = e · a + b · e, (xi) (b + a) · e = a · e + e · b, (xii) (b + a) · e = e · a + e · b, (xiii) (a + b) · e = b · e + a · e, (xiv) e · (a + b) = e · b + e · a.

(85) (i) e · (a − b) = a · e − e · b, (ii) e · (a − b) = e · a − b · e, (iii) e · (a − b) = a · e − b · e, (iv) (a − b) · e = e · a − b · e,

(v) (a − b) · e = a · e − e · b, (vi) (a − b) · e = e · a − e · b, (vii) (a − b) · e = (b − a) · (−e), (viii) (a − b) · e = −(b − a) · e, (ix) e · (a − b) = (−e) · (b − a),

(x) e · (a − b) = −e · (b − a).

(86) If a ≠ 0, then if \frac{1}{a} = 1 or a^{−1} = 1, then a = 1.

(87) If a ≠ 0, then if \frac{1}{a} = −1 or a^{−1} = −1, then a = −1.

(88) (i) 2 · a = a + a, (ii) a · 2 = a + a, (iii) 3 · a = (a + a) + a, (iv) a · 3 = (a + a) + a,

(v) 4 · a = ((a + a) + a) + a, (vi) a · 4 = ((a + a) + a) + a.

(89) \frac{a+a}{2} = a and \frac{(a+a)+a}{3} = a and \frac{((a+a)+a)+a}{4} = a and \frac{a+a}{4} = \frac{a}{2}.

(90) (i) \frac{a}{2} + \frac{a}{2} = a, (ii) (\frac{a}{3} + \frac{a}{3}) + \frac{a}{3} = a, (iii) ((\frac{a}{4} + \frac{a}{4}) + \frac{a}{4}) + \frac{a}{4} = a, (iv) \frac{a}{4} + \frac{a}{4} = \frac{a}{2}.

(91) If b ≠ 0, then \frac{a}{2·b} + \frac{a}{2·b} = \frac{a}{b} and (\frac{a}{3·b} + \frac{a}{3·b}) + \frac{a}{3·b} = \frac{a}{b}.

(92) Suppose e ≠ 0. Then (i) a + b = e · (\frac{a}{e} + \frac{b}{e}), (ii) b + a = e · (\frac{a}{e} + \frac{b}{e}), (iii) b + a = (\frac{a}{e} + \frac{b}{e}) · e, (iv) a + b = (\frac{a}{e} + \frac{b}{e}) · e.

(93) If e ≠ 0, then a − b = e · (\frac{a}{e} − \frac{b}{e}) and a − b = (\frac{a}{e} − \frac{b}{e}) · e. One can prove the following propositions:

(94) Suppose e ≠ 0. Then (i) a + b = \frac{a·e+b·e}{e}, (ii) a + b = \frac{a·e+e·b}{e}, (iii) a + b = \frac{e·a+e·b}{e}, (iv) a + b = \frac{e·a+b·e}{e},

(v) b + a = \frac{e·a+b·e}{e}, (vi) b + a = \frac{e·a+e·b}{e}, (vii) b + a = \frac{a·e+e·b}{e}, (viii) b + a = \frac{a·e+b·e}{e}.

(95) Suppose e ≠ 0. Then (i) a − b = \frac{a·e−b·e}{e}, (ii) a − b = \frac{a·e−e·b}{e}, (iii) a − b = \frac{e·a−e·b}{e}, (iv) a − b = \frac{e·a−b·e}{e}.

(96) Suppose a ≠ 0. Then (i) a + b = a · (1 + \frac{b}{a}), (ii) a + b = (1 + \frac{b}{a}) · a, (iii) a + b = (\frac{b}{a} + 1) · a, (iv) a + b = a · (\frac{b}{a} + 1),

(v) b + a = a · (1 + \frac{b}{a}), (vi) b + a = (1 + \frac{b}{a}) · a, (vii) b + a = (\frac{b}{a} + 1) · a, (viii) b + a = a · (\frac{b}{a} + 1).

(97) If a ≠ 0, then a − b = a · (1 − \frac{b}{a}) and a − b = (1 − \frac{b}{a}) · a.

(98) (a − b) · (e − d) = (b − a) · (d − e).

(99) (i) ((a + b) + e) · d = (a · d + b · d) + e · d, (ii) d · ((a + b) + e) = (d · a + d · b) + d · e, (iii) ((a + b) − e) · d = (a · d + b · d) − e · d, (iv) d · ((a + b) − e) = (d · a + d · b) − d · e,

(v) ((a − b) + e) · d = (a · d − b · d) + e · d, (vi) d · ((a − b) + e) = (d · a − d · b) + d · e, (vii) ((a − b) − e) · d = (a · d − b · d) − e · d, (viii) d · ((a − b) − e) = (d · a − d · b) − d · e.

(100) Suppose d ≠ 0. Then (i) \frac{(a+b)+e}{d} = (\frac{a}{d} + \frac{b}{d}) + \frac{e}{d}, (ii) \frac{(a+b)−e}{d} = (\frac{a}{d} + \frac{b}{d}) − \frac{e}{d}, (iii) \frac{(a−b)+e}{d} = (\frac{a}{d} − \frac{b}{d}) + \frac{e}{d}, (iv) \frac{(a−b)−e}{d} = (\frac{a}{d} − \frac{b}{d}) − \frac{e}{d}.

(101) (i) (a + b) · (e + d) = ((a · e + a · d) + b · e) + b · d, (ii) (a + b) · (e − d) = ((a · e − a · d) + b · e) − b · d, (iii) (a − b) · (e + d) = ((a · e + a · d) − b · e) − b · d, (iv) (a − b) · (e − d) = ((a · e − a · d) − b · e) + b · d.

(103)^{2} If a ≥ b, then a + e ≥ e + b and e + a ≥ e + b and e + a ≥ b + e.

(104) If a + e ≥ b + e or a + e ≥ e + b or e + a ≥ e + b or e + a ≥ b + e or a − e ≥ b − e, then a ≥ b.

(105) Suppose that

(i) a − b ≤ 0 or a + (−b) ≤ 0 or (−b) + a ≤ 0 or −a ≥ −b or b − a ≥ 0 or b + (−a) ≥ 0 or (−a) + b ≥ 0 or a − e ≤ b + (−e) or a − e ≤ (−e) + b or a + (−e) ≤ b − e or (−e) + a ≤ b − e or e − a ≥ e − b. Then a ≤ b.

(106) Suppose that

(i) a − b < 0 or a + (−b) < 0 or (−b) + a < 0 or −a > −b or b − a > 0 or b + (−a) > 0 or (−a) + b > 0 or a − e < b + (−e) or a − e < (−e) + b or a + (−e) < b − e or (−e) + a < b − e or e − a > e − b. Then a < b.

(107) Suppose a ≤ b. Then a − b ≤ 0 and a + (−b) ≤ 0 and (−b) + a ≤ 0 and b − a ≥ 0 and b + (−a) ≥ 0 and (−a) + b ≥ 0 and −a ≥ −b and e − a ≥ e − b.

(108) Suppose a < b. Then a − b < 0 and a + (−b) < 0 and (−b) + a < 0 and b − a > 0 and b + (−a) > 0 and (−a) + b > 0 and −a > −b and e − a > e − b.

(109) If a ≤ −b, then a + b ≤ 0 and b + a ≤ 0 and −a ≥ b.

(110) If a < −b, then a + b < 0 and b + a < 0 and −a > b.

(111) If −a ≤ b, then b + a ≥ 0 and a + b ≥ 0 and a ≥ −b.

(112) If −b < a, then a + b > 0 and b + a > 0 and b > −a.

(113) If a + b ≤ 0 or b + a ≤ 0, then a ≤ −b.

(114) If a + b < 0 or b + a < 0, then a < −b.

(115) If a + b ≥ 0 or b + a ≥ 0, then a ≥ −b.

2The proposition (102) was either repeated or obvious.

(116) If a + b > 0 or b + a > 0, then a > −b.

(117) Suppose b > 0. Then (i) if \frac{a}{b} > 1, then a > b, (ii) if \frac{a}{b} < 1, then a < b, (iii) if \frac{a}{b} > −1, then a > −b and b > −a, (iv) if \frac{a}{b} < −1, then a < −b and b < −a.

(118) Suppose b > 0. Then (i) if \frac{a}{b} ≥ 1, then a ≥ b, (ii) if \frac{a}{b} ≤ 1, then a ≤ b, (iii) if \frac{a}{b} ≥ −1, then a ≥ −b and b ≥ −a, (iv) if \frac{a}{b} ≤ −1, then a ≤ −b and b ≤ −a.

(119) Suppose b < 0. Then (i) if \frac{a}{b} > 1, then a < b, (ii) if \frac{a}{b} < 1, then a > b, (iii) if \frac{a}{b} > −1, then a < −b and b < −a, (iv) if \frac{a}{b} < −1, then a > −b and b > −a.

(120) Suppose b < 0. Then (i) if \frac{a}{b} ≥ 1, then a ≤ b, (ii) if \frac{a}{b} ≤ 1, then a ≥ b, (iii) if \frac{a}{b} ≥ −1, then a ≤ −b and b ≤ −a, (iv) if \frac{a}{b} ≤ −1, then a ≥ −b and b ≥ −a.

(121) If a ≥ 0 or a > 0 but b ≥ 0 or b > 0 or a ≤ 0 or a < 0 but b ≤ 0 or b < 0, then a · b ≥ 0 and b · a ≥ 0.

(122) If a < 0 and b < 0 or a > 0 and b > 0, then a · b > 0.

(123) If a ≥ 0 or a > 0 but b ≤ 0 or b < 0 or a ≤ 0 or a < 0 but b ≥ 0 or b > 0, then a · b ≤ 0 and b · a ≤ 0.

(124) If a > 0 and b < 0, then a · b < 0 and b · a < 0.

One can prove the following propositions:

(125) If a ≤ 0 and b < 0 or a ≥ 0 and b > 0, then \frac{a}{b} ≥ 0.

(126) If a ≥ 0 and b < 0 or a ≤ 0 and b > 0, then \frac{a}{b} ≤ 0.

(127) If a > 0 and b > 0 or a < 0 and b < 0, then \frac{a}{b} > 0.

(128) If a < 0 and b > 0, then \frac{a}{b} < 0 and \frac{b}{a} < 0.

(129) If a · b ≤ 0, then a ≥ 0 and b ≤ 0 or a ≤ 0 and b ≥ 0. (131)^{3} If a · b > 0, then a > 0 and b > 0 or a < 0 and b < 0.

(132) If a · b < 0, then a > 0 and b < 0 or a < 0 and b > 0.

(133) If b ≠ 0 and \frac{a}{b} ≤ 0, then b > 0 and a ≤ 0 or b < 0 and a ≥ 0.

(134) If b ≠ 0 and \frac{a}{b} ≥ 0, then b > 0 and a ≥ 0 or b < 0 and a ≤ 0.

(135) If b ≠ 0 and \frac{a}{b} < 0, then b < 0 and a > 0 or b > 0 and a < 0.

(136) If b ≠ 0 and \frac{a}{b} > 0, then b > 0 and a > 0 or b < 0 and a < 0.

3The proposition (130) was either repeated or obvious.

(137) If a > 1 but b > 1 or b ≥ 1 or a < −1 but b < −1 or b ≤ −1, then a · b > 1 and b · a > 1.

(138) If a ≥ 1 and b ≥ 1 or a ≤ −1 and b ≤ −1, then a · b ≥ 1.

(139) Suppose that

(i) 0 < a or 0 ≤ a but a < 1 but 0 < b or 0 ≤ b but b < 1 or b ≤ 1 or 0 > a or 0 ≥ a but a > −1 but 0 > b or 0 ≥ b but b > −1 or b ≥ −1. Then a · b < 1 and b · a < 1.

(140) If 0 ≤ a and a ≤ 1 and 0 ≤ b and b ≤ 1 or 0 ≥ a and a ≥ −1 and 0 ≥ b and b ≥ −1, then a · b ≤ 1.

(141) If e < 0 and a ≤ b or e > 0 and a ≥ b, then \frac{a}{e} ≥ \frac{b}{e}.

(142) If 0 < a and a < b or b < a and a < 0, then \frac{a}{b} < 1 and \frac{b}{a} > 1.

(143) If 0 < a and a ≤ b or b ≤ a and a < 0, then \frac{a}{b} ≤ 1 and \frac{b}{a} ≥ 1.

(144) If a > 0 and b > 1 or a < 0 and b < 1, then a · b > a and b · a > a.

(145) If a > 0 and b < 1 or a < 0 and b > 1, then a · b < a and b · a < a.

(146) If a > 0 or a ≥ 0 but b > 1 or b ≥ 1 or a < 0 or a ≤ 0 but b < 1 or b ≤ 1, then a · b ≥ a and b · a ≥ a.

(147) If a > 0 or a ≥ 0 but b < 1 or b ≤ 1 or a < 0 or a ≤ 0 but b > 1 or b ≥ 1, then a · b ≤ a and b · a ≤ a.

(148) a > 0 if and only if −a < 0 but a ≥ 0 if and only if −a ≤ 0 but a ≤ 0 if and only if −a ≥ 0.

(149) If a < 0, then \frac{1}{a} < 0 and a^{−1} < 0 but if a > 0, then \frac{1}{a} > 0.

(150) If a ≠ 0, then if \frac{1}{a} < 0, then a < 0 but if \frac{1}{a} > 0, then a > 0.

(151) If 0 < a or b < 0 but a < b, then \frac{1}{a} > \frac{1}{b}.

(152) If 0 < a or b < 0 but a ≤ b, then \frac{1}{a} ≥ \frac{1}{b}.

(153) If a < 0 and b > 0, then \frac{1}{a} < \frac{1}{b}.

(154) If a ≠ 0 and b ≠ 0 but \frac{1}{b} > 0 or \frac{1}{a} < 0 and \frac{1}{a} > \frac{1}{b}, then a < b.

(155) If a ≠ 0 and b ≠ 0 but \frac{1}{b} > 0 or \frac{1}{a} < 0 and \frac{1}{a} ≥ \frac{1}{b}, then a ≤ b. Next we state a number of propositions:

(156) If a ≠ 0 and b ≠ 0 and \frac{1}{a} < 0 and \frac{1}{b} > 0, then a < b.

(157) If a < −1, then 0 > \frac{1}{a} and \frac{1}{a} > −1.

(158) If a ≤ −1, then 0 > \frac{1}{a} and \frac{1}{a} ≥ −1.

(159) If −1 < a and a < 0, then \frac{1}{a} < −1.

(160) If −1 ≤ a and a < 0, then \frac{1}{a} ≤ −1.

(161) If 0 < a and a < 1, then \frac{1}{a} > 1.

(162) If 0 < a and a ≤ 1, then \frac{1}{a} ≥ 1.

(163) If 1 < a, then 0 < \frac{1}{a} and \frac{1}{a} < 1.

(164) If 1 ≤ a, then 0 < \frac{1}{a} and \frac{1}{a} ≤ 1.

(165) If b ≤ e − a, then a ≤ e − b but if b ≥ e − a, then a ≥ e − b.

(166) If b < e − a, then a < e − b but if b > e − a, then a > e − b.

(167) If a + b ≤ e + d, then a − e ≤ d − b and e − a ≥ b − d and a − d ≤ e − b and d − a ≥ b − e.

(168) If a + b < e + d, then a − e < d − b and e − a > b − d and a − d < e − b and d − a > b − e.

(169) Suppose a − b ≤ e − d. Then a + d ≤ e + b and d + a ≤ e + b and d + a ≤ b + e and a + d ≤ b + e and a − e ≤ b − d and e − a ≥ d − b and b − a ≥ d − e.

(170) Suppose a − b < e − d. Then a + d < e + b and d + a < e + b and d + a < b + e and a + d < b + e and a − e < b − d and e − a > d − b and b − a > d − e.

(171) (i) If a + b ≤ e − d or b + a ≤ e − d, then a + d ≤ e − b and d + a ≤ e − b, (ii) if a + b ≥ e − d or b + a ≥ e − d, then a + d ≥ e − b and d + a ≥ e − b.

(172) (i) If a + b < e − d or b + a < e − d, then a + d < e − b and d + a < e − b, (ii) if a + b > e − d or b + a > e − d, then a + d > e − b and d + a > e − b.

(173) If a < 0, then b + a < b and a + b < b and b − a > b but if a + b < b or b + a < b or b − a > b, then a < 0.

(174) If a ≤ 0, then b + a ≤ b and a + b ≤ b and b − a ≥ b but if b + a ≤ b or a + b ≤ b or b − a ≥ b, then a ≤ 0.

(175) If a > 0, then b + a > b and a + b > b and b − a < b but if b + a > b or a + b > b or b − a < b, then a > 0.

(176) If a ≥ 0, then b + a ≥ b and a + b ≥ b and b − a ≤ b but if b + a ≥ b or a + b ≥ b or b − a ≤ b, then a ≥ 0.

(177) (i) If b > 0 but a · b ≤ e or b · a ≤ e, then a ≤ \frac{e}{b}, (ii) if b < 0 but a · b ≤ e or b · a ≤ e, then a ≥ \frac{e}{b}, (iii) if b > 0 but a · b ≥ e or b · a ≥ e, then a ≥ \frac{e}{b}, (iv) if b < 0 but a · b ≥ e or b · a ≥ e, then a ≤ \frac{e}{b}.

(178) (i) If b > 0 but a · b < e or b · a < e, then a < \frac{e}{b}, (ii) if b < 0 but a · b < e or b · a < e, then a > \frac{e}{b}, (iii) if b > 0 but a · b > e or b · a > e, then a > \frac{e}{b}, (iv) if b < 0 but a · b > e or b · a > e, then a < \frac{e}{b}.

(179) (i) If b > 0 and a ≥ \frac{e}{b}, then a · b ≥ e and b · a ≥ e, (ii) if b > 0 and a ≤ \frac{e}{b}, then a · b ≤ e and b · a ≤ e, (iii) if b < 0 and a ≥ \frac{e}{b}, then a · b ≤ e and b · a ≤ e, (iv) if b < 0 and a ≤ \frac{e}{b}, then a · b ≥ e and b · a ≥ e.

(180) (i) If b > 0 and a > \frac{e}{b}, then a · b > e and b · a > e, (ii) if b > 0 and a < \frac{e}{b}, then a · b < e and b · a < e, (iii) if b < 0 and a > \frac{e}{b}, then a · b < e and b · a < e, (iv) if b < 0 and a < \frac{e}{b}, then a · b > e and b · a > e.

(181) If for every a such that a > 0 holds b + a ≥ e or for every a such that a < 0 holds b − a ≥ e, then b ≥ e.

(182) If for every a such that a > 0 holds b − a ≤ e or for every a such that a < 0 holds b + a ≤ e, then b ≤ e.

(183) If for every a such that a > 1 holds b · a ≥ e or for every a such that 0 < a and a < 1 holds \frac{b}{a} ≥ e, then b ≥ e.

(184) If for every a such that 0 < a and a < 1 holds b · a ≤ e or for every a such that a > 1 holds \frac{b}{a} ≤ e, then b ≤ e.

(185) Suppose b > 0 and d > 0 or b < 0 and d < 0 but a·d < e·b or d·a < e·b or d · a < b · e or a · d < b · e. Then \frac{a}{b} < \frac{e}{d}.

(186) Suppose b > 0 and d < 0 or b < 0 and d > 0 but a·d < e·b or d·a < e·b

or d · a < b · e or a · d < b · e. Then \frac{a}{b} > \frac{e}{d}. The following propositions are true:

(187) Suppose b > 0 and d > 0 or b < 0 and d < 0 but a·d ≤ e·b or d·a ≤ e·b or d · a ≤ b · e or a · d ≤ b · e. Then \frac{a}{b} ≤ \frac{e}{d}.

(188) Suppose b > 0 and d < 0 or b < 0 and d > 0 but a·d ≤ e·b or d·a ≤ e·b or d · a ≤ b · e or a · d ≤ b · e. Then \frac{a}{b} ≥ \frac{e}{d}.

(189) Suppose b > 0 and d > 0 or b < 0 and d < 0 but \frac{a}{b} < \frac{e}{d}. Then a·d < e·b and d · a < e · b and d · a < b · e and a · d < b · e.

(190) Suppose b < 0 and d > 0 or b > 0 and d < 0 but \frac{a}{b} < \frac{e}{d}. Then a·d > e·b and d · a > e · b and d · a > b · e and a · d > b · e.

(191) Suppose b > 0 and d > 0 or b < 0 and d < 0 but \frac{a}{b} ≤ \frac{e}{d}. Then a·d ≤ e·b and d · a ≤ e · b and d · a ≤ b · e and a · d ≤ b · e.

(192) Suppose b < 0 and d > 0 or b > 0 and d < 0 but \frac{a}{b} ≤ \frac{e}{d}. Then a·d ≥ e·b and d · a ≥ e · b and d · a ≥ b · e and a · d ≥ b · e.

(193) Suppose b < 0 and d < 0 or b > 0 and d > 0. Then (i) if a · b < \frac{e}{d} or b · a < \frac{e}{d}, then a · d < \frac{e}{b} and d · a < \frac{e}{b}, (ii) if a · b > \frac{e}{d} or b · a > \frac{e}{d}, then a · d > \frac{e}{b} and d · a > \frac{e}{b}.

(194) Suppose b < 0 and d > 0 or b > 0 and d < 0. Then (i) if a · b < \frac{e}{d} or b · a < \frac{e}{d}, then a · d > \frac{e}{b} and d · a > \frac{e}{b}, (ii) if a · b > \frac{e}{d} or b · a > \frac{e}{d}, then a · d < \frac{e}{b} and d · a < \frac{e}{b}.

(195) Suppose b < 0 and d < 0 or b > 0 and d > 0. Then (i) if a · b ≤ \frac{e}{d} or b · a ≤ \frac{e}{d}, then a · d ≤ \frac{e}{b} and d · a ≤ \frac{e}{b}, (ii) if a · b ≥ \frac{e}{d} or b · a ≥ \frac{e}{d}, then a · d ≥ \frac{e}{b} and d · a ≥ \frac{e}{b}.

(196) Suppose b < 0 and d > 0 or b > 0 and d < 0. Then (i) if a · b ≤ \frac{e}{d} or b · a ≤ \frac{e}{d}, then a · d ≥ \frac{e}{b} and d · a ≥ \frac{e}{b}, (ii) if a · b ≥ \frac{e}{d} or b · a ≥ \frac{e}{d}, then a · d ≤ \frac{e}{b} and d · a ≤ \frac{e}{b}.

(197) Suppose 0 < a or 0 ≤ a but a < b or a ≤ b but 0 < e or 0 ≤ e and e ≤ d. Then a · e ≤ b · d and a · e ≤ d · b and e · a ≤ d · b and e · a ≤ b · d.

(198) Suppose 0 > a or 0 ≥ a but a > b or a ≥ b but 0 > e or 0 ≥ e and e ≥ d. Then a · e ≤ b · d and a · e ≤ d · b and e · a ≤ d · b and e · a ≤ b · d.

(199) Suppose 0 < a but a ≤ b or a < b and 0 < e and e < d or 0 > a but a ≥ b or a > b and 0 > e and e > d. Then a · e < b · d and a · e < d · b and e · a < d · b and e · a < b · d.

(200) If e > 0 but a > 0 or b < 0 and a < b, then \frac{e}{a} > \frac{e}{b}.

(201) If e > 0 or e ≥ 0 but a > 0 or b < 0 and a ≤ b, then \frac{e}{a} ≥ \frac{e}{b}.

Equalities and Inequalities in Real Numbers 63

(202) If e < 0 but a > 0 or b < 0 and a < b, then \frac{e}{a} < \frac{e}{b}.

(203) If e < 0 or e ≤ 0 but a > 0 or b < 0 and a ≤ b, then \frac{e}{a} ≤ \frac{e}{b}. Next we state the proposition

(204) For all subsets X, Y of such that X ≠ ∅ and Y ≠ ∅ and for all a, b such that a ∈ X and b ∈ Y holds a ≤ b there exists d such that for every a such that a ∈ X holds a ≤ d and for every b such that b ∈ Y holds d ≤ b.

References

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

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

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

[4] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

64

Countable Sets and Hessenberg’s Theorem

Grzegorz Bancerek Warsaw University Białystok

Summary. The concept of countable sets is introduced and there are shown some facts which deal with finite and countable sets. Besides, the article includes theorems and lemmas on the sum and the product of infinite cardinals. The most important of them is Hessenberg’s theorem which says that for every infinite cardinal m the product m · m is equal to m.

MML Identifier: CARD 4.

The papers [20], [16], [3], [11], [9], [15], [5], [8], [7], [21], [19], [2], [1], [10], [22], [12], [13], [18], [14], [17], [4], and [6] provide the terminology and notation for this paper. For simplicity we follow the rules: X, Y are sets, D is a non-empty set, m, n, n_{1}, n_{2}, n_{3}, m_{2}, m_{1} are natural numbers, A, B are ordinal numbers, L, K, M, N are cardinal numbers, x is arbitrary, and f is a function. Next we state a number of propositions:

(1) X is finite if and only if \overline{\overline{X}} is finite.

(2) X is finite if and only if \overline{\overline{X}} < ℵ_{0}.

(3) If X is finite, then \overline{\overline{X}} ∈ ℵ_{0} and \overline{\overline{X}} ∈ ω.

(4) X is finite if and only if there exists n such that \overline{\overline{X}} = \overline{\overline{n}}.

(5) succ A \ {A} = A.

(6) If A ≈ ord(n), then A = ord(n).

(7) A is finite if and only if A ∈ ω.

(8) A is not finite if and only if ω ⊆ A.

(9) M is finite if and only if M ∈ ℵ_{0}.

(10) M is finite if and only if M < ℵ_{0}.

(11) M is not finite if and only if ℵ_{0} ⊆ M.

(12) M is not finite if and only if ℵ_{0} ≤ M.

(13) If N is finite and M is not finite, then N < M and N ≤ M.

(14) X is not finite if and only if there exists Y such that Y ⊆ X and \overline{\overline{Y}} = ℵ_{0}.

(15) ω is not finite and is not finite.

(16) ℵ_{0} is not finite.

(17) X = ∅ if and only if \overline{\overline{X}} = \overline{0}.

(18) M ≠ \overline{0} if and only if \overline{0} < M.

(19) \overline{0} ≤ M.

(20) \overline{\overline{X}} = \overline{\overline{Y}} if and only if X^{+} = Y^{+}.

(21) M = N if and only if N^{+} = M^{+}.

(22) N < M if and only if N^{+} ≤ M.

(23) N < M^{+} if and only if N ≤ M.

(24) \overline{0} < M if and only if \overline{1} ≤ M.

(25) \overline{1} < M if and only if \overline{2} ≤ M.

(26) If M is finite but N ≤ M or N < M, then N is finite.

(27) A is a limit ordinal number if and only if for all B, n such that B ∈ A holds B + ord(n) ∈ A.

(28) A+succ ord(n) = succ A+ord(n) and A+ord(n+1) = succ A+ord(n).

(29) There exists n such that A · succ 1 = A + ord(n).

(30) If A is a limit ordinal number, then A · succ 1 = A.

(31) If ω ⊆ A, then 1 + A = A.

Next we state a number of propositions:

(32) If M is not finite, then ord(M) is a limit ordinal number.

(33) If M is not finite, then M + M = M.

(34) If M is not finite but N ≤ M or N < M, then M + N = M and N + M = M.

(35) If X is not finite but X ≈ Y or Y ≈ X, then X ∪ Y ≈ X and \overline{\overline{X ∪ Y}} = \overline{\overline{X}}.

(36) If X is not finite and Y is finite, then X ∪ Y ≈ X and \overline{\overline{X ∪ Y}} = \overline{\overline{X}}.

(37) If X is not finite but \overline{\overline{Y}} < \overline{\overline{X}} or \overline{\overline{Y}} ≤ \overline{\overline{X}}, then X ∪ Y ≈ X and \overline{\overline{X ∪ Y}} = \overline{\overline{X}}.

(38) If M is finite and N is finite, then M + N is finite.

(39) If M is not finite, then M + N is not finite and N + M is not finite.

(40) If M is finite and N is finite, then M · N is finite.

(41) If K < L and M < N or K ≤ L and M < N or K < L and M ≤ N or K ≤ L and M ≤ N, then K + M ≤ L + N and M + K ≤ L + N.

(42) If M < N or M ≤ N, then K + M ≤ K + N and K + M ≤ N + K and

M + K ≤ K + N and M + K ≤ N + K. Let us consider X. We say that X is countable if and only if:

(Def.1) \overline{\overline{X}} ≤ ℵ_{0}.

One can prove the following propositions:

(43) If X is finite, then X is countable.

(44) ω is countable and is countable.

(45) X is countable if and only if there exists f such that dom f = and X ⊆ rng f.

(46) If Y ⊆ X and X is countable, then Y is countable.

(47) If X is countable and Y is countable, then X ∪ Y is countable.

(48) If X is countable, then X ∩ Y is countable and Y ∩ X is countable.

(49) If X is countable, then X \ Y is countable.

(50) If X is countable and Y is countable, then X− . Y is countable. The scheme Lambda2N deals with a binary functor F yielding a natural

number and states that:

there exists a function f from [: , :] into such that for all n, m holds f(⟨⟨n, m⟩⟩) = F(n, m) for all values of the parameter.

In the sequel r will denote a real number. Next we state the proposition (51) r ≠ 0 or n = 0 if and only if r^{n} ≠ 0.

Let m, n be natural numbers. Then m^{n} is a natural number.

One can prove the following propositions:

(52) If 2^{n_{1}} · (2 · m_{1} + 1) = 2^{n_{2}} · (2 · m_{2} + 1), then n_{1} = n_{2} and m_{1} = m_{2}.

(53) [: , :] ≈ and \overline{\overline{}}= \overline{\overline{[: , :]}} .

(54) ℵ_{0} · ℵ_{0} = ℵ_{0}.

(55) If X is countable and Y is countable, then [: X, Y :] is countable.

(56) D^{1} ≈ D and \overline{\overline{D^{1}}} = \overline{\overline{D}}.

We now state a number of propositions:

(57) [: D^{n}, D^{m}:] ≈ D^{n+m} and \overline{\overline{[: D^{n}, D^{m}:]}} = \overline{\overline{D^{n+m}}}.

(58) If D is countable, then D^{n} is countable.

(59) If \overline{\overline{dom f}} ≤ M and for every x such that x ∈ dom f holds \overline{\overline{f(x)}} ≤ N, then \overline{\overline{⋃ f}} ≤ M · N.

(60) If \overline{\overline{X}} ≤ M and for every Y such that Y ∈ X holds \overline{\overline{Y}} ≤ N, then \overline{\overline{⋃ X}} ≤ M · N.

(61) For every f such that dom f is countable and for every x such that x ∈ dom f holds f(x) is countable holds ⋃ f is countable.

(62) If X is countable and for every Y such that Y ∈ X holds Y is countable, then ⋃ X is countable.

(63) For every f such that dom f is finite and for every x such that x ∈ dom f holds f(x) is finite holds ⋃ f is finite.

(64) If X is finite and for every Y such that Y ∈ X holds Y is finite, then ⋃ X is finite.

(65) If D is countable, then D^{∗} is countable.

(66) ℵ_{0} ≤ \overline{\overline{D^{∗}}}.

Now we present three schemes. The scheme FraenCoun1 deals with a unary functor F, and a unary predicate P, and states that:

{F(n) : P[n]} is countable for all values of the parameters.

The scheme FraenCoun2 concerns a binary functor F, and a binary predicate P, and states that:

{F(n_{1}, n_{2}) : P[n_{1}, n_{2}]} is countable for all values of the parameters.

The scheme FraenCoun3 concerns a ternary functor F, and a ternary predicate P, and states that:

{F(n_{1}, n_{2}, n_{3}) : P[n_{1}, n_{2}, n_{3}]} is countable for all values of the parameters.

The following propositions are true:

(67) ℵ_{0} · \overline{\overline{n}} ≤ ℵ_{0} and \overline{\overline{n}} · ℵ_{0} ≤ ℵ_{0}.

(68) If K < L and M < N or K ≤ L and M < N or K < L and M ≤ N or K ≤ L and M ≤ N, then K · M ≤ L · N and M · K ≤ L · N.

(69) If M < N or M ≤ N, then K · M ≤ K · N and K · M ≤ N · K and M · K ≤ K · N and M · K ≤ N · K.

(70) If K < L and M < N or K ≤ L and M < N or K < L and M ≤ N or K ≤ L and M ≤ N, then K = \overline{0} or K^{M} ≤ L^{N}.

(71) If M < N or M ≤ N, then K = \overline{0} or K^{M} ≤ K^{N} and M^{K} ≤ N^{K}.

(72) M ≤ M + N and N ≤ M + N.

(73) If N ≠ \overline{0}, then M ≤ M · N and M ≤ N · M.

(74) If K < L and M < N, then K + M < L + N and M + K < L + N.

(75) If K + M < K + N, then M < N.

(76) If \overline{\overline{X}} + \overline{\overline{Y}} = \overline{\overline{X}} and \overline{\overline{Y}} < \overline{\overline{X}}, then \overline{\overline{X \ Y}} = \overline{\overline{X}}.

One can prove the following propositions:

(77) If M is not finite, then M · M = M.

(78) If M is not finite and \overline{0} < N but N ≤ M or N < M, then M · N = M and N · M = M.

(79) If M is not finite but N ≤ M or N < M, then M · N ≤ M and N · M ≤ M.

(80) If X is not finite, then [: X, X:] ≈ X and \overline{\overline{[: X, X:]}} = \overline{\overline{X}}.

(81) If X is not finite and Y is finite and Y ≠ ∅, then [: X, Y :] ≈ X and \overline{\overline{[: X, Y :]}} = \overline{\overline{X}}.

(82) If K < L and M < N, then K · M < L · N and M · K < L · N.

(83) If K · M < K · N, then M < N.

(84) If X is not finite, then \overline{\overline{X}} = ℵ_{0} · \overline{\overline{X}}.

Countable Sets and Hessenberg’s Theorem 69

(85) If X ≠ ∅ and X is finite and Y is not finite, then \overline{\overline{Y}} · \overline{\overline{X}} = \overline{\overline{Y}} .

(86) If D is not finite and n ≠ 0, then D^{n} ≈ D and \overline{\overline{D^{n}}} = \overline{\overline{D}}.

(87) If D is not finite, then \overline{\overline{D}} = \overline{\overline{D^{∗}}}.

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

[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. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

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

[6] Grzegorz Bancerek. The reflection theorem. Formalized Mathematics, 1(5):973–977, 1990.

[7] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[8] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

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

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

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

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

[13] 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.

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

[15] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

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

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

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

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

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

70

The Limit of a Real Function at a Point

Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. We define the proper and the improper limit of a real function at a point. The main properties of the operations on the limit of a function are proved. The connection between the one-side limits and the limit of a function at a point are exposed. Equivalent Cauchy and Heine characterizations of the limit of a real function at a point are proved.

MML Identifier: LIMFUNC3.

The papers [17], [5], [1], [2], [3], [15], [13], [6], [8], [14], [18], [16], [4], [10], [11], [12], [7], and [9] provide the notation and terminology for this paper. For simplicity we adopt the following convention: r, r_{1}, r_{2}, g, g_{1}, g_{2}, x_{0} will be real numbers, n, k will be natural numbers, s_{1} will be a sequence of real numbers, and f, f_{1}, f_{2} will be partial functions from to . The following propositions are true:

(1) If rng s_{1} ⊆ dom f ∩]−∞, x_{0}[ or rng s_{1} ⊆ dom f ∩]x_{0}, +∞[, then rng s_{1} ⊆ dom f \ {x_{0}}.

(2) Suppose for every n holds 0 < |x_{0} − s_{1}(n)| and |x_{0} − s_{1}(n)| < \frac{1}{n+1} and s_{1}(n) ∈ dom f. Then s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f and rng s_{1} ⊆ dom f \ {x_{0}}.

(3) Suppose s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f \ {x_{0}}. Then for every r such that 0 < r there exists n such that for every k such that n ≤ k holds 0 < |x_{0} − s_{1}(k)| and |x_{0} − s_{1}(k)| < r and s_{1}(k) ∈ dom f.

(4) If 0 < r, then ]x_{0} − r, x_{0} + r[ \ {x_{0}} = ]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[.

(5) Suppose 0 < r_{2} and ]x_{0} − r_{2}, x_{0}[ ∪]x_{0}, x_{0} + r_{2}[ ⊆ dom f. Then for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f.

1Supported by RPBP.III-24.C8

(6) If for every n holds x_{0}− \frac{1}{n+1} < s_{1}(n) and s_{1}(n) < x_{0} and s_{1}(n) ∈ dom f, then s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f \ {x_{0}}.

(7) If s_{1} is convergent and lim s_{1} = x_{0} and 0 < g, then there exists k such that for every n such that k ≤ n holds x_{0} − g < s_{1}(n) and s_{1}(n) < x_{0} + g.

(8) The following conditions are equivalent:

(i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom f and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom f.

We now define three new predicates. Let us consider f, x_{0}. We say that f is convergent in x_{0} if and only if:

(Def.1) (i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) there exists g such that for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f \ {x_{0}} holds f · s_{1} is convergent and lim(f · s_{1}) = g.

We say that f is divergent to +∞ in x_{0} if and only if:

(Def.2) (i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f \ {x_{0}} holds f · s_{1} is divergent to +∞.

We say that f is divergent to −∞ in x_{0} if and only if:

(Def.3) (i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f \ {x_{0}} holds f · s_{1} is divergent to −∞.

The following propositions are true:

(9) f is convergent in x_{0} if and only if the following conditions are satisfied:

(i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) there exists g such that for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f \ {x_{0}} holds f · s_{1} is convergent and lim(f · s_{1}) = g.

(10) f is divergent to +∞ in x_{0} if and only if the following conditions are satisfied:

(i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆

dom f \ {x_{0}} holds f · s_{1} is divergent to +∞.

(11) f is divergent to −∞ in x_{0} if and only if the following conditions are

satisfied:

(i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆

dom f \ {x_{0}} holds f · s_{1} is divergent to −∞.

(12) f is convergent in x_{0} if and only if the following conditions are satisfied:

(i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) there exists g such that for every g_{1} such that 0 < g_{1} there exists g_{2} such that 0 < g_{2} and for every r_{1} such that 0 < |x_{0} −r_{1}| and |x_{0} −r_{1}| < g_{2} and r_{1} ∈ dom f holds |f(r_{1}) − g| < g_{1}.

(13) f is divergent to +∞ in x_{0} if and only if the following conditions are

satisfied:

(i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) for every g_{1} there exists g_{2} such that 0 < g_{2} and for every r_{1} such that

0 < |x_{0} − r_{1}| and |x_{0} − r_{1}| < g_{2} and r_{1} ∈ dom f holds g_{1} < f(r_{1}).

(14) f is divergent to −∞ in x_{0} if and only if the following conditions are

satisfied:

(i) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(ii) for every g_{1} there exists g_{2} such that 0 < g_{2} and for every r_{1} such that

0 < |x_{0} − r_{1}| and |x_{0} − r_{1}| < g_{2} and r_{1} ∈ dom f holds f(r_{1}) < g_{1}.

(15) f is divergent to +∞ in x_{0} if and only if f is left divergent to +∞ in x_{0} and f is right divergent to +∞ in x_{0}.

(16) f is divergent to −∞ in x_{0} if and only if f is left divergent to −∞ in x_{0} and f is right divergent to −∞ in x_{0}.

(17) Suppose that (i) f_{1} is divergent to +∞ in x_{0}, (ii) f_{2} is divergent to +∞ in x_{0}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f_{1} ∩ dom f_{2} and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f_{1} ∩ dom f_{2}. Then f_{1} + f_{2} is divergent to +∞ in x_{0} and f_{1}f_{2} is divergent to +∞ in x_{0}.

(18) Suppose that (i) f_{1} is divergent to −∞ in x_{0}, (ii) f_{2} is divergent to −∞ in x_{0}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f_{1} ∩ dom f_{2} and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f_{1} ∩ dom f_{2}. Then f_{1} + f_{2} is divergent to −∞ in x_{0} and f_{1}f_{2} is divergent to +∞ in x_{0}.

(19) Suppose that (i) f_{1} is divergent to +∞ in x_{0}, (ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{1} + f_{2}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{1} + f_{2}),

(iii) there exists r such that 0 < r and f_{2} is lower bounded on ]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[. Then f_{1} + f_{2} is divergent to +∞ in x_{0}.

(20) Suppose that (i) f_{1} is divergent to +∞ in x_{0}, (ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{1}f_{2}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{1}f_{2}),

(iii) there exist r, r_{1} such that 0 < r and 0 < r_{1} and for every g such that g ∈ dom f_{2} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) holds r_{1} ≤ f_{2}(g). Then f_{1}f_{2} is divergent to +∞ in x_{0}.

(21) (i) If f is divergent to +∞ in x_{0} and r > 0, then rf is divergent to +∞

in x_{0}, (ii) if f is divergent to +∞ in x_{0} and r < 0, then rf is divergent to −∞

in x_{0}, (iii) if f is divergent to −∞ in x_{0} and r > 0, then rf is divergent to −∞

in x_{0}, (iv) if f is divergent to −∞ in x_{0} and r < 0, then rf is divergent to +∞

in x_{0}.

(22) If f is divergent to +∞ in x_{0} or f is divergent to −∞ in x_{0}, then |f| is divergent to +∞ in x_{0}.

(23) Suppose that

(i) there exists r such that 0 < r and f is non-decreasing on ]x_{0} − r, x_{0}[ and f is non-increasing on ]x_{0}, x_{0} + r[ and f is not upper bounded on ]x_{0} − r, x_{0}[ and f is not upper bounded on ]x_{0}, x_{0} + r[,

(ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f. Then f is divergent to +∞ in x_{0}.

(24) Suppose that

(i) there exists r such that 0 < r and f is increasing on ]x_{0} − r, x_{0}[ and f is decreasing on ]x_{0}, x_{0} + r[ and f is not upper bounded on ]x_{0} − r, x_{0}[

and f is not upper bounded on ]x_{0}, x_{0} + r[, (ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f. Then f is divergent to +∞ in x_{0}.

(25) Suppose that

(i) there exists r such that 0 < r and f is non-increasing on ]x_{0} − r, x_{0}[ and f is non-decreasing on ]x_{0}, x_{0} + r[ and f is not lower bounded on ]x_{0} − r, x_{0}[ and f is not lower bounded on ]x_{0}, x_{0} + r[,

(ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f. Then f is divergent to −∞ in x_{0}.

(26) Suppose that

(i) there exists r such that 0 < r and f is decreasing on ]x_{0} − r, x_{0}[ and f is increasing on ]x_{0}, x_{0} + r[ and f is not lower bounded on ]x_{0} − r, x_{0}[ and f is not lower bounded on ]x_{0}, x_{0} + r[,

(ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f. Then f is divergent to −∞ in x_{0}.

(27) Suppose that (i) f_{1} is divergent to +∞ in x_{0}, (ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(iii) there exists r such that 0 < r and dom f ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) ⊆ dom f_{1} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) and for every g such that g ∈ dom f ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) holds f_{1}(g) ≤ f(g). Then f is divergent to +∞ in x_{0}.

(28) Suppose that (i) f_{1} is divergent to −∞ in x_{0}, (ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f,

(iii) there exists r such that 0 < r and dom f ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) ⊆ dom f_{1} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) and for every g such that g ∈ dom f ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) holds f(g) ≤ f_{1}(g). Then f is divergent to −∞ in x_{0}.

(29) Suppose that (i) f_{1} is divergent to +∞ in x_{0}, (ii) there exists r such that 0 < r and ]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[ ⊆ dom f ∩

dom f_{1} and for every g such that g ∈ ]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[ holds f_{1}(g) ≤ f(g).

Then f is divergent to +∞ in x_{0}.

(30) Suppose that (i) f_{1} is divergent to −∞ in x_{0}, (ii) there exists r such that 0 < r and ]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[ ⊆ dom f ∩

dom f_{1} and for every g such that g ∈ ]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[ holds f(g) ≤ f_{1}(g). Then f is divergent to −∞ in x_{0}.

Let us consider f, x_{0}. Let us assume that f is convergent in x_{0}. The functor lim_{x_{0}}f yields a real number and is defined by: (Def.4) for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆

dom f \ {x_{0}} holds f · s_{1} is convergent and lim(f · s_{1}) = lim_{x_{0}}f.

The following propositions are true:

(31) If f is convergent in x_{0}, then lim_{x_{0}}f = g if and only if for every s_{1} such that s_{1} is convergent and lim s_{1} = x_{0} and rng s_{1} ⊆ dom f \ {x_{0}} holds f · s_{1} is convergent and lim(f · s_{1}) = g.

(32) Suppose f is convergent in x_{0}. Then lim_{x_{0}}f = g if and only if for every g_{1} such that 0 < g_{1} there exists g_{2} such that 0 < g_{2} and for every r_{1} such that 0 < |x_{0} −r_{1}| and |x_{0} −r_{1}| < g_{2} and r_{1} ∈ dom f holds |f(r_{1})−g| < g_{1}.

(33) If f is convergent in x_{0}, then f is left convergent in x_{0} and f is right convergent in x_{0} and lim_{x_{0}}_{−} f = lim_{x_{0}}_{+} f and lim_{x_{0}}f = lim_{x_{0}}_{−} f and lim_{x_{0}}f = lim_{x_{0}}_{+} f.

(34) If f is left convergent in x_{0} and f is right convergent in x_{0} and lim_{x_{0}}_{−} f = lim_{x_{0}}_{+} f, then f is convergent in x_{0} and lim_{x_{0}}f = lim_{x_{0}}_{−} f and lim_{x_{0}}f = lim_{x_{0}}_{+} f.

(35) If f is convergent in x_{0}, then rf is convergent in x_{0} and lim_{x_{0}}(rf) = r · (lim_{x_{0}}f).

(36) If f is convergent in x_{0}, then −f is convergent in x_{0} and lim_{x_{0}}(−f) = −lim_{x_{0}}f.

(37) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is convergent in x_{0}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{1} + f_{2}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{1} + f_{2}). Then f_{1} + f_{2} is convergent in x_{0} and lim_{x_{0}}(f_{1} + f_{2}) = lim_{x_{0}}f_{1} + lim_{x_{0}}f_{2}.

(38) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is convergent in x_{0}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{1} − f_{2}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{1} − f_{2}). Then f_{1} − f_{2} is convergent in x_{0} and lim_{x_{0}}(f_{1} − f_{2}) = lim_{x_{0}}f_{1} − lim_{x_{0}}f_{2}.

(39) If f is convergent in x_{0} and f ^{−1} \{0\} = ∅ and lim_{x_{0}}f ≠ 0, then \frac{1}{f} is convergent in x_{0} and lim_{x_{0}} \frac{1}{f} = (lim_{x_{0}}f)^{−1}.

(40) If f is convergent in x_{0}, then |f| is convergent in x_{0} and lim_{x_{0}}|f| = |lim_{x_{0}}f|.

(41) Suppose that (i) f is convergent in x_{0}, (ii) lim_{x_{0}}f ≠ 0, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f and f(g_{1}) ≠ 0 and f(g_{2}) ≠ 0. Then \frac{1}{f} is convergent in x_{0} and lim_{x_{0}} \frac{1}{f} = (lim_{x_{0}}f)^{−1}.

(42) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is convergent in x_{0}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{1}f_{2}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{1}f_{2}). Then f_{1}f_{2} is convergent in x_{0} and lim_{x_{0}}(f_{1}f_{2}) = (lim_{x_{0}}f_{1}) · (lim_{x_{0}}f_{2}).

(43) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is convergent in x_{0}, (iii) lim_{x_{0}}f_{2} ≠ 0, (iv) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom \frac{f_{1}}{f_{2}} and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom \frac{f_{1}}{f_{2}} .

Then \frac{f_{1}}{f_{2}} is convergent in x_{0} and lim_{x_{0}} \frac{f_{1}}{f_{2}} = \frac{lim_{x_{0}}f_{1}}{lim_{x_{0}}f_{2}} .

(44) Suppose that (i) f_{1} is convergent in x_{0}, (ii) lim_{x_{0}}f_{1} = 0, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{1}f_{2}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{1}f_{2}),

(iv) there exists r such that 0 < r and f_{2} is bounded on ]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[. Then f_{1}f_{2} is convergent in x_{0} and lim_{x_{0}}(f_{1}f_{2}) = 0.

(45) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is convergent in x_{0}, (iii) lim_{x_{0}}f_{1} = lim_{x_{0}}f_{2}, (iv) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f, (v) there exists r such that 0 < r and for every g such that g ∈ dom f ∩

(]x_{0}−r, x_{0}[∪]x_{0}, x_{0}+r[) holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g) but dom f_{1}∩ (]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[) ⊆ dom f_{2} ∩ (]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[) and dom f ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) ⊆ dom f_{1} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) or dom f_{2} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) ⊆ dom f_{1} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) and dom f ∩(]x_{0} −r, x_{0}[∪]x_{0}, x_{0} +r[) ⊆ dom f_{2} ∩(]x_{0} −r, x_{0}[∪]x_{0}, x_{0} +r[). Then f is convergent in x_{0} and lim_{x_{0}}f = lim_{x_{0}}f_{1}.

(46) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is convergent in x_{0}, (iii) lim_{x_{0}}f_{1} = lim_{x_{0}}f_{2}, (iv) there exists r such that 0 < r and ]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[ ⊆ (dom f_{1} ∩

dom f_{2}) ∩ dom f and for every g such that g ∈ ]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[ holds f_{1}(g) ≤ f(g) and f(g) ≤ f_{2}(g). Then f is convergent in x_{0} and lim_{x_{0}}f = lim_{x_{0}}f_{1}.

(47) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is convergent in x_{0}, (iii) there exists r such that 0 < r but dom f_{1} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) ⊆

dom f_{2} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) and for every g such that g ∈ dom f_{1} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) holds f_{1}(g) ≤ f_{2}(g) or dom f_{2} ∩ (]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[) ⊆ dom f_{1} ∩ (]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[) and for every g such that g ∈ dom f_{2} ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) holds f_{1}(g) ≤ f_{2}(g). Then lim_{x_{0}}f_{1} ≤ lim_{x_{0}}f_{2}.

(48) Suppose that (i) f is divergent to +∞ in x_{0} or f is divergent to −∞ in x_{0}, (ii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f and f(g_{1}) ≠ 0 and f(g_{2}) ≠ 0. Then \frac{1}{f} is convergent in x_{0} and lim_{x_{0}} \frac{1}{f} = 0.

(49) Suppose that (i) f is convergent in x_{0}, (ii) lim_{x_{0}}f = 0, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom f and f(g_{1}) ≠ 0 and f(g_{2}) ≠ 0,

(iv) there exists r such that 0 < r and for every g such that g ∈ dom f ∩ (]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) holds 0 ≤ f(g). Then \frac{1}{f} is divergent to +∞ in x_{0}.

(50) Suppose that (i) f is convergent in x_{0}, (ii) lim_{x_{0}}f = 0, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such

that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom f and g_{2} < r_{2} and x_{0} < g_{2} and

The Limit of a Real Function at a Point 79

g_{2} ∈ dom f and f(g_{1}) ≠ 0 and f(g_{2}) ≠ 0, (iv) there exists r such that 0 < r and for every g such that g ∈ dom f ∩

(]x_{0} − r, x_{0}[ ∪]x_{0}, x_{0} + r[) holds f(g) ≤ 0. Then \frac{1}{f} is divergent to −∞ in x_{0}.

(51) If f is convergent in x_{0} and lim_{x_{0}}f = 0 and there exists r such that 0 < r and for every g such that g ∈ dom f ∩ (]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[) holds 0 < f(g), then \frac{1}{f} is divergent to +∞ in x_{0}.

(52) If f is convergent in x_{0} and lim_{x_{0}}f = 0 and there exists r such that 0 < r and for every g such that g ∈ dom f ∩ (]x_{0} − r, x_{0}[ ∪ ]x_{0}, x_{0} + r[) holds f(g) < 0, then \frac{1}{f} is divergent to −∞ in x_{0}.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 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 sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[7] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

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

[9] Jarosław Kotowicz. One-side limits of a real function at a point. Formalized Mathematics, 2(1):29–40, 1991.

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

[11] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

[12] Jarosław Kotowicz. Properties of real functions. Formalized Mathematics, 1(4):781–786, 1990.

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

[14] Andrzej Ne‘dzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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

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

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

80 Jarosław Kotowicz

[18] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized

Mathematics, 1(3):445–449, 1990.

Received September 5, 1990

The Limit of a Composition of Real Functions

Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. The theorem on the proper and the improper limit of a composition of real functions at a point, at infinity and one-side limits at a point are presented.

MML Identifier: LIMFUNC4.

The terminology and notation used in this paper have been introduced in the following articles: [17], [4], [1], [2], [15], [13], [5], [8], [14], [16], [3], [10], [11], [12], [7], [9], and [6]. We follow a convention: r, r_{1}, r_{2}, g, g_{1}, g_{2}, x_{0} will be real numbers and f_{1}, f_{2} will be partial functions from to . The following propositions are true:

(1) Let s be a sequence of real numbers. Then for every set X such that rng s ⊆ dom(f_{2} · f_{1}) ∩ X holds rng s ⊆ dom(f_{2} · f_{1}) and rng s ⊆ X and rng s ⊆ dom f_{1} and rng s ⊆ dom f_{1} ∩ X and rng(f_{1} · s) ⊆ dom f_{2}.

(2) For every sequence of real numbers s and for every set X such that rng s ⊆ dom(f_{2} · f_{1}) \ X holds rng s ⊆ dom(f_{2} · f_{1}) and rng s ⊆ dom f_{1} and rng s ⊆ dom f_{1} \ X and rng(f_{1} · s) ⊆ dom f_{2}.

(3) If f_{1} is divergent in +∞ to +∞ and f_{2} is divergent in +∞ to +∞ and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is divergent in +∞ to +∞.

(4) If f_{1} is divergent in +∞ to +∞ and f_{2} is divergent in +∞ to −∞ and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is divergent in +∞ to −∞.

(5) If f_{1} is divergent in +∞ to −∞ and f_{2} is divergent in −∞ to +∞ and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is divergent in +∞ to +∞.

1Supported by RPBP.III-24.C8

(6) If f_{1} is divergent in +∞ to −∞ and f_{2} is divergent in −∞ to −∞ and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is divergent in +∞ to −∞.

(7) If f_{1} is divergent in −∞ to +∞ and f_{2} is divergent in +∞ to +∞ and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is divergent in −∞ to +∞.

(8) If f_{1} is divergent in −∞ to +∞ and f_{2} is divergent in +∞ to −∞ and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is divergent in −∞ to −∞.

(9) If f_{1} is divergent in −∞ to −∞ and f_{2} is divergent in −∞ to +∞ and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is divergent in −∞ to +∞.

(10) If f_{1} is divergent in −∞ to −∞ and f_{2} is divergent in −∞ to −∞ and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is divergent in −∞ to −∞.

(11) If f_{1} is left divergent to +∞ in x_{0} and f_{2} is divergent in +∞ to +∞ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is left divergent to +∞ in x_{0}.

(12) If f_{1} is left divergent to +∞ in x_{0} and f_{2} is divergent in +∞ to −∞ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is left divergent to −∞ in x_{0}.

(13) If f_{1} is left divergent to −∞ in x_{0} and f_{2} is divergent in −∞ to +∞ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is left divergent to +∞ in x_{0}.

(14) If f_{1} is left divergent to −∞ in x_{0} and f_{2} is divergent in −∞ to −∞ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is left divergent to −∞ in x_{0}.

(15) If f_{1} is right divergent to +∞ in x_{0} and f_{2} is divergent in +∞ to +∞ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is right divergent to +∞ in x_{0}.

(16) If f_{1} is right divergent to +∞ in x_{0} and f_{2} is divergent in +∞ to −∞ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is right divergent to −∞ in x_{0}.

(17) If f_{1} is right divergent to −∞ in x_{0} and f_{2} is divergent in −∞ to +∞ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is right divergent to +∞ in x_{0}.

(18) If f_{1} is right divergent to −∞ in x_{0} and f_{2} is divergent in −∞ to −∞ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is right divergent to −∞ in x_{0}.

(19) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is left divergent to +∞ in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds f_{1}(r) < lim_{x_{0}}_{−} f_{1}. Then f_{2} · f_{1} is left divergent to +∞ in x_{0}.

(20) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is left divergent to −∞ in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds f_{1}(r) < lim_{x_{0}}_{−} f_{1}. Then f_{2} · f_{1} is left divergent to −∞ in x_{0}.

(21) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is right divergent to +∞ in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds lim_{x_{0}}_{−} f_{1} < f_{1}(r). Then f_{2} · f_{1} is left divergent to +∞ in x_{0}.

(22) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is right divergent to −∞ in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds lim_{x_{0}}_{−} f_{1} < f_{1}(r). Then f_{2} · f_{1} is left divergent to −∞ in x_{0}.

(23) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is right divergent to +∞ in lim_{x_{0}}_{+} f_{1}, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds lim_{x_{0}}_{+} f_{1} < f_{1}(r). Then f_{2} · f_{1} is right divergent to +∞ in x_{0}.

(24) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is right divergent to −∞ in lim_{x_{0}}_{+} f_{1}, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds lim_{x_{0}}_{+} f_{1} < f_{1}(r).

Then f_{2} · f_{1} is right divergent to −∞ in x_{0}.

(25) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is left divergent to +∞ in lim_{x_{0}}_{+} f_{1}, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds f_{1}(r) < lim_{x_{0}}_{+} f_{1}. Then f_{2} · f_{1} is right divergent to +∞ in x_{0}.

(26) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is left divergent to −∞ in lim_{x_{0}}_{+} f_{1}, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds f_{1}(r) < lim_{x_{0}}_{+} f_{1}. Then f_{2} · f_{1} is right divergent to −∞ in x_{0}.

(27) If f_{1} is convergent in +∞ and f_{2} is left divergent to +∞ in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds f_{1}(g) < lim_{+∞} f_{1}, then f_{2} · f_{1} is divergent in +∞ to +∞.

(28) If f_{1} is convergent in +∞ and f_{2} is left divergent to −∞ in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds f_{1}(g) < lim_{+∞} f_{1}, then f_{2} · f_{1} is divergent in +∞ to −∞.

(29) If f_{1} is convergent in +∞ and f_{2} is right divergent to +∞ in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds lim_{+∞} f_{1} < f_{1}(g), then f_{2} · f_{1} is divergent in +∞ to +∞.

(30) If f_{1} is convergent in +∞ and f_{2} is right divergent to −∞ in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds lim_{+∞} f_{1} < f_{1}(g), then f_{2} · f_{1} is divergent in +∞ to −∞.

(31) If f_{1} is convergent in −∞ and f_{2} is left divergent to +∞ in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds f_{1}(g) < lim_{−∞} f_{1}, then f_{2} · f_{1} is divergent in −∞ to +∞.

Next we state a number of propositions:

(32) If f_{1} is convergent in −∞ and f_{2} is left divergent to −∞ in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds f_{1}(g) < lim_{−∞} f_{1}, then f_{2} · f_{1} is divergent in −∞ to −∞.

(33) If f_{1} is convergent in −∞ and f_{2} is right divergent to +∞ in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds lim_{−∞} f_{1} < f_{1}(g), then f_{2} · f_{1} is divergent in −∞ to +∞.

(34) If f_{1} is convergent in −∞ and f_{2} is right divergent to −∞ in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds lim_{−∞} f_{1} < f_{1}(g), then f_{2} · f_{1} is divergent in −∞ to −∞.

(35) Suppose f_{1} is divergent to +∞ in x_{0} and f_{2} is divergent in +∞ to +∞ and for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}). Then f_{2} · f_{1} is divergent to +∞ in x_{0}.

(36) Suppose f_{1} is divergent to +∞ in x_{0} and f_{2} is divergent in +∞ to −∞ and for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}). Then f_{2} · f_{1} is divergent to −∞ in x_{0}.

(37) Suppose f_{1} is divergent to −∞ in x_{0} and f_{2} is divergent in −∞ to +∞ and for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}). Then f_{2} · f_{1} is divergent to +∞ in x_{0}.

(38) Suppose f_{1} is divergent to −∞ in x_{0} and f_{2} is divergent in −∞ to −∞ and for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}). Then f_{2} · f_{1} is divergent to −∞ in x_{0}.

(39) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is divergent to +∞ in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds f_{1}(r) ≠ lim_{x_{0}}f_{1}. Then f_{2} · f_{1} is divergent to +∞ in x_{0}.

(40) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is divergent to −∞ in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds f_{1}(r) ≠ lim_{x_{0}}f_{1}. Then f_{2} · f_{1} is divergent to −∞ in x_{0}.

(41) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is right divergent to +∞ in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds f_{1}(r) > lim_{x_{0}}f_{1}. Then f_{2} · f_{1} is divergent to +∞ in x_{0}.

(42) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is right divergent to −∞ in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds f_{1}(r) > lim_{x_{0}}f_{1}. Then f_{2} · f_{1} is divergent to −∞ in x_{0}.

(43) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is divergent to +∞ in lim_{x_{0}}_{+} f_{1}, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds f_{1}(r) ≠ lim_{x_{0}}_{+} f_{1}. Then f_{2} · f_{1} is right divergent to +∞ in x_{0}.

(44) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is divergent to −∞ in lim_{x_{0}}_{+} f_{1}, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds f_{1}(r) ≠ lim_{x_{0}}_{+} f_{1}. Then f_{2} · f_{1} is right divergent to −∞ in x_{0}.

(45) If f_{1} is convergent in +∞ and f_{2} is divergent to +∞ in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds f_{1}(g) ≠ lim_{+∞} f_{1}, then f_{2} · f_{1} is divergent in +∞ to +∞.

(46) If f_{1} is convergent in +∞ and f_{2} is divergent to −∞ in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds f_{1}(g) ≠ lim_{+∞} f_{1}, then f_{2} · f_{1} is divergent in +∞ to −∞.

(47) If f_{1} is convergent in −∞ and f_{2} is divergent to +∞ in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds f_{1}(g) ≠ lim_{−∞} f_{1}, then f_{2} · f_{1} is divergent in −∞ to +∞.

(48) If f_{1} is convergent in −∞ and f_{2} is divergent to −∞ in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds f_{1}(g) ≠ lim_{−∞} f_{1}, then f_{2} · f_{1} is divergent in −∞ to −∞.

(49) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is left divergent to +∞ in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds f_{1}(r) < lim_{x_{0}}f_{1}. Then f_{2} · f_{1} is divergent to +∞ in x_{0}.

(50) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is left divergent to −∞ in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds f_{1}(r) < lim_{x_{0}}f_{1}. Then f_{2} · f_{1} is divergent to −∞ in x_{0}.

(51) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is divergent to +∞ in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds f_{1}(r) ≠ lim_{x_{0}}_{−} f_{1}. Then f_{2} · f_{1} is left divergent to +∞ in x_{0}.

(52) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is divergent to −∞ in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds f_{1}(r) ≠ lim_{x_{0}}_{−} f_{1}. Then f_{2} · f_{1} is left divergent to −∞ in x_{0}.

(53) If f_{1} is divergent in +∞ to +∞ and f_{2} is convergent in +∞ and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is convergent in +∞ and lim_{+∞}(f_{2} · f_{1}) = lim_{+∞} f_{2}.

(54) If f_{1} is divergent in +∞ to −∞ and f_{2} is convergent in −∞ and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is convergent in +∞ and lim_{+∞}(f_{2} · f_{1}) = lim_{−∞} f_{2}.

(55) If f_{1} is divergent in −∞ to +∞ and f_{2} is convergent in +∞ and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is convergent in −∞ and lim_{−∞}(f_{2} · f_{1}) = lim_{+∞} f_{2}.

(56) If f_{1} is divergent in −∞ to −∞ and f_{2} is convergent in −∞ and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is convergent in −∞ and lim_{−∞}(f_{2} · f_{1}) = lim_{−∞} f_{2}.

(57) If f_{1} is left divergent to +∞ in x_{0} and f_{2} is convergent in +∞ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{2} · f_{1}) = lim_{+∞} f_{2}.

(58) If f_{1} is left divergent to −∞ in x_{0} and f_{2} is convergent in −∞ and for every r such that r < x_{0} there exists g such that r < g and g < x_{0} and g ∈ dom(f_{2} · f_{1}), then f_{2} · f_{1} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{2} · f_{1}) = lim_{−∞} f_{2}.

(59) If f_{1} is right divergent to +∞ in x_{0} and f_{2} is convergent in +∞ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{2} ·f_{1}), then f_{2} ·f_{1} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{2} ·f_{1}) = lim_{+∞} f_{2}.

(60) If f_{1} is right divergent to −∞ in x_{0} and f_{2} is convergent in −∞ and for every r such that x_{0} < r there exists g such that g < r and x_{0} < g and g ∈ dom(f_{2} ·f_{1}), then f_{2} ·f_{1} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{2} ·f_{1}) = lim_{−∞} f_{2}.

(61) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is left convergent in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds f_{1}(r) < lim_{x_{0}}_{−} f_{1}. Then f_{2} · f_{1} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{2} · f_{1}) = lim_{lim_{_{x_{0}}_{−}} f_{1}}_{−} f_{2}.

(62) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is right convergent in lim_{x_{0}}_{+} f_{1}, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds lim_{x_{0}}_{+} f_{1} < f_{1}(r). Then f_{2} ·f_{1} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{2} ·f_{1}) = lim_{lim_{_{x_{0}}_{+}} f_{1}}_{+} f_{2}.

One can prove the following propositions:

(63) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is right convergent in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds lim_{x_{0}}_{−} f_{1} < f_{1}(r). Then f_{2} · f_{1} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{2} · f_{1}) = lim_{lim_{_{x_{0}}_{−}} f_{1}}_{+} f_{2}.

(64) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is left convergent in lim_{x_{0}}_{+} f_{1}, (iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds f_{1}(r) < lim_{x_{0}}_{+} f_{1}. Then f_{2} ·f_{1} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{2} ·f_{1}) = lim_{lim_{_{x_{0}}_{+}} f_{1}}_{−} f_{2}.

(65) Suppose f_{1} is convergent in +∞ and f_{2} is left convergent in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds f_{1}(g) < lim_{+∞} f_{1}. Then f_{2} · f_{1} is convergent in +∞ and lim_{+∞}(f_{2} · f_{1}) = lim_{lim_{+∞} f_{1}}_{−} f_{2}.

(66) Suppose f_{1} is convergent in +∞ and f_{2} is right convergent in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds lim_{+∞} f_{1} < f_{1}(g). Then f_{2} · f_{1} is convergent in +∞ and lim_{+∞}(f_{2} · f_{1}) = lim_{lim_{+∞} f_{1}}_{+} f_{2}.

(67) Suppose f_{1} is convergent in −∞ and f_{2} is left convergent in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds f_{1}(g) < lim_{−∞} f_{1}. Then f_{2} · f_{1} is convergent in −∞ and lim_{−∞}(f_{2} · f_{1}) = lim_{lim_{−∞} f_{1}}_{−} f_{2}.

(68) Suppose f_{1} is convergent in −∞ and f_{2} is right convergent in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds lim_{−∞} f_{1} < f_{1}(g). Then f_{2} · f_{1} is convergent in −∞ and lim_{−∞}(f_{2} · f_{1}) = lim_{lim_{−∞} f_{1}}_{+} f_{2}.

(69) Suppose f_{1} is divergent to +∞ in x_{0} and f_{2} is convergent in +∞ and for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}). Then f_{2} · f_{1} is convergent in x_{0} and lim_{x_{0}}(f_{2} · f_{1}) = lim_{+∞} f_{2}.

(70) Suppose f_{1} is divergent to −∞ in x_{0} and f_{2} is convergent in −∞ and for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}). Then f_{2} · f_{1} is convergent in x_{0} and lim_{x_{0}}(f_{2} · f_{1}) = lim_{−∞} f_{2}.

(71) Suppose f_{1} is convergent in +∞ and f_{2} is convergent in lim_{+∞} f_{1} and for every r there exists g such that r < g and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]r, +∞[ holds f_{1}(g) ≠ lim_{+∞} f_{1}. Then f_{2} · f_{1} is convergent in +∞ and lim_{+∞}(f_{2} · f_{1}) = lim_{lim_{+∞} f_{1}}f_{2}.

(72) Suppose f_{1} is convergent in −∞ and f_{2} is convergent in lim_{−∞} f_{1} and for every r there exists g such that g < r and g ∈ dom(f_{2} · f_{1}) and there exists r such that for every g such that g ∈ dom f_{1} ∩]−∞, r[ holds f_{1}(g) ≠ lim_{−∞} f_{1}. Then f_{2} · f_{1} is convergent in −∞ and lim_{−∞}(f_{2} · f_{1}) = lim_{lim_{−∞} f_{1}}f_{2}.

(73) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is left convergent in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds f_{1}(r) < lim_{x_{0}}f_{1}. Then f_{2} · f_{1} is convergent in x_{0} and lim_{x_{0}}(f_{2} · f_{1}) = lim_{lim_{x_{0}}f_{1}}_{−} f_{2}.

(74) Suppose that (i) f_{1} is left convergent in x_{0}, (ii) f_{2} is convergent in lim_{x_{0}}_{−} f_{1}, (iii) for every r such that r < x_{0} there exists g such that r < g and g < x_{0}

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0} − g, x_{0}[ holds f_{1}(r) ≠ lim_{x_{0}}_{−} f_{1}. Then f_{2} · f_{1} is left convergent in x_{0} and lim_{x_{0}}_{−}(f_{2} · f_{1}) = lim_{lim_{_{x_{0}}_{−}} f_{1}}f_{2}.

(75) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is right convergent in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds lim_{x_{0}}f_{1} < f_{1}(r). Then f_{2} · f_{1} is convergent in x_{0} and lim_{x_{0}}(f_{2} · f_{1}) = lim_{lim_{x_{0}}f_{1}}_{+} f_{2}.

(76) Suppose that (i) f_{1} is right convergent in x_{0}, (ii) f_{2} is convergent in lim_{x_{0}}_{+} f_{1},

The Limit of a Composition of Real Functions 91

(iii) for every r such that x_{0} < r there exists g such that g < r and x_{0} < g

and g ∈ dom(f_{2} · f_{1}), (iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩

]x_{0}, x_{0} + g[ holds f_{1}(r) ≠ lim_{x_{0}}_{+} f_{1}. Then f_{2} · f_{1} is right convergent in x_{0} and lim_{x_{0}}_{+}(f_{2} · f_{1}) = lim_{lim_{_{x_{0}}_{+}} f_{1}}f_{2}.

(77) Suppose that (i) f_{1} is convergent in x_{0}, (ii) f_{2} is convergent in lim_{x_{0}}f_{1}, (iii) for all r_{1}, r_{2} such that r_{1} < x_{0} and x_{0} < r_{2} there exist g_{1}, g_{2} such that

r_{1} < g_{1} and g_{1} < x_{0} and g_{1} ∈ dom(f_{2} · f_{1}) and g_{2} < r_{2} and x_{0} < g_{2} and g_{2} ∈ dom(f_{2} · f_{1}),

(iv) there exists g such that 0 < g and for every r such that r ∈ dom f_{1} ∩ (]x_{0} − g, x_{0}[ ∪]x_{0}, x_{0} + g[) holds f_{1}(r) ≠ lim_{x_{0}}f_{1}. Then f_{2} · f_{1} is convergent in x_{0} and lim_{x_{0}}(f_{2} · f_{1}) = lim_{lim_{x_{0}}f_{1}}f_{2}.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 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 sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[6] Jarosław Kotowicz. The limit of a real function at a point. Formalized Mathematics, 2(1):71–80, 1991.

[7] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

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

[9] Jarosław Kotowicz. One-side limits of a real function at a point. Formalized Mathematics, 2(1):29–40, 1991.

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

[11] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

[12] Jarosław Kotowicz. Properties of real functions. Formalized Mathematics, 1(4):781–786, 1990.

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

[14] Andrzej Ne‘dzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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

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

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

Locally Connected Spaces

Beata Padlewska^{1} Technical University of Białystok

Summary. This article is a continuation of [6]. We define a neighbourhood of a point and a neighbourhood of a set and prove some facts about them. Then the definitions of a locally connected space and a locally connected set are introduced. Some theorems on locally connected spaces are given (based on [5]). We also define a quasi-component of a point and prove some of its basic properties.

MML Identifier: CONNSP 2.

The papers [11], [10], [2], [13], [7], [1], [12], [9], [3], [8], [14], [6], and [4] provide the terminology and notation for this paper. Let X be a topological space, and let x be a point of X. A subset of X is called a neighborhood of x if: (Def.1) x ∈ Int it.

Let X be a topological space, and let A be a subset of X. A subset of X is called a neighborhood of A if: (Def.2) A ⊆ Int it.

In the sequel X will denote a topological space, x will denote a point of X, and A, U_{1} will denote subsets of X. We now state a number of propositions:

(2)^{2} A is a neighborhood of U_{1} if and only if U_{1} ⊆ Int A.

(3) For every x and for all subsets A, B of X such that A is a neighborhood of x and B is a neighborhood of x holds A ∪ B is a neighborhood of x.

(4) For every x and for all subsets A, B of X holds A is a neighborhood of x and B is a neighborhood of x if and only if A ∩ B is a neighborhood of x.

(5) For every subset U_{1} of X and for every point x of X such that U_{1} is open and x ∈ U_{1} holds U_{1} is a neighborhood of x.

(6) For every subset U_{1} of X and for every point x of X such that U_{1} is a neighborhood of x holds x ∈ U_{1}.

1Supported by RPBP.III-24.C1

2The proposition (1) was either repeated or obvious.

(7) For all U_{1}, x such that U_{1} is a neighborhood of x there exists a subset V of X such that V is a neighborhood of x and V is open and V ⊆ U_{1}.

(8) For all U_{1}, x holds U_{1} is a neighborhood of x if and only if there exists a subset V of X such that V is open and V ⊆ U_{1} and x ∈ V.

(9) U_{1} is open if and only if for every x such that x ∈ U_{1} there exists a subset A of X such that A is a neighborhood of x and A ⊆ U_{1}.

(10) For every subset V of X holds V is a neighborhood of {x} if and only if V is a neighborhood of x.

(11) For every subset B of X and for every point x of X B and for every subset A of X B and for every subset A_{1} of X and for every point x_{1} of X such that B ≠ ∅_{X} and B is open and A is a neighborhood of x and A = A_{1} and x = x_{1} holds A_{1} is a neighborhood of x_{1}.

(12) For every subset B of X and for every point x of X B and for every subset A of X B and for every subset A_{1} of X and for every point x_{1} of X such that A_{1} is a neighborhood of x_{1} and A = A_{1} and x = x_{1} holds A is a neighborhood of x.

(13) For all subsets A, B of X such that A is a component of X and A ⊆ B holds A is a component of B.

(14) For every subspace X_{1} of X and for every point x of X and for every

point x_{1} of X_{1} such that x = x_{1} holds Component(x_{1}) ⊆ Component(x). Let X be a topological space, and let x be a point of X. We say that X is

locally connected in x if and only if: (Def.3) for every subset U_{1} of X such that U_{1} is a neighborhood of x there exists

a subset V of X such that V is a neighborhood of x and V is connected and V ⊆ U_{1}.

Let X be a topological space. We say that X is locally connected if and only if: (Def.4) for every point x of X holds X is locally connected in x.

Let X be a topological space, and let A be a subset of X, and let x be a point of X. We say that A is locally connected in x if and only if: (Def.5) there exists a point x_{1} of X A such that x_{1} = x and X A is locally

connected in x_{1}.

The following proposition is true (17)^{3} A is locally connected in x if and only if there exists a point x_{1} of X A

such that x_{1} = x and X A is locally connected in x_{1}. Let X be a topological space, and let A be a subset of X. We say that A is

locally connected if and only if: (Def.6) X A is locally connected.

One can prove the following propositions:

3The propositions (15)–(16) were either repeated or obvious.

(19)^{4} For every x holds X is locally connected in x if and only if for all subsets V, S of X such that V is a neighborhood of x and S is a component of V and x ∈ S holds S is a neighborhood of x.

(20) For every x holds X is locally connected in x if and only if for every subset U_{1} of X such that U_{1} is open and x ∈ U_{1} there exists a point x_{1} of X U_{1} such that x_{1} = x and x ∈ Int Component(x_{1}).

(21) If X is locally connected, then for every subset S of X such that S is a component of X holds S is open.

(22) If X is locally connected in x, then for every subset A of X such that A is open and x ∈ A holds A is locally connected in x.

(23) If X is locally connected, then for every subset A of X such that A ≠ ∅_{X} and A is open holds A is locally connected.

(24) X is locally connected if and only if for all subsets A, B of X such that A ≠ ∅_{X} and A is open and B is a component of A holds B is open.

(25) If X is locally connected, then for every subset E of X and for every subset C of X E such that E ≠ ∅_{X} and C ≠ ∅_{X E} and C is connected and C is open there exists a subset H of X such that H is open and H is connected and C = E ∩ H.

(26) X is a T_{4} space if and only if for all subsets A, C of X such that A ≠ ∅ and C ≠ Ω_{X} and A ⊆ C and A is closed and C is open there exists a subset G of X such that G is open and A ⊆ G and \overline{G} ⊆ C.

(27) Suppose X is locally connected and X is a T_{4} space. Let A, B be subsets of X. Suppose A ≠ ∅ and B ≠ ∅ and A is closed and B is closed and A ∩ B = ∅. Then if A is connected and B is connected, then there exist subsets R, S of X such that R is connected and S is connected and R is open and S is open and A ⊆ R and B ⊆ S and R ∩ S = ∅.

(28) For every point x of X and for every family F of subsets of X such that for every subset A of X holds A ∈ F if and only if A is open closed and x ∈ A holds F ≠ ∅.

Let X be a topological space, and let x be a point of X. The quasi-component of x is a subset of X defined by:

(Def.7) there exists a family F of subsets of X such that for every subset A of X holds A ∈ F if and only if A is open closed and x ∈ A and ⋂ F = the quasi-component of x.

We now state several propositions:

(29) A = the quasi-component of x if and only if there exists a family F of subsets of X such that for every subset A of X holds A ∈ F if and only if A is open closed and x ∈ A and ⋂ F = A.

(30) x ∈ the quasi-component of x.

4The proposition (18) was either repeated or obvious.

96 Beata Padlewska

(31) If A is open closed and x ∈ A, then if A ⊆ the quasi-component of x, then A = the quasi-component of x.

(32) The quasi-component of x is closed.

(33) Component(x) ⊆ the quasi-component of x.

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

[5] Kazimierz Kuratowski. Wste‘p do teorii mnogości i topologii. PWN, Warszawa, 1977.

[6] Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239–244, 1990.

[7] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

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

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

[10] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

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

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

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

[14] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231–237, 1990.

Construction of Finite Sequences over Ring and Left-, Right-,

and Bi-Modules over a Ring ^{1}

Michał Muzalewski Warsaw University Białystok Lesław W. Szczerba Siedlce University

Summary. This text includes definitions of finite sequences over rings and left-, right-, and bi-module over a ring, treated as functions defined for all natural numbers, but almost everywhere equal to zero. Some elementary theorems are proved, in particular for each category of sequences the schema of existence is proved. In all four cases, i.e for rings, left-, right-, and bi-modules are almost exactly the same, hovewer we do not know how to do the job in Mizar in a different way. The paper is mostly based on [2]. In particular the notion of initial segment of natural numbers is introduced which differs from that of [2] by starting with zero. This proved to be more convenient for algebraic purposes.

MML Identifier: ALGSEQ 1.

The notation and terminology used in this paper are introduced in the following papers: [8], [3], [5], [1], [4], [6], and [7]. We adopt the following rules: i, k, l, m, n will be natural numbers and x will be arbitrary. We now state four propositions:

(2)^{2} If m < n + 1, then m < n or m = n. (4)^{3} If k < 2, then k = 0 or k = 1.

(5) For every real number x holds x < x + 1. (7)^{4} If k < l and l ≤ k + 1, then l = k + 1. Let us consider n. The functor PSeg n yields a set and is defined by:

(Def.1) PSeg n = {k : k < n}.

1Supported by RPBP.III-24.C3 2The proposition (1) was either repeated or obvious. 3The proposition (3) was either repeated or obvious. 4The proposition (6) was either repeated or obvious.

Let us consider n. Then PSeg n is sets of natural numbers.

We now state a number of propositions:

(8) PSeg n = {k : k < n}.

(9) If x ∈ PSeg n, then x is a natural number.

(10) k ∈ PSeg n if and only if k < n.

(11) PSeg 0 = ∅ and PSeg 1 = {0} and PSeg 2 = {0, 1}.

(12) n ∈ PSeg(n + 1).

(13) n ≤ m if and only if PSeg n ⊆ PSeg m.

(14) If PSeg n = PSeg m, then n = m.

(15) If k ≤ n, then PSeg k = PSeg k ∩ PSeg n and PSeg k = PSeg n ∩ PSeg k.

(16) If PSeg k = PSeg k ∩ PSeg n or PSeg k = PSeg n ∩ PSeg k, then k ≤ n.

(17) PSeg n ∪ {n} = PSeg(n + 1).

In the sequel R is a field structure and x is a scalar of R. Let us consider R. A function from into the carrier of R is said to be an algebraic sequence of R if: (Def.2) there exists n such that for every i such that i ≥ n holds it(i) = 0_{R}.

In the sequel p, q denote algebraic sequences of R. Next we state the proposition (19)^{5} dom p = .

Let us consider R, p, k. We say that the length of p is at most k if and only if: (Def.3) for every i such that i ≥ k holds p(i) = 0_{R}.

We now state the proposition

(20) the length of p is at most k if and only if for every i such that i ≥ k

holds p(i) = 0_{R}. Let us consider R, p. The functor len p yielding a natural number is defined

as follows: (Def.4) the length of p is at most len p and for every m such that the length of

p is at most m holds len p ≤ m. We now state several propositions:

(21) i = len p if and only if the length of p is at most i and for every m such that the length of p is at most m holds i ≤ m.

(22) For every i such that i ≥ len p holds p(i) = 0_{R}.

(23) If p(k) ≠ 0_{R}, then len p > k.

(24) If for every i such that i < k holds p(i) ≠ 0_{R}, then len p ≥ k.

(25) If len p = k + 1, then p(k) ≠ 0_{R}.

Let us consider R, p. The functor support p yields sets of natural numbers and is defined as follows:

5The proposition (18) was either repeated or obvious.

(Def.5) support p = PSeg(len p).

Next we state two propositions:

(26) For every y being sets of natural numbers holds y = support p if and only if y = PSeg(len p).

(27) k = len p if and only if PSeg k = support p.

The scheme AlgSeqLambdaF concerns field structure A, a natural number B, and a unary functor F yielding a scalar of A and states that:

there exists an algebraic sequence p of A such that len p ≤ B and for every k such that k < B holds p(k) = F(k) for all values of the parameters.

One can prove the following proposition

(28) If len p = len q and for every k such that k < len p holds p(k) = q(k),

then p = q. The following proposition is true

(29) For every R such that the carrier of R ≠ {0_{R}} for every k there exists

an algebraic sequence p of R such that len p = k. Let us consider R, x. The functor ⟨x⟩ yielding an algebraic sequence of R is

defined by: (Def.6) len⟨x⟩ ≤ 1 and ⟨x⟩(0) = x.

One can prove the following propositions:

(30) p = ⟨x⟩ if and only if len p ≤ 1 and p(0) = x.

(31) p = ⟨0_{R}⟩ if and only if len p = 0.

(32) p = ⟨0_{R}⟩ if and only if support p = ∅.

(33) ⟨0_{R}⟩(i) = 0_{R}.

(34) p = ⟨0_{R}⟩ if and only if rng p = {0_{R}}.

In the sequel R will be an associative ring and V will be a left module over R. Let us consider R, V. The functor Θ_{V} yields a vector of V and is defined by: (Def.7) Θ_{V} = 0_{the carrier of V}.

One can prove the following proposition

(35) Θ_{V} = 0_{the carrier of V}.

In the sequel x denotes a vector of V. Let us consider R, V. A function from into the carrier of the carrier of V is said to be an algebraic sequence of V if: (Def.8) there exists n such that for every i such that i ≥ n holds it(i) = Θ_{V}.

In the sequel p, q will denote algebraic sequences of V. The following proposition is true (37)^{6} dom p = .

Let us consider R, V, p, k. We say that the length of p is at most k if and only if:

6The proposition (36) was either repeated or obvious.

(Def.9) for every i such that i ≥ k holds p(i) = Θ_{V}.

We now state the proposition

(38) the length of p is at most k if and only if for every i such that i ≥ k

holds p(i) = Θ_{V}. Let us consider R, V, p. The functor len p yields a natural number and is

defined as follows: (Def.10) the length of p is at most len p and for every m such that the length of

p is at most m holds len p ≤ m. One can prove the following propositions:

(39) i = len p if and only if the length of p is at most i and for every m such that the length of p is at most m holds i ≤ m.

(40) For every i such that i ≥ len p holds p(i) = Θ_{V}.

(41) If p(k) ≠ Θ_{V}, then len p > k.

(42) If for every i such that i < k holds p(i) ≠ Θ_{V}, then len p ≥ k.

(43) If len p = k + 1, then p(k) ≠ Θ_{V}.

Let us consider R, V, p. The functor support p yields sets of natural numbers and is defined by:

(Def.11) support p = PSeg(len p).

We now state two propositions:

(44) For every y being sets of natural numbers holds y = support p if and only if y = PSeg(len p).

(45) k = len p if and only if PSeg k = support p.

The scheme AlgSeqLambdaLM deals with an associative ring A, a left module B over A, a natural number C, and a unary functor F yielding a vector of B and states that:

there exists an algebraic sequence p of B such that len p ≤ C and for every k such that k < C holds p(k) = F(k) for all values of the parameters.

The following proposition is true

(46) If len p = len q and for every k such that k < len p holds p(k) = q(k),

then p = q. We now state the proposition

(47) For all R, V such that the carrier of the carrier of V ≠ {Θ_{V} } for every

k there exists an algebraic sequence p of V such that len p = k. Let us consider R, V, x. The functor ⟨x⟩ yielding an algebraic sequence of

V is defined as follows: (Def.12) len⟨x⟩ ≤ 1 and ⟨x⟩(0) = x.

One can prove the following propositions:

(48) p = ⟨x⟩ if and only if len p ≤ 1 and p(0) = x.

(49) p = ⟨Θ_{V} ⟩ if and only if len p = 0.

(50) p = ⟨Θ_{V} ⟩ if and only if support p = ∅.

(51) ⟨Θ_{V} ⟩(i) = Θ_{V}.

(52) p = ⟨Θ_{V} ⟩ if and only if rng p = {Θ_{V} }.

In the sequel V will denote a right module over R. Let us consider R, V. The functor Θ_{V} yields a vector of V and is defined as follows:

(Def.13) Θ_{V} = 0_{the carrier of V}.

The following proposition is true

(53) Θ_{V} = 0_{the carrier of V}.

Let us consider R, V. The functor Θ_{V} yields a vector of V and is defined as follows:

(Def.14) Θ_{V} = 0_{the carrier of V}.

The following proposition is true

(54) Θ_{V} = 0_{the carrier of V}.

In the sequel x will denote a vector of V. Let us consider R, V. A function from into the carrier of the carrier of V is called an algebraic sequence of V if:

(Def.15) there exists n such that for every i such that i ≥ n holds it(i) = Θ_{V}.

In the sequel p, q will be algebraic sequences of V. We now state the proposition (56)^{7} dom p = .

Let us consider R, V, p, k. We say that the length of p is at most k if and only if:

(Def.16) for every i such that i ≥ k holds p(i) = Θ_{V}.

Next we state the proposition

(57) the length of p is at most k if and only if for every i such that i ≥ k

holds p(i) = Θ_{V}. Let us consider R, V, p. The functor len p yields a natural number and is

defined by: (Def.17) the length of p is at most len p and for every m such that the length of

p is at most m holds len p ≤ m. Next we state several propositions:

(58) i = len p if and only if the length of p is at most i and for every m such that the length of p is at most m holds i ≤ m.

(59) For every i such that i ≥ len p holds p(i) = Θ_{V}.

(60) If p(k) ≠ Θ_{V}, then len p > k.

(61) If for every i such that i < k holds p(i) ≠ Θ_{V}, then len p ≥ k.

(62) If len p = k + 1, then p(k) ≠ Θ_{V}.

7The proposition (55) was either repeated or obvious.

Let us consider R, V, p. The functor support p yielding sets of natural numbers is defined by:

(Def.18) support p = PSeg(len p).

The following propositions are true:

(63) For every y being sets of natural numbers holds y = support p if and only if y = PSeg(len p).

(64) k = len p if and only if PSeg k = support p.

The scheme AlgSeqLambdaRM deals with an associative ring A, a right module B over A, a natural number C, and a unary functor F yielding a vector of B and states that:

there exists an algebraic sequence p of B such that len p ≤ C and for every k such that k < C holds p(k) = F(k) for all values of the parameters.

The following proposition is true

(65) If len p = len q and for every k such that k < len p holds p(k) = q(k),

then p = q. One can prove the following proposition

(66) For all R, V such that the carrier of the carrier of V ≠ {Θ_{V} } for every

k there exists an algebraic sequence p of V such that len p = k. Let us consider R, V, x. The functor ⟨x⟩ yielding an algebraic sequence of

V is defined by: (Def.19) len⟨x⟩ ≤ 1 and ⟨x⟩(0) = x.

We now state several propositions:

(67) p = ⟨x⟩ if and only if len p ≤ 1 and p(0) = x.

(68) p = ⟨Θ_{V} ⟩ if and only if len p = 0.

(69) p = ⟨Θ_{V} ⟩ if and only if support p = ∅.

(70) ⟨Θ_{V} ⟩(i) = Θ_{V}.

(71) p = ⟨Θ_{V} ⟩ if and only if rng p = {Θ_{V} }.

In the sequel V is a bimodule over R. Let us consider R, V. The functor Θ_{V} yields a vector of V and is defined as follows:

(Def.20) Θ_{V} = 0_{the carrier of V}.

One can prove the following proposition

(72) Θ_{V} = 0_{the carrier of V}.

Let us consider R, V. The functor Θ_{V} yields a vector of V and is defined as follows:

(Def.21) Θ_{V} = 0_{the carrier of V}.

We now state the proposition

(73) Θ_{V} = 0_{the carrier of V}.

In the sequel x will denote a vector of V. Let us consider R, V. A function from into the carrier of the carrier of V is said to be an algebraic sequence of V if:

(Def.22) there exists n such that for every i such that i ≥ n holds it(i) = Θ_{V}.

In the sequel p, q will be algebraic sequences of V. We now state the proposition (75)^{8} dom p = .

Let us consider R, V, p, k. We say that the length of p is at most k if and only if:

(Def.23) for every i such that i ≥ k holds p(i) = Θ_{V}.

Next we state the proposition

(76) the length of p is at most k if and only if for every i such that i ≥ k

holds p(i) = Θ_{V}. Let us consider R, V, p. The functor len p yielding a natural number is

defined by: (Def.24) the length of p is at most len p and for every m such that the length of

p is at most m holds len p ≤ m.

One can prove the following propositions:

(77) i = len p if and only if the length of p is at most i and for every m such that the length of p is at most m holds i ≤ m.

(78) For every i such that i ≥ len p holds p(i) = Θ_{V}.

(79) If p(k) ≠ Θ_{V}, then len p > k.

(80) If for every i such that i < k holds p(i) ≠ Θ_{V}, then len p ≥ k.

(81) If len p = k + 1, then p(k) ≠ Θ_{V}.

Let us consider R, V, p. The functor support p yielding sets of natural numbers is defined by:

(Def.25) support p = PSeg(len p).

We now state two propositions:

(82) For every y being sets of natural numbers holds y = support p if and only if y = PSeg(len p).

(83) k = len p if and only if PSeg k = support p.

The scheme AlgSeqLambdaBM concerns an associative ring A, a bimodule B over A, a natural number C, and a unary functor F yielding a vector of B and states that:

there exists an algebraic sequence p of B such that len p ≤ C and for every k such that k < C holds p(k) = F(k) for all values of the parameters.

We now state the proposition 104 Michał Muzalewski et al.

8The proposition (74) was either repeated or obvious.

(84) If len p = len q and for every k such that k < len p holds p(k) = q(k),

then p = q. The following proposition is true

(85) For all R, V such that the carrier of the carrier of V ≠ {Θ_{V} } for every

k there exists an algebraic sequence p of V such that len p = k. Let us consider R, V, x. The functor ⟨x⟩ yields an algebraic sequence of V

and is defined by: (Def.26) len⟨x⟩ ≤ 1 and ⟨x⟩(0) = x.

Next we state several propositions:

(86) p = ⟨x⟩ if and only if len p ≤ 1 and p(0) = x.

(87) p = ⟨Θ_{V} ⟩ if and only if len p = 0.

(88) p = ⟨Θ_{V} ⟩ if and only if support p = ∅.

(89) ⟨Θ_{V} ⟩(i) = Θ_{V}.

(90) p = ⟨Θ_{V} ⟩ if and only if rng p = {Θ_{V} }.

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

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

[7] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

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

Relations of Tolerance ^{1}

Krzysztof Hryniewiecki Warsaw University

Summary. Introduces notions of relations of tolerance, tolerance set and neighbourhood of an element. The basic properties of relations of tolerance are proved.

MML Identifier: TOLER 1.

The notation and terminology used here have been introduced in the following papers: [2], [3], [4], [5], and [1]. We adopt the following rules: X, Y, Z denote sets, x, y are arbitrary, and R denotes a relation between X and X. The following propositions are true:

(1) field = ∅.

(2) is pseudo reflexive.

(3) is symmetric.

(4) is irreflexive.

(5) is antisymmetric.

(6) is asymmetric.

(7) is connected.

(8) is strongly connected.

(9) is transitive. Let us consider X. The functor ∇_{X} yielding a relation between X and X is

defined by: (Def.1) ∇_{X} = [: X, X:].

Let us consider X, R, Y. Then R |^{2} Y is a relation between Y and Y. The following propositions are true:

(10) For every relation R between X and X holds R = ∇_{X} if and only if R = [: X, X:].

1Supported by Philippe le Hodey Foundation. This work had been done on Mizar Workshop ’89 (Fourdrain, France) in Summer ’89.

(11) ∇_{X} = [: X, X:].

(12) dom ∇_{X} = X.

(13) rng ∇_{X} = X.

(14) field ∇_{X} = X.

(15) For all x, y such that x ∈ X and y ∈ X holds ⟨⟨x, y⟩⟩ ∈ ∇_{X}.

(16) For all x, y such that x ∈ field ∇_{X} and y ∈ field ∇_{X} holds ⟨⟨x, y⟩⟩ ∈ ∇_{X}.

(17) ∇_{X} is pseudo reflexive.

(18) ∇_{X} is symmetric.

(19) ∇_{X} is strongly connected.

(20) ∇_{X} is transitive.

(21) ∇_{X} is connected.

Let us consider X. A relation between X and X is said to be a tolerance of X if: (Def.2) it is pseudo reflexive and it is symmetric and field it = X.

In the sequel T, R denote tolerances of X. The following propositions are true: (23)^{2} For every tolerance R of X holds R is pseudo reflexive and R is sym-

metric and field R = X.

(24) For every tolerance T of X holds dom T = X.

(25) For every tolerance T of X holds rng T = X.

(26) For every tolerance T of X holds field T = X.

(27) For every tolerance T of X holds x ∈ X if and only if ⟨⟨x, x⟩⟩ ∈ T.

(28) For every tolerance T of X holds T is reflexive in X.

(29) For every tolerance T of X holds T is symmetric in X.

(30) For every tolerance T of X such that ⟨⟨x, y⟩⟩ ∈ T holds ⟨⟨y, x⟩⟩ ∈ T.

(31) For every tolerance T of X and for all x, y such that ⟨⟨x, y⟩⟩ ∈ T holds x ∈ X and y ∈ X.

(32) For every relation R between X and Y such that R is symmetric holds

R |^{2} Z is symmetric. Let us consider X, T, and let Y be a subset of X. Then T |^{2} Y is a tolerance

of Y.

Next we state the proposition

(33) If Y ⊆ X, then T |^{2} Y is a tolerance of Y.

Let us consider X, and let T be a tolerance of X. A set is called a set of mutually elements w.r.t. T if: (Def.3) for all x, y such that x ∈ it and y ∈ it holds ⟨⟨x, y⟩⟩ ∈ T.

We now state the proposition

(34) ∅ is a set of mutually elements w.r.t. T.

2The proposition (22) was either repeated or obvious.

Let us consider X, and let T be a tolerance of X. A set of mutually elements w.r.t. T is called a tolerance class of T if: (Def.4) for every x such that x ∉ it and x ∈ X there exists y such that y ∈ it

and ⟨⟨x, y⟩⟩ ∉ T. Next we state a number of propositions:

(36)^{3} Y is a set of mutually elements w.r.t. T if and only if for all x, y such

that x ∈ Y and y ∈ Y holds ⟨⟨x, y⟩⟩ ∈ T. (38)^{4} For every tolerance T of X such that ∅ is a tolerance class of T holds

T = .

(39) is a tolerance of ∅.

(40) For all x, y such that ⟨⟨x, y⟩⟩ ∈ T holds {x, y} is a set of mutually elements w.r.t. T.

(41) For every x such that x ∈ X holds {x} is a set of mutually elements w.r.t. T.

(42) For all Y, Z such that Y is a set of mutually elements w.r.t. T and Z is a set of mutually elements w.r.t. T holds Y ∩ Z is a set of mutually elements w.r.t. T.

(43) If Y is a set of mutually elements w.r.t. T, then Y ⊆ X.

(44) If Y is a tolerance class of T, then Y ⊆ X.

(45) For every set Y of mutually elements w.r.t. T there exists a tolerance class Z of T such that Y ⊆ Z.

(46) For all x, y such that ⟨⟨x, y⟩⟩ ∈ T there exists a tolerance class Z of T such that x ∈ Z and y ∈ Z.

(47) For every x such that x ∈ X there exists a tolerance class Z of T such

that x ∈ Z. Let us consider X. Then △_{X} is a tolerance of X. We now state three propositions:

(48) ∇_{X} is a tolerance of X.

(49) T ⊆ ∇_{X}.

(50) △_{X} ⊆ T.

The scheme ToleranceEx concerns a set A, and a binary predicate P, and states that:

there exists a tolerance T of A such that for all x, y such that x ∈ A and y ∈ A holds ⟨⟨x, y⟩⟩ ∈ T if and only if P[x, y] provided the parameters satisfy the following conditions:

• for every x such that x ∈ A holds P[x, x],

• for all x, y such that x ∈ A and y ∈ A and P[x, y] holds P[y, x].

One can prove the following propositions:

3The proposition (35) was either repeated or obvious.

4The proposition (37) was either repeated or obvious.

(51) For every Y there exists a tolerance T of ⋃ Y such that for every Z such that Z ∈ Y holds Z is a set of mutually elements w.r.t. T.

(52) Let Y be a set. Let T, R be tolerances of ⋃ Y. Then if for all x, y holds ⟨⟨x, y⟩⟩ ∈ T if and only if there exists Z such that Z ∈ Y and x ∈ Z and y ∈ Z and for all x, y holds ⟨⟨x, y⟩⟩ ∈ R if and only if there exists Z such that Z ∈ Y and x ∈ Z and y ∈ Z, then T = R.

(53) For all tolerances T, R of X such that for every Z holds Z is a tolerance

class of T if and only if Z is a tolerance class of R holds T = R. Let us consider X, and let T be a tolerance of X, and let us consider x. The

functor neighbourhood(x, T) yielding a set is defined by: (Def.5) for every y holds y ∈ neighbourhood(x, T) if and only if ⟨⟨x, y⟩⟩ ∈ T.

One can prove the following propositions:

(54) For every tolerance T of X and for every x and for every set Y holds Y = neighbourhood(x, T) if and only if for every y holds y ∈ Y if and only if ⟨⟨x, y⟩⟩ ∈ T.

(55) For every tolerance T of X holds y ∈ neighbourhood(x, T) if and only if ⟨⟨x, y⟩⟩ ∈ T.

(56) If x ∈ X, then x ∈ neighbourhood(x, T).

(57) neighbourhood(x, T) ⊆ X.

(58) For every Y such that for every set Z holds Z ∈ Y if and only if x ∈ Z and Z is a tolerance class of T holds neighbourhood(x, T) = ⋃ Y.

(59) For every Y such that for every Z holds Z ∈ Y if and only if x ∈ Z and

Z is a set of mutually elements w.r.t. T holds neighbourhood(x, T) = ⋃ Y. We now define two new functors. Let us consider X, and let T be a tolerance

of X. The functor TolSets T yields a set and is defined by: (Def.6) for every Y holds Y ∈ TolSets T if and only if Y is a set of mutually

elements w.r.t. T. The functor TolClasses T yields a set and is defined by: (Def.7) for every Y holds Y ∈ TolClasses T if and only if Y is a tolerance class

of T.

The following propositions are true:

(60) For every set Y and for every tolerance T of X holds Y = TolSets T if and only if for every Z holds Z ∈ Y if and only if Z is a set of mutually elements w.r.t. T.

(61) For every tolerance T of X and for every Z holds Z ∈ TolSets T if and only if Z is a set of mutually elements w.r.t. T.

(62) For every set Y and for every tolerance T of X holds Y = TolClasses T if and only if for every Z holds Z ∈ Y if and only if Z is a tolerance class of T.

(63) For every tolerance T of X holds Z ∈ TolClasses T if and only if Z is a tolerance class of T.

(64) If TolClasses R ⊆ TolClasses T, then R ⊆ T.

(65) For all tolerances T, R of X such that TolClasses T = TolClasses R holds T = R.

(66) ⋃(TolClasses T) = X.

(67) ⋃(TolSets T) = X.

(68) If for every x such that x ∈ X holds neighbourhood(x, T) is a set of mutually elements w.r.t. T, then T is transitive.

(69) If T is transitive, then for every x such that x ∈ X holds neighbourhood(x, T) is a tolerance class of T.

(70) For every x and for every tolerance class Y of T such that x ∈ Y holds Y ⊆ neighbourhood(x, T).

(71) TolSets R ⊆ TolSets T if and only if R ⊆ T.

(72) TolClasses T ⊆ TolSets T.

(73) If for every x such that x ∈ X holds neighbourhood(x, R) ⊆ neighbourhood(x, T), then R ⊆ T.

(74) T ⊆ T · T.

(75) If T = T · T, then T is transitive.

References

[1] Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.

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

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

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

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

110

Real Normed Space

Jan Popiołek^{1} Warsaw University Białystok

Summary. We construct a real normed space ⟨V, ∥.∥⟩, where V is a real vector space and ∥.∥ is a norm. Auxillary properties of the norm are proved. Next, we introduce the notion of sequence in the real normed space. The basic operations on sequences (addition, substraction, multiplication by real number) are defined. We study some properties of sequences in the real normed space and the operations on them.

MML Identifier: NORMSP 1.

The notation and terminology used in this paper have been introduced in the following papers: [5], [13], [16], [3], [4], [1], [2], [17], [11], [12], [9], [7], [8], [10], [15], [14], and [6]. We consider normed structures which are systems

⟨vectors, a norm⟩, where the vectors constitute a real linear space and the norm is a function from the vectors of the vectors into .

In the sequel X is a normed structure and a, b are real numbers. Let us consider X. A point of X is an element of the vectors of the vectors of X.

In the sequel x denotes a point of X. Let us consider X, x. The functor ∥x∥ yields a real number and is defined as follows: (Def.1) ∥x∥ = (the norm of X)(x).

A normed structure is said to be a real normed space if: (Def.2) for all points x, y of it and for every a holds ∥x∥ = 0 if and only if

x = 0_{the vectors of it} but ∥a · x∥ = |a| · ∥x∥ and ∥x + y∥ ≤ ∥x∥ + ∥y∥.

We adopt the following rules: R_{1} is a real normed space and x, y, z, g are points of R_{1}. The following propositions are true:

(2)^{2} ∥x∥ = 0 if and only if x = 0_{the vectors of R_{1}}.

1Supported by RPBP.III-24.C8

2The proposition (1) was either repeated or obvious.

(3) ∥a · x∥ = |a| · ∥x∥.

(4) ∥x + y∥ ≤ ∥x∥ + ∥y∥.

(5) ∥0_{the vectors of R_{1}}∥ = 0.

(6) ∥−x∥ = ∥x∥.

(7) ∥x − y∥ ≤ ∥x∥ + ∥y∥.

(8) 0 ≤ ∥x∥.

(9) ∥a · x + b · y∥ ≤ |a| · ∥x∥ + |b| · ∥y∥.

(10) ∥x − y∥ = 0 if and only if x = y.

(11) ∥x − y∥ = ∥y − x∥.

(12) ∥x∥ − ∥y∥ ≤ ∥x − y∥.

(13) |∥x∥ − ∥y∥| ≤ ∥x − y∥.

(14) ∥x − z∥ ≤ ∥x − y∥ + ∥y − z∥.

(15) If x ≠ y, then ∥x − y∥ ≠ 0.

Let us consider R_{1}. A subset of R_{1} is a subset of the vectors of the vectors of R_{1}.

Let us consider R_{1}. A function is called a sequence of R_{1} if: (Def.3) dom it = and rng it ⊆ the vectors of the vectors of R_{1}.

For simplicity we adopt the following rules: S, S_{1}, S_{2}, T are sequences of R_{1}, k, n, m are natural numbers, r is a real number, f is a function, and d is arbitrary. We now state several propositions: (17)^{3} f is a sequence of R_{1} if and only if dom f = and for every d such that

d ∈ holds f(d) is a point of R_{1}.

(18) For all S, T such that for every n holds S(n) = T (n) holds S = T.

(19) For every x there exists S such that rng S = {x}.

(20) If there exists x such that for every n holds S(n) = x, then there exists x such that rng S = {x}.

(21) If there exists x such that rng S = {x}, then for every n holds S(n) = S(n + 1).

(22) If for every n holds S(n) = S(n + 1), then for all n, k holds S(n) = S(n + k).

(23) If for all n, k holds S(n) = S(n+k), then for all n, m holds S(n) = S(m).

(24) If for all n, m holds S(n) = S(m), then there exists x such that for every n holds S(n) = x.

(25) There exists S such that rng S = {0_{the vectors of R_{1}}}.

Let us consider R_{1}, S. We say that S is constant if and only if: (Def.4) there exists x such that for every n holds S(n) = x.

The following propositions are true: (27)^{4} S is constant if and only if there exists x such that rng S = {x}.

3The proposition (16) was either repeated or obvious.

4The proposition (26) was either repeated or obvious.

(28) For every n holds S(n) is a point of R_{1}.

Let us consider R_{1}, S, n. Then S(n) is a point of R_{1}. The scheme ExRNSSeq concerns a real normed space 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 n holds S(n) = F(n) for all values of the parameters.

Let us consider R_{1}, S_{1}, S_{2}. The functor S_{1} + S_{2} yielding a sequence of R_{1} is defined as follows: (Def.5) for every n holds (S_{1} + S_{2})(n) = S_{1}(n) + S_{2}(n).

One can prove the following proposition

(29) S = S_{1} + S_{2} if and only if for every n holds S(n) = S_{1}(n) + S_{2}(n). Let us consider R_{1}, S_{1}, S_{2}. The functor S_{1} − S_{2} yielding a sequence of R_{1} is

defined as follows: (Def.6) for every n holds (S_{1} − S_{2})(n) = S_{1}(n) − S_{2}(n).

The following proposition is true

(30) S = S_{1} − S_{2} if and only if for every n holds S(n) = S_{1}(n) − S_{2}(n). Let us consider R_{1}, S, x. The functor S − x yields a sequence of R_{1} and is

defined by: (Def.7) for every n holds (S − x)(n) = S(n) − x.

Next we state the proposition

(31) T = S − x if and only if for every n holds T (n) = S(n) − x. Let us consider R_{1}, S, a. The functor a · S yields a sequence of R_{1} and is

defined by: (Def.8) for every n holds (a · S)(n) = a · S(n).

We now state the proposition

(32) T = a · S if and only if for every n holds T (n) = a · S(n).

Let us consider R_{1}, S. We say that S is convergent if and only if: (Def.9) there exists g such that for every r such that 0 < r there exists m such

that for every n such that m ≤ n holds ∥S(n) − g∥ < r. One can prove the following propositions:

(34)^{5} If S_{1} is convergent and S_{2} is convergent, then S_{1} + S_{2} is convergent.

(35) If S_{1} is convergent and S_{2} is convergent, then S_{1} − S_{2} is convergent.

(36) If S is convergent, then S − x is convergent.

(37) If S is convergent, then a · S is convergent.

Let us consider R_{1}, S. The functor ∥S∥ yielding a sequence of real numbers is defined by:

(Def.10) for every n holds ∥S∥(n) = ∥S(n)∥. Next we state two propositions:

5The proposition (33) was either repeated or obvious.

114 Jan Popiołek

(38) ∥S∥ is a sequence of real numbers if and only if for every n holds ∥S∥(n) = ∥S(n)∥.

(39) If S is convergent, then ∥S∥ is convergent.

Let us consider R_{1}, S. Let us assume that S is convergent. The functor lim S yielding a point of R_{1} is defined by:

(Def.11) for every r such that 0 < r there exists m such that for every n such that m ≤ n holds ∥S(n) − (lim S)∥ < r.

The following propositions are true:

(40) If S is convergent, then lim S = g if and only if for every r such that 0 < r there exists m such that for every n such that m ≤ n holds ∥S(n) − g∥ < r.

(41) If S is convergent and lim S = g, then ∥S−g∥ is convergent and lim∥S− g∥ = 0.

(42) If S_{1} is convergent and S_{2} is convergent, then lim(S_{1} + S_{2}) = lim S_{1} + lim S_{2}.

(43) If S_{1} is convergent and S_{2} is convergent, then lim(S_{1} − S_{2}) = lim S_{1} − lim S_{2}.

(44) If S is convergent, then lim(S − x) = lim S − x.

(45) If S is convergent, then lim(a · S) = a · (lim S).

References

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

[2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[6] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 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. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

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

[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.

Real Normed Space 115

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

[14] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

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

[16] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

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

116

Schemes of Existence of some Types of Functions

Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. We prove some useful shemes of existence of real sequences, partial functions from a domain into a domain, partial functions from a set to a set and functions from a domain into a domain. At the begining we prove some related auxiliary theorems to the article [1].

MML Identifier: SCHEME1.

The notation and terminology used here are introduced in the following articles: [9], [5], [1], [2], [3], [8], [6], [4], and [7]. We adopt the following convention: x, y will be arbitrary, n, m will denote natural numbers, and r will denote a real number. Next we state four propositions:

(1) For every n there exists m such that n = 2 · m or n = 2 · m + 1.

(2) For every n there exists m such that n = 3 · m or n = 3 · m + 1 or n = 3 · m + 2.

(3) For every n there exists m such that n = 4 · m or n = 4 · m + 1 or n = 4 · m + 2 or n = 4 · m + 3.

(4) For every n there exists m such that n = 5 · m or n = 5 · m + 1 or n = 5 · m + 2 or n = 5 · m + 3 or n = 5 · m + 4.

In this article we present several logical schemes. The scheme ExRealSubseq concerns a sequence of real numbers A, and a unary predicate P, and states that:

there exists a sequence of real numbers q such that q is a subsequence of A and for every n holds P[q(n)] and for every n such that for every r such that r = A(n) holds P[r] there exists m such that A(n) = q(m) provided the following requirement is met:

• for every n there exists m such that n ≤ m and P[A(m)].

1Supported by RPBP.III-24.C8

The scheme ExRealSeq2 deals with a unary functor F yielding a real number and a unary functor G yielding a real number and states that:

there exists a sequence of real numbers s such that for every n holds s(2·n) = F(n) and s(2 · n + 1) = G(n) for all values of the parameters.

The scheme ExRealSeq3 deals with a unary functor F yielding a real number, a unary functor G yielding a real number, and a unary functor H yielding a real number and states that:

there exists a sequence of real numbers s such that for every n holds s(3·n) = F(n) and s(3 · n + 1) = G(n) and s(3 · n + 2) = H(n) for all values of the parameters.

The scheme ExRealSeq4 deals with a unary functor F yielding a real number, a unary functor G yielding a real number, a unary functor H yielding a real number, and a unary functor I yielding a real number and states that:

there exists a sequence of real numbers s such that for every n holds s(4·n) = F(n) and s(4 · n + 1) = G(n) and s(4 · n + 2) = H(n) and s(4 · n + 3) = I(n) for all values of the parameters.

The scheme ExRealSeq5 deals with a unary functor F yielding a real number, a unary functor G yielding a real number, a unary functor H yielding a real number, a unary functor I yielding a real number, and a unary functor J yielding a real number and states that:

there exists a sequence of real numbers s such that for every n holds s(5·n) = F(n) and s(5 · n + 1) = G(n) and s(5 · n + 2) = H(n) and s(5 · n + 3) = I(n) and s(5 · n + 4) = J (n) for all values of the parameters.

The scheme PartFuncExD2 deals with a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, a unary functor G yielding an element of B, and two unary predicates P and Q, and states that:

there exists a partial function f from A to B such that for every element c of A holds c ∈ dom f if and only if P[c] or Q[c] and for every element c of A such that c ∈ dom f holds if P[c], then f(c) = F(c) but if Q[c], then f(c) = G(c) provided the following condition is met:

• for every element c of A such that P[c] holds not Q[c].

The scheme PartFuncExD2’ concerns a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, a unary functor G yielding an element of B, and two unary predicates P and Q, and states that:

there exists a partial function f from A to B such that for every element c of A holds c ∈ dom f if and only if P[c] or Q[c] and for every element c of A such that c ∈ dom f holds if P[c], then f(c) = F(c) but if Q[c], then f(c) = G(c) provided the following requirement is met:

• for every element c of A such that P[c] and Q[c] holds F(c) = G(c).

The scheme PartFuncExD2” deals with a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, a unary functor G yielding an element of B, and a unary predicate P, and states that:

there exists a partial function f from A to B such that f is total and for every element c of A such that c ∈ dom f holds if P[c], then f(c) = F(c) but if not P[c], then f(c) = G(c) for all values of the parameters.

The scheme PartFuncExD3 deals with a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, a unary functor G yielding an element of B, a unary functor H yielding an element of B, and three unary predicates P, Q, and R, and states that:

there exists a partial function f from A to B such that for every element c of A holds c ∈ dom f if and only if P[c] or Q[c] or R[c] and for every element c of A such that c ∈ dom f holds if P[c], then f(c) = F(c) but if Q[c], then f(c) = G(c) but if R[c], then f(c) = H(c) provided the parameters satisfy the following condition:

• for every element c of A holds if P[c], then not Q[c] but if P[c], then not R[c] but if Q[c], then not R[c].

The scheme PartFuncExD3’ concerns a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, a unary functor G yielding an element of B, a unary functor H yielding an element of B, and three unary predicates P, Q, and R, and states that:

there exists a partial function f from A to B such that for every element c of A holds c ∈ dom f if and only if P[c] or Q[c] or R[c] and for every element c of A such that c ∈ dom f holds if P[c], then f(c) = F(c) but if Q[c], then f(c) = G(c) but if R[c], then f(c) = H(c) provided the following requirement is met:

• for every element c of A holds if P[c] and Q[c], then F(c) = G(c) but if P[c] and R[c], then F(c) = H(c) but if Q[c] and R[c], then G(c) = H(c).

The scheme PartFuncExD4 deals with a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, a unary functor G yielding an element of B, a unary functor H yielding an element of B, a unary functor I yielding an element of B, and four unary predicates P, Q, R, and S, and states that:

there exists a partial function f from A to B such that for every element c of A holds c ∈ dom f if and only if P[c] or Q[c] or R[c] or S[c] and for every element c of A such that c ∈ dom f holds if P[c], then f(c) = F(c) but if Q[c], then f(c) = G(c) but if R[c], then f(c) = H(c) but if S[c], then f(c) = I(c) provided the parameters satisfy the following condition:

• for every element c of A holds if P[c], then not Q[c] but if P[c], then not R[c] but if P[c], then not S[c] but if Q[c], then not R[c] but if Q[c], then not S[c] but if R[c], then not S[c].

The scheme PartFuncExS2 deals with a set A, a set B, a unary functor F, a unary functor G, and two unary predicates P and Q, and states that:

there exists a partial function f from A to B such that for every x holds x ∈ dom f if and only if x ∈ A but P[x] or Q[x] and for every x such that x ∈ dom f holds if P[x], then f(x) = F(x) but if Q[x], then f(x) = G(x) provided the parameters satisfy the following conditions:

• for every x such that x ∈ A holds if P[x], then not Q[x],

• for every x such that x ∈ A and P[x] holds F(x) ∈ B,

• for every x such that x ∈ A and Q[x] holds G(x) ∈ B.

The scheme PartFuncExS3 deals with a set A, a set B, a unary functor F, a unary functor G, a unary functor H, and three unary predicates P, Q, and R, and states that:

there exists a partial function f from A to B such that for every x holds x ∈ dom f if and only if x ∈ A but P[x] or Q[x] or R[x] and for every x such that x ∈ dom f holds if P[x], then f(x) = F(x) but if Q[x], then f(x) = G(x) but if R[x], then f(x) = H(x) provided the parameters meet the following conditions:

• for every x such that x ∈ A holds if P[x], then not Q[x] but if P[x], then not R[x] but if Q[x], then not R[x],

• for every x such that x ∈ A and P[x] holds F(x) ∈ B,

• for every x such that x ∈ A and Q[x] holds G(x) ∈ B,

• for every x such that x ∈ A and R[x] holds H(x) ∈ B.

The scheme PartFuncExS4 deals with a set A, a set B, a unary functor F, a unary functor G, a unary functor H, a unary functor I, and four unary predicates P, Q, R, and S, and states that:

there exists a partial function f from A to B such that for every x holds x ∈ dom f if and only if x ∈ A but P[x] or Q[x] or R[x] or S[x] and for every x such that x ∈ dom f holds if P[x], then f(x) = F(x) but if Q[x], then f(x) = G(x) but if R[x], then f(x) = H(x) but if S[x], then f(x) = I(x) provided the parameters meet the following requirements:

• for every x such that x ∈ A holds if P[x], then not Q[x] but if P[x], then not R[x] but if P[x], then not S[x] but if Q[x], then not R[x] but if Q[x], then not S[x] but if R[x], then not S[x],

• for every x such that x ∈ A and P[x] holds F(x) ∈ B,

• for every x such that x ∈ A and Q[x] holds G(x) ∈ B,

• for every x such that x ∈ A and R[x] holds H(x) ∈ B,

• for every x such that x ∈ A and S[x] holds I(x) ∈ B.

The scheme PartFuncExC D2 concerns a non-empty set A, a non-empty set B, a non-empty set C, a binary functor F yielding an element of C, a binary functor G yielding an element of C, and two binary predicates P and Q, and states that:

there exists a partial function f from [: A, B:] to C such that for every element c of A and for every element d of B holds ⟨⟨c, d⟩⟩ ∈ dom f if and only if P[c, d] or Q[c, d] and for every element c of A and for every element d of B such that ⟨⟨c, d⟩⟩ ∈ dom f holds if P[c, d], then f(⟨⟨c, d⟩⟩) = F(c, d) but if Q[c, d], then f(⟨⟨c, d⟩⟩) = G(c, d) provided the parameters meet the following requirement:

• for every element c of A and for every element d of B such that P[c, d] holds not Q[c, d].

The scheme PartFuncExC D3 concerns a non-empty set A, a non-empty set B, a non-empty set C, a binary functor F yielding an element of C, a binary functor G yielding an element of C, a binary functor H yielding an element of C, and three binary predicates P, Q, and R, and states that:

there exists a partial function f from [: A, B:] to C such that for every element c of A and for every element d of B holds ⟨⟨c, d⟩⟩ ∈ dom f if and only if P[c, d] or Q[c, d] or R[c, d] and for every element c of A and for every element r of B such that ⟨⟨c, r⟩⟩ ∈ dom f holds if P[c, r], then f(⟨⟨c, r⟩⟩) = F(c, r) but if Q[c, r], then f(⟨⟨c, r⟩⟩) = G(c, r) but if R[c, r], then f(⟨⟨c, r⟩⟩) = H(c, r) provided the following requirement is met:

• for every element c of A and for every element s of B holds if P[c, s], then not Q[c, s] but if P[c, s], then not R[c, s] but if Q[c, s], then not R[c, s].

The scheme PartFuncExC S2 concerns a set A, a set B, a set C, a binary functor F, a binary functor G, and two binary predicates P and Q, and states that:

there exists a partial function f from [: A, B:] to C such that for all x, y holds ⟨⟨x, y⟩⟩ ∈ dom f if and only if x ∈ A and y ∈ B but P[x, y] or Q[x, y] and for all x, y such that ⟨⟨x, y⟩⟩ ∈ dom f holds if P[x, y], then f(⟨⟨x, y⟩⟩) = F(x, y) but if Q[x, y], then f(⟨⟨x, y⟩⟩) = G(x, y) provided the following conditions are met:

• for all x, y such that x ∈ A and y ∈ B holds if P[x, y], then not Q[x, y],

• for all x, y such that x ∈ A and y ∈ B and P[x, y] holds F(x, y) ∈ C,

• for all x, y such that x ∈ A and y ∈ B and Q[x, y] holds G(x, y) ∈ C.

The scheme PartFuncExC S3 concerns a set A, a set B, a set C, a binary functor F, a binary functor G, a binary functor H, and three binary predicates P, Q, and R, and states that:

there exists a partial function f from [: A, B:] to C such that for all x, y holds ⟨⟨x, y⟩⟩ ∈ dom f if and only if x ∈ A and y ∈ B but P[x, y] or Q[x, y] or R[x, y] and for all x, y such that ⟨⟨x, y⟩⟩ ∈ dom f holds if P[x, y], then f(⟨⟨x, y⟩⟩) = F(x, y) but if Q[x, y], then f(⟨⟨x, y⟩⟩) = G(x, y) but if R[x, y], then f(⟨⟨x, y⟩⟩) = H(x, y) provided the following conditions are met:

• for all x, y such that x ∈ A and y ∈ B holds if P[x, y], then not Q[x, y] but if P[x, y], then not R[x, y] but if Q[x, y], then not R[x, y],

• for all x, y such that x ∈ A and y ∈ B holds if P[x, y], then F(x, y) ∈ C,

• for all x, y such that x ∈ A and y ∈ B holds if Q[x, y], then G(x, y) ∈ C,

• for all x, y such that x ∈ A and y ∈ B holds if R[x, y], then H(x, y) ∈ C.

The scheme ExFuncD3 concerns a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, a unary functor G yielding an element of B, a unary functor H yielding an element of B, and three unary predicates P, Q, and R, and states that:

there exists a function f from A into B such that for every element c of A holds if P[c], then f(c) = F(c) but if Q[c], then f(c) = G(c) but if R[c], then f(c) = H(c) provided the parameters satisfy the following conditions:

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

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

The scheme ExFuncD4 concerns a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, a unary functor G yielding an element of B, a unary functor H yielding an element of B, a unary functor I yielding an element of B, and four unary predicates P, Q, R, and S, and states that:

there exists a function f from A into B such that for every element c of A holds if P[c], then f(c) = F(c) but if Q[c], then f(c) = G(c) but if R[c], then f(c) = H(c) but if S[c], then f(c) = I(c) provided the following conditions are met:

• for every element c of A holds if P[c], then not Q[c] but if P[c], then not R[c] but if P[c], then not S[c] but if Q[c], then not R[c] but if Q[c], then not S[c] but if R[c], then not S[c],

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

The scheme FuncExC D2 deals with a non-empty set A, a non-empty set B, a non-empty set C, a binary functor F yielding an element of C, a binary functor G yielding an element of C, and a binary predicate P, and states that:

there exists a function f from [: A, B:] into C such that for every element c of A and for every element d of B such that ⟨⟨c, d⟩⟩ ∈ dom f holds if P[c, d], then f(⟨⟨c, d⟩⟩) = F(c, d) but if not P[c, d], then f(⟨⟨c, d⟩⟩) = G(c, d) for all values of the parameters.

The scheme FuncExC D3 deals with a non-empty set A, a non-empty set B, a non-empty set C, a binary functor F yielding an element of C, a binary functor G yielding an element of C, a binary functor H yielding an element of C, and three binary predicates P, Q, and R, and states that:

there exists a function f from [: A, B:] into C such that for every element c of A and for every element d of B holds ⟨⟨c, d⟩⟩ ∈ dom f if and only if P[c, d] or Q[c, d] or R[c, d] and for every element c of A and for every element d of B such that ⟨⟨c, d⟩⟩ ∈ dom f holds if P[c, d], then f(⟨⟨c, d⟩⟩) = F(c, d) but if Q[c, d], then f(⟨⟨c, d⟩⟩) = G(c, d) but if R[c, d], then f(⟨⟨c, d⟩⟩) = H(c, d) provided the parameters have the following properties:

• for every element c of A and for every element d of B holds if P[c, d], then not Q[c, d] but if P[c, d], then not R[c, d] but if Q[c, d], then not R[c, d],

• for every element c of A and for every element d of B holds P[c, d] or Q[c, d] or R[c, d].

Schemes of Existence of some Types of … 123

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 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. 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] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

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

124

Integer and Rational Exponents

Konrad Raczkowski^{1} Warsaw University Białystok

Summary. The article includes definitions and theorems which are needed to define real exponent. The following notions are defined: natural exponent, integer exponent and rational exponent.

MML Identifier: PREPOWER.

The terminology and notation used in this paper are introduced in the following papers: [12], [15], [4], [10], [1], [2], [3], [9], [7], [8], [14], [11], [13], [6], and [5]. For simplicity we follow the rules: a, b, c will be real numbers, m, n will be natural numbers, k, l, i will be integers, p, q will be rational numbers, and s_{1}, s_{2} will be sequences of real numbers. The following propositions are true:

(2)^{2} If s_{1} is convergent and for every n holds s_{1}(n) ≥ a, then lim s_{1} ≥ a.

(3) If s_{1} is convergent and for every n holds s_{1}(n) ≤ a, then lim s_{1} ≤ a. Let us consider a. The functor (a^{κ})_{κ∈} yielding a sequence of real numbers

is defined as follows: (Def.1) ((a^{κ})_{κ∈} )(0) = 1 and for every m holds ((a^{κ})_{κ∈} )(m+1) = ((a^{κ})_{κ∈} )(m)·

a.

Next we state two propositions:

(4) For every sequence of real numbers s and for every a holds s = (a^{κ})_{κ∈} if and only if s(0) = 1 and for every m holds s(m + 1) = s(m) · a.

(5) For every a such that a ≠ 0 for every m holds (a^{κ})_{κ∈} (m) ≠ 0.

Let us consider a, n. The functor an_{}yields a real number and is defined by:

(Def.2) an_{}= (a^{κ})_{κ∈} (n).

Next we state a number of propositions:

(6) an_{}= (a^{κ})_{κ∈} (n).

1Supported by RPBP.III-24.C8

2The proposition (1) was either repeated or obvious.

(7) an_{}· a = an+1_{}.

(8) 1n_{}= 1.

(9) an+m_{}= an_{}· am_{}.

(10) (a · b)n_{}= an_{}· bn_{}.

(11) an·m_{}= (an_{})m_{}.

(12) If 0 ≠ a, then 0 ≠ an_{}.

(13) If 0 < a, then 0 < an_{}.

(14) If a ≠ 0, then \frac{1}{a} n_{}= \frac{1}{an_{}}.

(15) If a ≠ 0, then \frac{b}{a} n_{}= \frac{bn_{}}{an_{}}.

(16) If n ≥ 1, then 0n_{}= 0.

(17) If 0 < a and a ≤ b, then an_{}≤ bn_{}.

(18) If 0 ≤ a and a < b and 1 ≤ n, then an_{}< bn_{}.

(19) If a ≥ 1, then an_{}≥ 1.

(20) If 1 ≤ a and 1 ≤ n, then a ≤ an_{}.

(21) If 1 < a and 2 ≤ n, then a < an_{}.

(22) If 0 < a and a ≤ 1 and 1 ≤ n, then an_{}≤ a.

(23) If 0 < a and a < 1 and 2 ≤ n, then an_{}< a.

(24) If −1 < a, then (1 + a)n_{}≥ 1 + n · a.

(25) If 0 < a and a < 1, then (1 + a)n_{}≤ 1 + 3n_{}· a.

(26) If s_{1} is convergent and for every n holds s_{2}(n) = (s_{1}(n))m_{}, then s_{2} is

convergent and lim s_{2} = (lim s_{1})m_{}. Let us consider n, a. Let us assume that 1 ≤ n. The functor root_{n}(a) yields

a real number and is defined as follows: (Def.3) (root_{n}(a))n_{}= a and root_{n}(a) > 0 if a > 0, root_{n}(a) = 0 if a = 0.

Next we state a number of propositions:

(27) For all a, b, n such that 1 ≤ n holds if a > 0, then b = root_{n}(a) if and only if bn_{}= a and b > 0 but if a = 0, then root_{n}(a) = 0.

(28) If a ≥ 0 and n ≥ 1, then (root_{n}(a))n_{}= a and root_{n}(an_{}) = a.

(29) If n ≥ 1, then root_{n}(1) = 1.

(30) If a ≥ 0, then root_{1}(a) = a.

(31) If a ≥ 0 and b ≥ 0 and n ≥ 1, then root_{n}(a · b) = root_{n}(a) · root_{n}(b).

(32) If a > 0 and n ≥ 1, then root_{n}(\frac{1}{a}) = \frac{1}{root_{n}(a)}.

(33) If a ≥ 0 and b > 0 and n ≥ 1, then root_{n}(\frac{a}{b}) = \frac{root_{n}(a)}{root_{n}(b)}.

(34) If a ≥ 0 and n ≥ 1 and m ≥ 1, then root_{n}(root_{m}(a)) = root_{n·m}(a).

(35) If a ≥ 0 and n ≥ 1 and m ≥ 1, then root_{n}(a)·root_{m}(a) = root_{n·m}(an+m_{}).

(36) If 0 ≤ a and a ≤ b and n ≥ 1, then root_{n}(a) ≤ root_{n}(b).

(37) If a ≥ 0 and a < b and n ≥ 1, then root_{n}(a) < root_{n}(b).

(38) If a ≥ 1 and n ≥ 1, then root_{n}(a) ≥ 1 and a ≥ root_{n}(a).

(39) If 0 ≤ a and a < 1 and n ≥ 1, then a ≤ root_{n}(a) and root_{n}(a) < 1.

(40) If a > 0 and n ≥ 1, then root_{n}(a) − 1 ≤ \frac{a−1}{n}.

(41) If a ≥ 0, then root_{2}(a) = \sqrt{a}.

(42) For every sequence of real numbers s and for every a such that a > 0 and for every n such that n ≥ 1 holds s(n) = root_{n}(a) holds s is convergent and lim s = 1.

Let us consider a, k. Let us assume that a ≠ 0. The functor ak_{}yields a real number and is defined as follows:

(Def.4) ak_{}= a|k|_{}if k ≥ 0, ak_{}= (a|k|_{})^{−1} if k < 0.

We now state a number of propositions:

(43) If a ≠ 0, then if k ≥ 0, then ak_{}= a|k|_{}but if k < 0, then ak_{}= (a|k|_{})^{−1}.

(44) If a ≠ 0, then for every i such that i = 0 holds ai_{}= 1.

(45) If a ≠ 0, then for every i such that i = 1 holds ai_{}= a.

(46) If a ≠ 0 and i = n, then ai_{}= an_{}.

(47) 1k_{}= 1.

(48) If a ≠ 0, then ak_{}≠ 0.

(49) If a > 0, then ak_{}> 0.

(50) If a ≠ 0 and b ≠ 0, then (a · b)k_{}= ak_{}· bk_{}.

(51) If a ≠ 0, then a−k_{}= \frac{1}{ak_{}}.

(52) If a ≠ 0, then \frac{1}{a} k_{}= \frac{1}{ak_{}}.

(53) If a ≠ 0, then am−n_{}= \frac{am_{}}{an_{}}.

(54) If a ≠ 0, then ak+l_{}= ak_{}· al_{}.

(55) If a ≠ 0, then (ak_{})l_{}= ak·l_{}.

(56) If a > 0 and n ≥ 1, then (root_{n}(a))k_{}= root_{n}(ak_{}).

Let us consider a, p. Let us assume that a > 0. The functor ap_{}yielding a real number is defined by: (Def.5) ap_{}= root_{den p}(anum p_{}).

We now state a number of propositions:

(57) If a > 0, then ap_{}= root_{den p}(anum p_{}).

(58) If a > 0 and p = 0, then ap_{}= 1.

(59) If a > 0 and p = 1, then ap_{}= a.

(60) If a > 0 and p = n, then ap_{}= an_{}.

(61) If a > 0 and n ≥ 1 and p = n^{−1}, then ap_{}= root_{n}(a).

(62) 1p_{}= 1.

(63) If a > 0, then ap_{}> 0.

(64) If a > 0, then ap_{}· aq_{}= ap+q_{}.

(65) If a > 0, then \frac{1}{ap_{}}= a−p_{}.

(66) If a > 0, then \frac{ap_{}}{aq_{}}= ap−q_{}.

(67) If a > 0 and b > 0, then (a · b)p_{}= ap_{}· bp_{}.

(68) If a > 0, then \frac{1}{a} p_{}= \frac{1}{ap_{}}.

(69) If a > 0 and b > 0, then \frac{a}{b} p_{}= \frac{ap_{}}{bp_{}}.

(70) If a > 0, then (ap_{})q_{}= ap·q_{}.

(71) If a ≥ 1 and p ≥ 0, then ap_{}≥ 1.

(72) If a ≥ 1 and p ≤ 0, then ap_{}≤ 1.

(73) If a > 1 and p > 0, then ap_{}> 1.

(74) If a ≥ 1 and p ≥ q, then ap_{}≥ aq_{}.

(75) If a > 1 and p > q, then ap_{}> aq_{}.

(76) If a > 0 and a < 1 and p > 0, then ap_{}< 1.

(77) If a > 0 and a ≤ 1 and p ≤ 0, then ap_{}≥ 1.

A sequence of real numbers is called a rational sequence if: (Def.6) for every n holds it(n) is a rational number.

Let s be a rational sequence, and let us consider n. Then s(n) is a rational number.

Next we state two propositions: (79)^{3} For every a there exists a rational sequence s such that s is convergent

and lim s = a and for every n holds s(n) ≤ a.

(80) For every a there exists a rational sequence s such that s is convergent

and lim s = a and for every n holds s(n) ≥ a. Let us consider a, and let s be a rational sequence. Let us assume that a > 0.

The functor as_{}yields a sequence of real numbers and is defined as follows: (Def.7) for every n holds (as_{})(n) = as(n)_{}.

The following propositions are true:

(81) For every a and for every rational sequence s and for every s_{1} such that a > 0 holds s_{1} = as_{}if and only if for every n holds s_{1}(n) = as(n)_{}.

(82) For every rational sequence s and for every a such that s is convergent and a > 0 holds as_{}is convergent.

(83) For all rational sequences s_{1}, s_{2} and for every a such that s_{1} is convergent and s_{2} is convergent and lim s_{1} = lim s_{2} and a > 0 holds as_{1}_{}is convergent and as_{2}_{}is convergent and lim as_{1}_{}= lim as_{2}_{}.

Let us consider a, b. Let us assume that a > 0. The functor ab_{}yielding a real number is defined by: (Def.8) there exists a rational sequence s such that s is convergent and lim s = b

and as_{}is convergent and lim as_{}= ab_{}. We now state a number of propositions:

3The proposition (78) was either repeated or obvious.

(84) For all a, b, c such that a > 0 holds c = ab_{}if and only if there exists a rational sequence s such that s is convergent and lim s = b and as_{}is convergent and lim as_{}= c.

(85) If a > 0, then a0_{}= 1.

(86) If a > 0, then a1_{}= a.

(87) 1a_{}= 1.

(88) If a > 0, then ap_{}= ap_{}.

(89) If a > 0, then ab+c_{}= ab_{}· ac_{}.

(90) If a > 0, then a−c_{}= \frac{1}{ac_{}}.

(91) If a > 0, then ab−c_{}= \frac{ab_{}}{ac_{}}.

(92) If a > 0 and b > 0, then (a · b)c_{}= ac_{}· bc_{}.

(93) If a > 0, then \frac{1}{a} c_{}= \frac{1}{ac_{}}.

(94) If a > 0 and b > 0, then \frac{a}{b} c_{}= \frac{ac_{}}{bc_{}}.

(95) If a > 0, then ab_{}> 0.

(96) If a ≥ 1 and c ≥ b, then ac_{}≥ ab_{}.

(97) If a > 1 and c > b, then ac_{}> ab_{}.

(98) If a > 0 and a ≤ 1 and c ≥ b, then ac_{}≤ ab_{}.

(99) If a ≥ 1 and b ≥ 0, then ab_{}≥ 1.

(100) If a > 1 and b > 0, then ab_{}> 1.

(101) If a ≥ 1 and b ≤ 0, then ab_{}≤ 1.

(102) If a > 1 and b < 0, then ab_{}< 1.

(103) If s_{1} is convergent and s_{2} is convergent and lim s_{1} > 0 and for every n holds s_{1}(n) > 0 and s_{2}(n) = (s_{1}(n))p_{}, then lim s_{2} = (lim s_{1})p_{}.

(104) If a > 0 and s_{1} is convergent and s_{2} is convergent and for every n holds s_{2}(n) = as_{1}(n)_{}, then lim s_{2} = alim s_{1}_{}.

(105) If a > 0, then (ab_{})c_{}= ab·c_{}.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 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] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[5] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.

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

130 Konrad Raczkowski

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

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

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

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

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

[14] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.

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

Homotheties and Shears in Affine Planes ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. We study connections between Major Desargues Axiom and the transitivity of group of homotheties. A formal proof of the theorem which establishes an equivalence of these two properties of affine planes is given. We also study connections between the trapezium version of Major Desargues Axiom and the existence of the shears in affine planes. The article contains investigations on ”Scherungssatz”.

MML Identifier: HOMOTHET.

The papers [9], [1], [2], [10], [3], [4], [6], [7], [5], and [8] provide the terminology and notation for this paper. For simplicity we adopt the following rules: A_{1} will be an affine plane, a, b, o, p, p′, q, q′, x, y will be elements of the points of A_{1}, M, K will be subsets of the points of A_{1}, and f will be a permutation of the points of A_{1}. We now state four propositions:

(1) Suppose that

(i) not L(o, a, p), (ii) L(o, a, b), (iii) L(o, a, x), (iv) L(o, a, y),

(v) L(o, p, p′), (vi) L(o, p, q), (vii) L(o, p, q′), (viii) p ≠ q, (ix) a ≠ x,

(x) o ≠ q,

(xi) o ≠ x,

(xii) a, p ∥ b, p′, (xiii) a, q ∥ b, q′, (xiv) x, p ∥ y, p′, (xv) A_{1} satisfies DES.

1Supported by RPBP.III-24.C2

Then x, q ∥ y, q′.

(2) If for all o, a, b such that o ≠ a and o ≠ b and L(o, a, b) there exists f such that f is a dilatation and f(o) = o and f(a) = b, then A_{1} satisfies DES.

(3) If A_{1} satisfies DES, then for all o, a, b such that o ≠ a and o ≠ b and L(o, a, b) there exists f such that f is a dilatation and f(o) = o and f(a) = b.

(4) A_{1} satisfies DES if and only if for all o, a, b such that o ≠ a and o ≠ b and L(o, a, b) there exists f such that f is a dilatation and f(o) = o and f(a) = b.

Let us consider A_{1}, f, K. We say that f is Sc K if and only if:

(Def.1) f is a collineation and K is a line and for every x such that x ∈ K holds f(x) = x and for every x holds x, f(x) ∥ K.

One can prove the following propositions:

(5) If f is Sc K and f(p) = p and p ∉ K, then f = id_{the points of A_{1}}.

(6) If for all a, b, K such that a, b ∥ K and a ∉ K there exists f such that f is Sc K and f(a) = b, then A_{1} satisfies TDES.

(7) Suppose that

(i) K ∥ M, (ii) p ∈ K, (iii) q ∈ K, (iv) p′ ∈ K,

(v) q′ ∈ K, (vi) A_{1} satisfies TDES, (vii) a ∈ M, (viii) b ∈ M, (ix) x ∈ M,

(x) y ∈ M, (xi) a ≠ b, (xii) q ≠ p, (xiii) p, a ∥ p′, x, (xiv) p, b ∥ p′, y, (xv) q, a ∥ q′, x.

Then q, b ∥ q′, y.

(8) If a, b ∥ K and a ∉ K and A_{1} satisfies TDES, then there exists f such that f is Sc K and f(a) = b.

(9) A_{1} satisfies TDES if and only if for all a, b, K such that a, b ∥ K and a ∉ K there exists f such that f is Sc K and f(a) = b.

Homotheties and Shears in Affine Planes 133

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

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[5] Henryk Oryszczyszyn and Krzysztof Prażmowski. Classical configurations in affine planes. Formalized Mathematics, 1(4):625–633, 1990.

[6] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[7] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[8] Henryk Oryszczyszyn and Krzysztof Prażmowski. Transformations in affine spaces. Formalized Mathematics, 1(4):715–723, 1990.

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

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

134

Directed Geometrical Bundles and Their Analytical Representation ^{1}

Grzegorz Lewandowski Siedlce Agricultural

and Pedagogical University Krzysztof Prażmowski Warsaw University Białystok

Bożena Lewandowska Siedlce Agricultural and Pedagogical University

Summary. We introduce the notion of weak directed geometrical bundle. We prove representation theorems for directed and weak directed geometrical bundles which establish a one-to-one correspondence between such structures and appropriate 2-divisible abelian groups. To this aim we construct over an arbitrary weak directed geometrical bundle a group defined entirely in terms of geometrical notions - the group of (abstract) “free vectors”.

MML Identifier: AFVECT0.

The terminology and notation used here have been introduced in the following articles: [8], [3], [4], [10], [11], [7], [5], [6], [1], [9], and [2]. An affine structure is said to be a weak affine vector space if: (Def.1) (i) there exist elements a, b of the points of it such that a ≠ b,

(ii) for all elements a, b, c of the points of it such that a, b c, c holds

a = b, (iii) for all elements a, b, c, d, p, q of the points of it such that a, b p, q

and c, d p, q holds a, b c, d, (iv) for every elements a, b, c of the points of it there exists an element d

of the points of it such that a, b c, d,

(v) for all elements a, b, c, a′, b′, c′ of the points of it such that a, b a′, b′ and a, c a′, c′ holds b, c b′, c′,

(vi) for every elements a, c of the points of it there exists an element b of the points of it such that a, b b, c, (vii) for all elements a, b, c, d of the points of it such that a, b c, d holds

1Supported by RPBP.III-24.C3

a, c b, d. We see that the space of free vectors is a weak affine vector space. We adopt the following convention: A_{1} will be a weak affine vector space and

a, b, c, d, f, a′, b′, c′, d′, f′, p, q, r, o will be elements of the points of A_{1}. The following propositions are true:

(2)^{2} a, b a, b.

(3) a, a a, a.

(4) If a, b c, d, then c, d a, b.

(5) If a, b a, c, then b = c.

(6) If a, b c, d and a, b c, d′, then d = d′.

(7) For all a, b holds a, a b, b.

(8) If a, b c, d, then b, a d, c.

(9) If a, b c, d and a, c b′, d, then b = b′.

(10) If b, c b′, c′ and a, d b, c and a, d′ b′, c′, then d = d′.

(11) If a, b a′, b′ and c, d b, a and c, d′ b′, a′, then d = d′.

(12) If a, b a′, b′ and c, d c′, d′ and b, f c, d and b′, f′ c′, d′, then a, f a′, f′.

(13) If a, b a′, b′ and a, c c′, b′, then b, c c′, a′.

Let us consider A_{1}, a, b. We say that a, b are in a maximal distance if and only if: (Def.2) a, b b, a and a ≠ b.

One can prove the following propositions: (15)^{3} a, a are not in a maximal distance.

(16) There exist a, b such that a ≠ b and a, b are not in a maximal distance.

(17) If a, b are in a maximal distance, then b, a are in a maximal distance.

(18) If a, b are in a maximal distance and a, c are in a maximal distance, then b = c or b, c are in a maximal distance.

(19) If a, b are in a maximal distance and a, b c, d, then c, d are in a

maximal distance. Let us consider A_{1}, a, b, c. We say that b is a midpoint of a, c if and only if:

(Def.3) a, b b, c.

We now state a number of propositions: (21)^{4} If b is a midpoint of a, c, then b is a midpoint of c, a.

(22) b is a midpoint of a, b if and only if a = b.

(23) b is a midpoint of a, a if and only if a = b or a, b are in a maximal distance.

2The proposition (1) was either repeated or obvious.

3The proposition (14) was either repeated or obvious.

4The proposition (20) was either repeated or obvious.

(24) There exists b such that b is a midpoint of a, c.

(25) If b is a midpoint of a, c and b′ is a midpoint of a, c, then b = b′ or b, b′ are in a maximal distance.

(26) There exists c such that b is a midpoint of a, c.

(27) If b is a midpoint of a, c and b is a midpoint of a, c′, then c = c′.

(28) If b is a midpoint of a, c and b, b′ are in a maximal distance, then b′ is a midpoint of a, c.

(29) If b is a midpoint of a, c and b′ is a midpoint of a, c′ and b, b′ are in a maximal distance, then c = c′.

(30) If p is a midpoint of a, a′ and p is a midpoint of b, b′, then a, b b′, a′.

(31) If p is a midpoint of a, a′ and q is a midpoint of b, b′ and p, q are in a

maximal distance, then a, b b′, a′. Let us consider A_{1}, a, b. The functor PSym(a, b) yields an element of the

points of A_{1} and is defined as follows: (Def.4) a is a midpoint of b, PSym(a, b).

One can prove the following propositions:

(32) PSym(p, a) = b if and only if p is a midpoint of a, b.

(33) PSym(p, a) = b if and only if a, p p, b.

(34) p is a midpoint of a, PSym(p, a).

(35) PSym(p, a) = a if and only if a = p or a, p are in a maximal distance.

(36) PSym(p, PSym(p, a)) = a.

(37) If PSym(p, a) = PSym(p, b), then a = b.

(38) There exists a such that PSym(p, a) = b.

(39) a, b PSym(p, b), PSym(p, a).

(40) a, b c, d if and only if PSym(p, a), PSym(p, b) PSym(p, c), PSym(p, d).

(41) a, b are in a maximal distance if and only if PSym(p, a), PSym(p, b) are in a maximal distance.

(42) b is a midpoint of a, c if and only if PSym(p, b) is a midpoint of PSym(p, a), PSym(p, c).

(43) PSym(p, a) = PSym(q, a) if and only if p = q or p, q are in a maximal distance.

(44) PSym(q, PSym(p, PSym(q, a))) = PSym(PSym(q, p), a).

(45) PSym(p, PSym(q, a)) = PSym(q, PSym(p, a)) if and only if p = q or p, q are in a maximal distance or q, PSym(p, q) are in a maximal distance.

(46) PSym(p, PSym(q, PSym(r, a))) = PSym(r, PSym(q, PSym(p, a))).

(47) There exists d such that PSym(a, PSym(b, PSym(c, p))) = PSym(d, p).

(48) There exists c such that PSym(a, PSym(c, p)) = PSym(c, PSym(b, p)). Let us consider A_{1}, o, a, b. The functor Padd(o, a, b) yielding an element of

the points of A_{1} is defined as follows:

(Def.5) o, a b, Padd(o, a, b).

Next we state the proposition

(49) Padd(o, a, b) = c if and only if o, a b, c.

Let us consider A_{1}, o, a. The functor Pcom(o, a) yielding an element of the points of A_{1} is defined as follows: (Def.6) o is a midpoint of a, Pcom(o, a).

One can prove the following propositions:

(50) Pcom(o, a) = b if and only if o is a midpoint of a, b.

(51) Pcom(o, a) = b if and only if a, o o, b.

Let us consider A_{1}, o. The functor Padd o yielding a binary operation on the points of A_{1} is defined as follows: (Def.7) for all a, b holds (Padd o)(a, b) = Padd(o, a, b).

Let us consider A_{1}, o. The functor Pcom o yielding a unary operation on the points of A_{1} is defined as follows: (Def.8) for every a holds (Pcom o)(a) = Pcom(o, a).

The following propositions are true:

(52) For every binary operation O on the points of A_{1} holds O = Padd o if and only if for all a, b holds O(a, b) = Padd(o, a, b).

(53) For every unary operation O on the points of A_{1} holds O = Pcom o if

and only if for every a holds O(a) = Pcom(o, a). Let us consider A_{1}, o. The functor GroupVect(A_{1}, o) yields a group structure

and is defined by: (Def.9) GroupVect(A_{1}, o) = ⟨ the points of A_{1}, Padd o, Pcom o, o⟩.

The following two propositions are true:

(54) For every X being a group structure holds X = GroupVect(A_{1}, o) if and only if X = ⟨ the points of A_{1}, Padd o, Pcom o, o⟩.

(55) For all A_{1}, o holds the carrier of GroupVect(A_{1}, o) = the points of A_{1} and the addition of GroupVect(A_{1}, o) = Padd o and the reverse-map of GroupVect(A_{1}, o) = Pcom o and the zero of GroupVect(A_{1}, o) = o.

In the sequel a, b, c will denote elements of GroupVect(A_{1}, o). One can prove the following propositions:

(56) For an arbitrary x holds x is an element of the points of A_{1} if and only if x is an element of GroupVect(A_{1}, o).

(57) For all elements a, b of GroupVect(A_{1}, o) and for all elements a′, b′ of the points of A_{1} such that a = a′ and b = b′ holds a + b = (Padd o)(a′, b′).

(58) For every element a of GroupVect(A_{1}, o) and for every element a′ of the points of A_{1} such that a = a′ holds −a = (Pcom o)(a′).

(59) 0_{GroupVect(A_{1},o)} = o.

(60) For every uniquely 2-divisible group A_{2} and for all elements a, b of A_{2} and for all elements a′, b′ of the carrier of A_{2} such that a = a′ and b = b′ holds a + b = a′#b′.

(61) a + b = b + a.

(62) (a + b) + c = a + (b + c).

(63) a + 0_{GroupVect(A_{1},o)} = a.

(64) a + (−a) = 0_{GroupVect(A_{1},o)}.

(65) GroupVect(A_{1}, o) is an Abelian group.

Let us consider A_{1}, o. Then GroupVect(A_{1}, o) is an Abelian group.

In the sequel a, b will be elements of the carrier of GroupVect(A_{1}, o). Next we state the proposition

(66) For every a there exists b such that (the addition of GroupVect(A_{1}, o))(b,

b) = a. Let us consider A_{1}, o. Then GroupVect(A_{1}, o) is a 2-divisible group. In the sequel A_{1} will denote a space of free vectors and o will denote an

element of the points of A_{1}. One can prove the following proposition

(67) For every element a of the carrier of GroupVect(A_{1}, o) such that (the addition of GroupVect(A_{1}, o))(a, a) = 0_{GroupVect(A_{1},o)} holds a = 0_{GroupVect(A_{1},o)}.

Let us consider A_{1}, o. Then GroupVect(A_{1}, o) is a uniquely 2-divisible group. A uniquely 2-divisible group is said to be a proper uniquely two divisible

group if: (Def.10) there exist elements a, b of the carrier of it such that a ≠ b.

The following proposition is true (69)^{5} GroupVect(A_{1}, o) is a proper uniquely two divisible group.

Let us consider A_{1}, o. Then GroupVect(A_{1}, o) is a proper uniquely two divisible group.

Next we state the proposition

(70) For every proper uniquely two divisible group A_{2} holds Vectors(A_{2}) is

a space of free vectors. Let A_{2} be a proper uniquely two divisible group. Then Vectors(A_{2}) is a space

of free vectors.

We now state two propositions:

(71) For every A_{1} and for every element o of the points of A_{1} holds A_{1} = Vectors(GroupVect(A_{1}, o)).

(72) For every A_{3} being an affine structure holds A_{3} is a space of free vectors if and only if there exists a proper uniquely two divisible group A_{2} such that A_{3} = Vectors(A_{2}).

5The proposition (68) was either repeated or obvious.

Let X, Y be group structures, and let f be a function from the carrier of X into the carrier of Y. We say that f is an isomorphism of X and Y if and only if:

(Def.11) f is one-to-one and rng f = the carrier of Y and for all elements a, b of X holds f(a + b) = f(a) + f(b) and f(0_{X}) = 0_{Y} and f(−a) = −f(a).

Let X, Y be group structures. We say that X, Y are isomorph if and only if:

(Def.12) there exists a function f from the carrier of X into the carrier of Y such that f is an isomorphism of X and Y.

In the sequel A_{2} will be a proper uniquely two divisible group and f will be a function from the carrier of A_{2} into the carrier of A_{2}. The following propositions are true: (75)^{6} Let o′ be an element of A_{2}. Let o be an element of the points of

Vectors(A_{2}). Suppose for every element x of A_{2} holds f(x) = o′ + x and o = o′. Then for all elements a, b of A_{2} holds f(a + b) = (Padd o)(f(a), f(b)) and f(0_{A_{2}}) = 0_{GroupVect(Vectors(A_{2}),o)} and f(−a) = (Pcom o)(f(a)).

(76) For every element o′ of A_{2} such that for every element b of A_{2} holds f(b) = o′ + b holds f is one-to-one.

(77) For every element o′ of A_{2} and for every element o of the points of Vectors(A_{2}) such that for every element b of A_{2} holds f(b) = o′ + b and o = o′ holds rng f = the carrier of GroupVect(Vectors(A_{2}), o).

(78) For every proper uniquely two divisible group A_{2} and for every element o′ of A_{2} and for every element o of the points of Vectors(A_{2}) such that o = o′ holds A_{2}, GroupVect(Vectors(A_{2}), o) are isomorph.

References

[1] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537–541, 1990.

[2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

[6] Grzegorz Lewandowski and Krzysztof Prażmowski. A construction of an abstract space of congruence of vectors. Formalized Mathematics, 1(4):685–688, 1990.

[7] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

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

6The propositions (73)–(74) were either repeated or obvious.

Directed Geometrical Bundles and Their … 141

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

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

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

142

Definable Functions

Grzegorz Bancerek Warsow University Białystok

Summary. The article is contituation of [6] and [5]. It deals with concepts of variables occuring in a formula and free variables, replacement of variables in a formula and definable functions. The goal is to create a base of facts which are neccesary to show that every model of ZF set theory is a good model, i.e. it is closed under fundamental settheoretical operations (union, intersection, Cartesian product ect.). The base includes the facts concerning the composition and conditional sum of two definable functions.

MML Identifier: ZFMODEL2.

The notation and terminology used here are introduced in the following articles: [12], [1], [11], [8], [7], [10], [4], [9], [2], [3], [5], and [6]. For simplicity we follow a convention: x, y, z, x_{1}, x_{2}, x_{3}, x_{4} will denote variables, M will denote a nonempty set, i, j will denote natural numbers, m, m_{1}, m_{2}, m_{3}, m_{4} will denote elements of M, H, H_{1}, H_{2} will denote ZF-formulae, and v, v_{1}, v_{2} will denote functions from VAR into M. One can prove the following propositions:

(1) Free(H(\frac{x}{y})) ⊆ (Free H \ \{x\}) ∪ \{y\}.

(2) If y ∉ Var_{H}, then if x ∈ Free H, then Free(H(\frac{x}{y})) = (Free H \\{x\})∪\{y\} but if x ∉ Free H, then Free(H(\frac{x}{y})) = Free H.

(3) Var_{H} is finite.

(4) There exists i such that for every j such that x_{j} ∈ Var_{H} holds j < i and there exists x such that x ∉ Var_{H}.

(5) If x ∉ Var_{H}, then M, v ⊧ H if and only if M, v ⊧ ∀_{x}H.

(6) If x ∉ Var_{H}, then M, v ⊧ H if and only if M, v(\frac{x}{m}) ⊧ H.

(7) Suppose x ≠ y and y ≠ z and z ≠ x. Then ((v(\frac{x}{m_{1}} ))(\frac{y}{m_{2}} ))(\frac{z}{m_{3}} ) = ((v(\frac{z}{m_{3}} ))(\frac{y}{m_{2}} ))(\frac{x}{m_{1}} ) and ((v(\frac{x}{m_{1}} ))(\frac{y}{m_{2}} ))(\frac{z}{m_{3}} ) = ((v(\frac{y}{m_{2}} ))(\frac{z}{m_{3}} ))(\frac{x}{m_{1}} ).

(8) Suppose x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4} and x_{3} ≠ x_{4}. Then (i) (((v(\frac{x_{1}}{m_{1}} ))(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{4}}{m_{4}} ) = (((v(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{4}}{m_{4}} ))(\frac{x_{1}}{m_{1}} ), (ii) (((v(\frac{x_{1}}{m_{1}} ))(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{4}}{m_{4}} ) = (((v(\frac{x_{3}}{m_{3}} ))(\frac{x_{4}}{m_{4}} ))(\frac{x_{1}}{m_{1}} ))(\frac{x_{2}}{m_{2}} ), (iii) (((v(\frac{x_{1}}{m_{1}} ))(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{4}}{m_{4}} ) = (((v(\frac{x_{4}}{m_{4}} ))(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{1}}{m_{1}} ).

(9) (i) ((v(\frac{x_{1}}{m_{1}} ))(\frac{x_{2}}{m_{2}} ))(\frac{x_{1}}{m}) = (v(\frac{x_{2}}{m_{2}} ))(\frac{x_{1}}{m}), (ii) (((v(\frac{x_{1}}{m_{1}} ))(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{1}}{m}) = ((v(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{1}}{m}), (iii) ((((v(\frac{x_{1}}{m_{1}} ))(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{4}}{m_{4}} ))(\frac{x_{1}}{m}) = (((v(\frac{x_{2}}{m_{2}} ))(\frac{x_{3}}{m_{3}} ))(\frac{x_{4}}{m_{4}} ))(\frac{x_{1}}{m}).

(10) If x ∉ Free H, then M, v ⊧ H if and only if M, v(\frac{x}{m}) ⊧ H.

(11) Suppose x_{0} ∉ Free H and M, v ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H ⇔ x_{4}=x_{0})). Then for all m_{1}, m_{2} holds f_{H}[v](m_{1}) = m_{2} if and only if M, (v(\frac{x_{3}}{m_{1}} ))(\frac{x_{4}}{m_{2}} ) ⊧ H.

(12) If Free H ⊆ {x_{3}, x_{4}} and M ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H ⇔ x_{4}=x_{0})), then f_{H}[v] = f_{H}[M].

(13) If x ∉ Var_{H}, then M, v ⊧ H(\frac{y}{x}) if and only if M, v(\frac{y}{v(x)}) ⊧ H.

(14) If x ∉ Var_{H} and M, v ⊧ H, then M, v(\frac{x}{v(y)}) ⊧ H(\frac{y}{x}).

(15) Suppose that (i) x_{0} ∉ Free H, (ii) M, v ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H ⇔ x_{4}=x_{0})), (iii) x ∉ Var_{H}, (iv) y ≠ x_{3},

(v) y ≠ x_{4}, (vi) y ∉ Free H, (vii) x ≠ x_{0}, (viii) x ≠ x_{3}, (ix) x ≠ x_{4}.

Then

(x) x_{0} ∉ Free(H(\frac{y}{x})), (xi) M, v(\frac{x}{v(y)}) ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}(H(\frac{y}{x})) ⇔ x_{4}=x_{0})), (xii) f_{H}[v] = f_{H( \frac{y}{x})}[v(\frac{x}{v(y)})].

(16) If x ∉ Var_{H}, then M ⊧ H(\frac{y}{x}) if and only if M ⊧ H.

(17) Suppose x_{0} ∉ Free H_{1} and M, v_{1} ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H_{1} ⇔ x_{4}=x_{0})). Then there exist H_{2}, v_{2} such that for every j such that j < i and x_{j} ∈ Var_{H_{2}} holds j = 3 or j = 4 and x_{0} ∉ Free H_{2} and M, v_{2} ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H_{2} ⇔ x_{4}=x_{0})) and f_{H_{1}}[v_{1}] = f_{H_{2}}[v_{2}].

(18) Suppose x_{0} ∉ Free H_{1} and M, v_{1} ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H_{1} ⇔ x_{4}=x_{0})). Then there exist H_{2}, v_{2} such that Free H_{1}∩Free H_{2} ⊆ {x_{3}, x_{4}} and x_{0} ∉ Free H_{2} and M, v_{2} ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H_{2} ⇔ x_{4}=x_{0})) and f_{H_{1}}[v_{1}] = f_{H_{2}}[v_{2}].

In the sequel F, G are functions. One can prove the following propositions:

(19) If F is definable in M and G is definable in M, then F · G is definable in M.

(20) If x_{0} ∉ Free H, then M, v ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H ⇔ x_{4}=x_{0})) if and only if for every m_{1} there exists m_{2} such that for every m_{3} holds M, (v(\frac{x_{3}}{m_{1}} ))(\frac{x_{4}}{m_{3}} ) ⊧ H if and only if m_{3} = m_{2}.

Definable Functions 145

(21) Suppose F is definable in M and G is definable in M and Free H ⊆ {x_{3}}. Let F_{1} be a function. Then if dom F_{1} = M and for every v holds if M, v ⊧ H, then F_{1}(v(x_{3})) = F (v(x_{3})) but if M, v ⊧ ¬H, then F_{1}(v(x_{3})) = G(v(x_{3})), then F_{1} is definable in M.

(22) If F is parametrically definable in M and G is parametrically definable in M, then G · F is parametrically definable in M.

(23) Suppose that (i) {x_{0}, x_{1}, x_{2}} misses Free H_{1}, (ii) M, v ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H_{1} ⇔ x_{4}=x_{0})), (iii) {x_{0}, x_{1}, x_{2}} misses Free H_{2}, (iv) M, v ⊧ ∀_{x_{3}}(∃_{x_{0}}(∀_{x_{4}}H_{2} ⇔ x_{4}=x_{0})),

(v) {x_{0}, x_{1}, x_{2}} misses Free H, (vi) x_{4} ∉ Free H.

Let F_{1} be a function. Then if dom F_{1} = M and for every m holds if M, v(\frac{x_{3}}{m}) ⊧ H, then F_{1}(m) = f_{H_{1}}[v](m) but if M, v(\frac{x_{3}}{m}) ⊧ ¬H, then F_{1}(m) = f_{H_{2}}[v](m), then F_{1} is parametrically definable in M.

(24) id_{M} is definable in M.

(25) id_{M} is parametrically definable in M.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Grzegorz Bancerek. A model of ZF set theory language. Formalized Mathematics, 1(1):131–145, 1990.

[3] Grzegorz Bancerek. Models and satisfiability. Formalized Mathematics, 1(1):191–199, 1990.

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

[5] Grzegorz Bancerek. Properties of ZF models. Formalized Mathematics, 1(2):277–280, 1990.

[6] Grzegorz Bancerek. Replacing of variables in formulas of ZF theory. Formalized Mathematics, 1(5):963–972, 1990.

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

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

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

[10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[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.

146

Propositional Calculus ^{1}

Grzegorz Bancerek Warsaw University

Białystok Agata Darmochwał Warsaw University Białystok Andrzej Trybulec Warsaw University Białystok

Summary. We develop the classical propositional calculus, following [3].

MML Identifier: LUKASI 1.

The notation and terminology used here are introduced in the articles [1] and [2]. We follow the rules: p, q, r, s are elements of CQC−WFF and X is a subset of CQC−WFF. We now state a number of propositions:

(1) (p ⇒ q) ⇒ ((q ⇒ r) ⇒ (p ⇒ r)) ∈ Taut.

(2) If p ⇒ q ∈ Taut, then (q ⇒ r) ⇒ (p ⇒ r) ∈ Taut.

(3) If p ⇒ q ∈ Taut and q ⇒ r ∈ Taut, then p ⇒ r ∈ Taut.

(4) p ⇒ p ∈ Taut.

(5) q ⇒ (p ⇒ q) ∈ Taut.

(6) ((p ⇒ q) ⇒ r) ⇒ (q ⇒ r) ∈ Taut.

(7) q ⇒ ((q ⇒ p) ⇒ p) ∈ Taut.

(8) (s ⇒ (q ⇒ p)) ⇒ (q ⇒ (s ⇒ p)) ∈ Taut.

(9) (q ⇒ r) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ Taut.

(10) (q ⇒ (q ⇒ r)) ⇒ (q ⇒ r) ∈ Taut.

(11) (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)) ∈ Taut.

(12) ¬ VERUM ⇒ p ∈ Taut.

(13) If q ∈ Taut, then p ⇒ q ∈ Taut.

(14) If p ∈ Taut, then (p ⇒ q) ⇒ q ∈ Taut.

(15) If s ⇒ (q ⇒ p) ∈ Taut, then q ⇒ (s ⇒ p) ∈ Taut.

(16) If s ⇒ (q ⇒ p) ∈ Taut and q ∈ Taut, then s ⇒ p ∈ Taut.

(17) If s ⇒ (q ⇒ p) ∈ Taut and q ∈ Taut and s ∈ Taut, then p ∈ Taut.

1Supported by RPBP.III-24.C1

(18) If q ⇒ (q ⇒ r) ∈ Taut, then q ⇒ r ∈ Taut.

(19) If p ⇒ (q ⇒ r) ∈ Taut, then (p ⇒ q) ⇒ (p ⇒ r) ∈ Taut.

(20) If p ⇒ (q ⇒ r) ∈ Taut and p ⇒ q ∈ Taut, then p ⇒ r ∈ Taut.

(21) If p ⇒ (q ⇒ r) ∈ Taut and p ⇒ q ∈ Taut and p ∈ Taut, then r ∈ Taut.

(22) If p ⇒ (q ⇒ r) ∈ Taut and p ⇒ (r ⇒ s) ∈ Taut, then p ⇒ (q ⇒ s) ∈ Taut.

(23) p ⇒ VERUM ∈ Taut.

(24) (¬p ⇒ ¬q) ⇒ (q ⇒ p) ∈ Taut.

(25) ¬(¬p) ⇒ p ∈ Taut.

(26) (p ⇒ q) ⇒ (¬q ⇒ ¬p) ∈ Taut.

(27) p ⇒ ¬(¬p) ∈ Taut.

(28) (¬(¬p) ⇒ q) ⇒ (p ⇒ q) ∈ Taut and (p ⇒ q) ⇒ (¬(¬p) ⇒ q) ∈ Taut.

(29) (p ⇒ ¬(¬q)) ⇒ (p ⇒ q) ∈ Taut and (p ⇒ q) ⇒ (p ⇒ ¬(¬q)) ∈ Taut.

(30) (p ⇒ ¬q) ⇒ (q ⇒ ¬p) ∈ Taut.

(31) (¬p ⇒ q) ⇒ (¬q ⇒ p) ∈ Taut.

We now state a number of propositions:

(32) (p ⇒ ¬p) ⇒ ¬p ∈ Taut.

(33) ¬p ⇒ (p ⇒ q) ∈ Taut.

(34) p ⇒ q ∈ Taut if and only if ¬q ⇒ ¬p ∈ Taut.

(35) If ¬p ⇒ ¬q ∈ Taut, then q ⇒ p ∈ Taut.

(36) p ∈ Taut if and only if ¬(¬p) ∈ Taut.

(37) p ⇒ q ∈ Taut if and only if p ⇒ ¬(¬q) ∈ Taut.

(38) p ⇒ q ∈ Taut if and only if ¬(¬p) ⇒ q ∈ Taut.

(39) If p ⇒ ¬q ∈ Taut, then q ⇒ ¬p ∈ Taut.

(40) If ¬p ⇒ q ∈ Taut, then ¬q ⇒ p ∈ Taut.

(41) ⊢ (p ⇒ q) ⇒ ((q ⇒ r) ⇒ (p ⇒ r)).

(42) If ⊢ p ⇒ q, then ⊢ (q ⇒ r) ⇒ (p ⇒ r).

(43) If ⊢ p ⇒ q and ⊢ q ⇒ r, then ⊢ p ⇒ r.

(44) ⊢ p ⇒ p.

(45) ⊢ p ⇒ (q ⇒ p).

(46) If ⊢ p, then ⊢ q ⇒ p.

(47) ⊢ (s ⇒ (q ⇒ p)) ⇒ (q ⇒ (s ⇒ p)).

(48) If ⊢ p ⇒ (q ⇒ r), then ⊢ q ⇒ (p ⇒ r).

(49) If ⊢ p ⇒ (q ⇒ r) and ⊢ q, then ⊢ p ⇒ r.

(50) ⊢ p ⇒ VERUM and ⊢ ¬ VERUM ⇒ p.

(51) ⊢ p ⇒ ((p ⇒ q) ⇒ q).

(52) ⊢ (q ⇒ (q ⇒ r)) ⇒ (q ⇒ r).

(53) If ⊢ q ⇒ (q ⇒ r), then ⊢ q ⇒ r.

(54) ⊢ (p ⇒ (q ⇒ r)) ⇒ ((p ⇒ q) ⇒ (p ⇒ r)).

(55) If ⊢ p ⇒ (q ⇒ r), then ⊢ (p ⇒ q) ⇒ (p ⇒ r).

(56) If ⊢ p ⇒ (q ⇒ r) and ⊢ p ⇒ q, then ⊢ p ⇒ r.

(57) ⊢ ((p ⇒ q) ⇒ r) ⇒ (q ⇒ r).

(58) If ⊢ (p ⇒ q) ⇒ r, then ⊢ q ⇒ r.

(59) ⊢ (p ⇒ q) ⇒ ((r ⇒ p) ⇒ (r ⇒ q)).

(60) If ⊢ p ⇒ q, then ⊢ (r ⇒ p) ⇒ (r ⇒ q).

(61) ⊢ (p ⇒ q) ⇒ (¬q ⇒ ¬p).

(62) ⊢ (¬p ⇒ ¬q) ⇒ (q ⇒ p).

The following propositions are true:

(63) ⊢ ¬p ⇒ ¬q if and only if ⊢ q ⇒ p.

(64) ⊢ p ⇒ ¬(¬p).

(65) ⊢ ¬(¬p) ⇒ p.

(66) ⊢ ¬(¬p) if and only if ⊢ p.

(67) ⊢ (¬(¬p) ⇒ q) ⇒ (p ⇒ q).

(68) ⊢ ¬(¬p) ⇒ q if and only if ⊢ p ⇒ q.

(69) ⊢ (p ⇒ ¬(¬q)) ⇒ (p ⇒ q).

(70) ⊢ p ⇒ ¬(¬q) if and only if ⊢ p ⇒ q.

(71) ⊢ (p ⇒ ¬q) ⇒ (q ⇒ ¬p).

(72) If ⊢ p ⇒ ¬q, then ⊢ q ⇒ ¬p.

(73) ⊢ (¬p ⇒ q) ⇒ (¬q ⇒ p).

(74) If ⊢ ¬p ⇒ q, then ⊢ ¬q ⇒ p.

(75) If X ⊢ p ⇒ q, then X ⊢ (q ⇒ r) ⇒ (p ⇒ r).

(76) If X ⊢ p ⇒ q and X ⊢ q ⇒ r, then X ⊢ p ⇒ r.

(77) X ⊢ p ⇒ p.

(78) If X ⊢ p, then X ⊢ q ⇒ p.

(79) If X ⊢ p, then X ⊢ (p ⇒ q) ⇒ q.

(80) If X ⊢ p ⇒ (q ⇒ r), then X ⊢ q ⇒ (p ⇒ r).

(81) If X ⊢ p ⇒ (q ⇒ r) and X ⊢ q, then X ⊢ p ⇒ r.

(82) If X ⊢ p ⇒ (p ⇒ q), then X ⊢ p ⇒ q.

(83) If X ⊢ (p ⇒ q) ⇒ r, then X ⊢ q ⇒ r.

(84) If X ⊢ p ⇒ (q ⇒ r), then X ⊢ (p ⇒ q) ⇒ (p ⇒ r).

(85) If X ⊢ p ⇒ (q ⇒ r) and X ⊢ p ⇒ q, then X ⊢ p ⇒ r.

(86) X ⊢ ¬p ⇒ ¬q if and only if X ⊢ q ⇒ p.

(87) X ⊢ ¬(¬p) if and only if X ⊢ p.

(88) X ⊢ p ⇒ ¬(¬q) if and only if X ⊢ p ⇒ q.

(89) X ⊢ ¬(¬p) ⇒ q if and only if X ⊢ p ⇒ q.

(90) If X ⊢ p ⇒ ¬q, then X ⊢ q ⇒ ¬p.

(91) If X ⊢ ¬p ⇒ q, then X ⊢ ¬q ⇒ p.

(92) If X ⊢ p ⇒ ¬q and X ⊢ q, then X ⊢ ¬p.

150 Grzegorz Bancerek et al.

(93) If X ⊢ ¬p ⇒ q and X ⊢ ¬q, then X ⊢ p.

References

[1] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

[2] Agata Darmochwał. A first–order predicate calculus. Formalized Mathematics, 1(4):689– 695, 1990.

[3] Jan Łukasiewicz. Elementy logiki matematycznej. PWN, Warszawa, 1958.

Complex Spaces ^{1}

Czesław Byliński Warsaw University Białystok Andrzej Trybulec Warsaw University Białystok

Summary. We introduce the concept of n-dimensional complex space. We prove a number of simple but useful theorems concerning addition, multiplication by scalars and similar basic concepts. We introduce metric and topology. We prove that an n-dimensional complex space is a Hausdorf space and that it is regular.

MML Identifier: COMPLSP1.

The articles [20], [16], [12], [1], [21], [5], [22], [7], [8], [3], [17], [11], [2], [18], [19], [6], [4], [9], [10], [15], [14], and [13] provide the notation and terminology for this paper. We follow the rules: k, n will be natural numbers, r, r′, r_{1} will be real numbers, and c, c′, c_{1}, c_{2} will be elements of . In this article we present several logical schemes. The scheme FuncDefUniq concerns a non-empty set A, a non-empty set B, and a unary functor F yielding an element of B and states that:

for all functions f_{1}, f_{2} from A into B such that for every element x of A holds f_{1}(x) = F(x) and for every element x of A holds f_{2}(x) = F(x) holds f_{1} = f_{2} for all values of the parameters.

The scheme UnOpDefuniq deals with a non-empty set A and a unary functor F yielding an element of A and states that:

for all unary operations u_{1}, u_{2} on A such that for every element x of A holds u_{1}(x) = F(x) and for every element x of A holds u_{2}(x) = F(x) holds u_{1} = u_{2} for all values of the parameters.

The scheme BinOpDefuniq deals with a non-empty set A and a binary functor F yielding an element of A and states that:

for all binary operations o_{1}, o_{2} on A such that for all elements a, b of A holds o_{1}(a, b) = F(a, b) and for all elements a, b of A holds o_{2}(a, b) = F(a, b) holds o_{1} = o_{2} for all values of the parameters.

1Supported by RPBP.III-24.C1

The binary operation +_{}on is defined as follows: (Def.1) for all c_{1}, c_{2} holds +_{}(c_{1}, c_{2}) = c_{1} + c_{2}.

The following propositions are true:

(1) +_{}is commutative.

(2) +_{}is associative.

(3) 0_{}is a unity w.r.t. +_{}.

(4) 1_{+_{}}= 0_{}.

(5) +_{}has a unity. The unary operation −_{}on is defined as follows:

(Def.2) for every c holds −_{}(c) = −c.

Next we state three propositions:

(6) −_{}is an inverse operation w.r.t. +_{}.

(7) +_{}has an inverse operation.

(8) The inverse operation w.r.t. +_{}= −_{}. The binary operation −_{}on is defined by:

(Def.3) −_{}= +_{}∘ (id_{}, −_{}).

The following proposition is true

(9) −_{}(c_{1}, c_{2}) = c_{1} − c_{2}. The binary operation ·_{}on is defined by:

(Def.4) for all c_{1}, c_{2} holds ·_{}(c_{1}, c_{2}) = c_{1} · c_{2}.

The following propositions are true:

(10) ·_{}is commutative.

(11) ·_{}is associative.

(12) 1_{}is a unity w.r.t. ·_{}.

(13) 1_{·_{}}= 1_{}.

(14) ·_{}has a unity.

(15) ·_{}is distributive w.r.t. +_{}. Let us consider c. The functor ·c_{}yields a unary operation on and is defined

by:

(Def.5) ·c_{}= ·∘_{}(c, id_{}).

We now state two propositions:

(16) ·c_{}(c′) = c · c′.

(17) ·c_{}is distributive w.r.t. +_{}.

The function | · |_{}from into is defined by: (Def.6) for every c holds | · |_{}(c) = |c|.

In the sequel z, z_{1}, z_{2} will be finite sequences of elements of . We now define two new functors. Let us consider z_{1}, z_{2}. The functor z_{1} + z_{2} yields a finite sequence of elements of and is defined by:

(Def.7) z_{1} + z_{2} = +∘_{}(z_{1}, z_{2}).

The functor z_{1} − z_{2} yielding a finite sequence of elements of is defined as follows:

(Def.8) z_{1} − z_{2} = −∘_{}(z_{1}, z_{2}).

Let us consider z. The functor −z yielding a finite sequence of elements of is defined by: (Def.9) −z = −_{}· z.

Let us consider c, z. The functor c · z yielding a finite sequence of elements of is defined by:

(Def.10) c · z = ·c_{}·z.

Let us consider z. The functor |z| yields a finite sequence of elements of and is defined as follows:

(Def.11) |z| = | · |_{}· z.

Let us consider n. The functor ^{n} yielding a non-empty set of finite sequences of is defined by:

(Def.12) ^{n} = ^{n}.

We follow a convention: x, z, z_{1}, z_{2}, z_{3} will denote elements of ^{n} and A, B will denote subsets of ^{n}. One can prove the following propositions:

(18) len z = n.

(19) For every element z of ^{0} holds z = ε_{}.

(20) ε_{}is an element of ^{0}.

(21) If k ∈ Seg n, then z(k) ∈ .

(22) If k ∈ Seg n, then z(k) is an element of .

(23) If for every k such that k ∈ Seg n holds z_{1}(k) = z_{2}(k), then z_{1} = z_{2}. Let us consider n, z_{1}, z_{2}. Then z_{1} + z_{2} is an element of ^{n}. Next we state three propositions:

(24) If k ∈ Seg n and c_{1} = z_{1}(k) and c_{2} = z_{2}(k), then (z_{1} + z_{2})(k) = c_{1} + c_{2}.

(25) z_{1} + z_{2} = z_{2} + z_{1}.

(26) z_{1} + (z_{2} + z_{3}) = (z_{1} + z_{2}) + z_{3}.

Let us consider n. The functor 0n_{}yielding a finite sequence of elements of is defined by:

(Def.13) 0n_{}= n ⟼ 0_{}.

Let us consider n. Then 0n_{}is an element of ^{n}. Next we state two propositions:

(27) If k ∈ Seg n, then 0n_{}(k) = 0_{}.

(28) z + 0n_{}= z and z = 0n_{}+ z.

Let us consider n, z. Then −z is an element of ^{n}. Next we state several propositions:

(29) If k ∈ Seg n and c = z(k), then (−z)(k) = −c.

(30) z + (−z) = 0n_{}and (−z) + z = 0n_{}.

(31) If z_{1} + z_{2} = 0n_{}, then z_{1} = −z_{2} and z_{2} = −z_{1}.

(32) −(−z) = z.

(33) If −z_{1} = −z_{2}, then z_{1} = z_{2}.

(34) If z_{1} + z = z_{2} + z or z_{1} + z = z + z_{2}, then z_{1} = z_{2}.

(35) −(z_{1} + z_{2}) = (−z_{1}) + (−z_{2}).

Let us consider n, z_{1}, z_{2}. Then z_{1} − z_{2} is an element of ^{n}.

Next we state a number of propositions:

(36) If k ∈ Seg n and c_{1} = z_{1}(k) and c_{2} = z_{2}(k), then (z_{1} − z_{2})(k) = c_{1} − c_{2}.

(37) z_{1} − z_{2} = z_{1} + (−z_{2}).

(38) z − 0n_{}= z.

(39) 0n_{}− z = −z.

(40) z_{1} − (−z_{2}) = z_{1} + z_{2}.

(41) −(z_{1} − z_{2}) = z_{2} − z_{1}.

(42) −(z_{1} − z_{2}) = (−z_{1}) + z_{2}.

(43) z − z = 0n_{}.

(44) If z_{1} − z_{2} = 0n_{}, then z_{1} = z_{2}.

(45) (z_{1} − z_{2}) − z_{3} = z_{1} − (z_{2} + z_{3}).

(46) z_{1} + (z_{2} − z_{3}) = (z_{1} + z_{2}) − z_{3}.

(47) z_{1} − (z_{2} − z_{3}) = (z_{1} − z_{2}) + z_{3}.

(48) (z_{1} − z_{2}) + z_{3} = (z_{1} + z_{3}) − z_{2}.

(49) z_{1} = (z_{1} + z) − z.

(50) z_{1} + (z_{2} − z_{1}) = z_{2}.

(51) z_{1} = (z_{1} − z) + z.

Let us consider n, c, z. Then c · z is an element of ^{n}.

One can prove the following propositions:

(52) If k ∈ Seg n and c′ = z(k), then (c · z)(k) = c · c′.

(53) c_{1} · (c_{2} · z) = (c_{1} · c_{2}) · z.

(54) (c_{1} + c_{2}) · z = c_{1} · z + c_{2} · z.

(55) c · (z_{1} + z_{2}) = c · z_{1} + c · z_{2}.

(56) 1_{}· z = z.

(57) 0_{}· z = 0n_{}.

(58) (−1_{}) · z = −z.

Let us consider n, z. Then |z| is an element of ^{n}.

Next we state four propositions:

(59) If k ∈ Seg n and c = z(k), then |z|(k) = |c|.

(60) |0n_{}| = n ⟼ 0.

(61) |−z| = |z|.

(62) |c · z| = |c| · |z|.

Let z be a finite sequence of elements of . The functor |z| yields a real number and is defined by:

(Def.14) |z| = \sqrt{∑(^{2}|z|)}.

One can prove the following propositions:

(63) |0n_{}| = 0.

(64) If |z| = 0, then z = 0n_{}.

(65) 0 ≤ |z|.

(66) | − z| = |z|.

(67) |c · z| = |c| · |z|.

(68) |z_{1} + z_{2}| ≤ |z_{1}| + |z_{2}|.

(69) |z_{1} − z_{2}| ≤ |z_{1}| + |z_{2}|.

(70) |z_{1}| − |z_{2}| ≤ |z_{1} + z_{2}|.

(71) |z_{1}| − |z_{2}| ≤ |z_{1} − z_{2}|.

(72) |z_{1} − z_{2}| = 0 if and only if z_{1} = z_{2}.

(73) If z_{1} ≠ z_{2}, then 0 < |z_{1} − z_{2}|.

(74) |z_{1} − z_{2}| = |z_{2} − z_{1}|.

(75) |z_{1} − z_{2}| ≤ |z_{1} − z| + |z − z_{2}|.

Let us consider n, and let A be an element of 2^{^{n}} . We say that A is open if and only if:

(Def.15) for every x such that x ∈ A there exists r such that 0 < r and for every

z such that |z| < r holds x + z ∈ A. Let us consider n, and let A be an element of 2^{^{n}} . We say that A is closed

if and only if: (Def.16) for every x such that for every r such that r > 0 there exists z such

that |z| < r and x + z ∈ A holds x ∈ A. We now state four propositions:

(76) For every element A of 2^{^{n}} such that A = ∅ holds A is open.

(77) For every element A of 2^{^{n}} such that A = ^{n} holds A is open.

(78) For every family A_{1} of subsets of ^{n} such that for every element A of 2^{^{n}} such that A ∈ A_{1} holds A is open for every element A of 2^{^{n}} such that A = ⋃ A_{1} holds A is open.

(79) For all subsets A, B of ^{n} such that A is open and B is open for every

element C of 2^{^{n}} such that C = A ∩ B holds C is open. Let us consider n, x, r. The functor Ball(x, r) yielding a subset of ^{n} is

defined by: (Def.17) Ball(x, r) = {z : |z − x| < r}.

The following three propositions are true:

(80) z ∈ Ball(x, r) if and only if |x − z| < r.

(81) If 0 < r, then x ∈ Ball(x, r).

(82) Ball(z_{1}, r_{1}) is open.

Now we present two schemes. The scheme SubsetFD deals with a non-empty set A, a non-empty set B, a unary functor F yielding an element of B, and a unary predicate P, and states that:

{F(x) : P[x]}, where x is an element of A, is a subset of B for all values of the parameters.

The scheme SubsetFD2 deals with a non-empty set A, a non-empty set B, a non-empty set C, a binary functor F yielding an element of C, and a binary predicate P, and states that:

{F(x, y) : P[x, y]}, where x is an element of A, and y is an element of B, is a subset of C for all values of the parameters.

Let us consider n, x, A. The functor ρ(x, A) yielding a real number is defined by:

(Def.18) for every X being sets of real numbers such that X = {|x − z| : z ∈ A} holds ρ(x, A) = inf X.

Let us consider n, A, r. The functor Ball(A, r) yields a subset of ^{n} and is defined as follows:

(Def.19) Ball(A, r) = {z : ρ(z, A) < r}.

Next we state a number of propositions:

(83) If for every r′ such that r′ > 0 holds r + r′ > r_{1}, then r ≥ r_{1}.

(84) For every X being sets of real numbers and for every r such that X ≠ ∅ and for every r′ such that r′ ∈ X holds r ≤ r′ holds inf X ≥ r.

(85) If A ≠ ∅, then ρ(x, A) ≥ 0.

(86) If A ≠ ∅, then ρ(x + z, A) ≤ ρ(x, A) + |z|.

(87) If x ∈ A, then ρ(x, A) = 0.

(88) If x ∉ A and A ≠ ∅ and A is closed, then ρ(x, A) > 0.

(89) If A ≠ ∅, then |z_{1} − x| + ρ(x, A) ≥ ρ(z_{1}, A).

(90) z ∈ Ball(A, r) if and only if ρ(z, A) < r.

(91) If 0 < r and x ∈ A, then x ∈ Ball(A, r).

(92) If 0 < r, then A ⊆ Ball(A, r).

(93) If A ≠ ∅, then Ball(A, r_{1}) is open.

Let us consider n, A, B. The functor ρ(A, B) yields a real number and is defined as follows:

(Def.20) for every X being sets of real numbers such that X = {|x − z| : x ∈ A ∧ z ∈ B} holds ρ(A, B) = inf X.

Let X, Y be sets of real numbers. The functor X + Y yields sets of real numbers and is defined as follows:

(Def.21) X + Y = {r + r_{1} : r ∈ X ∧ r_{1} ∈ Y }.

Next we state several propositions:

(94) For all X, Y being sets of real numbers such that X ≠ ∅ and Y ≠ ∅ holds X + Y ≠ ∅.

(95) For all X, Y being sets of real numbers such that X ≠ ∅ and X is lower bounded and Y ≠ ∅ and Y is lower bounded holds X + Y is lower bounded.

(96) For all X, Y being sets of real numbers such that X ≠ ∅ and X is lower bounded and Y ≠ ∅ and Y is lower bounded holds inf(X + Y) = inf X + inf Y.

(97) For all X, Y being sets of real numbers such that Y is lower bounded and X ≠ ∅ and for every r such that r ∈ X there exists r_{1} such that r_{1} ∈ Y and r_{1} ≤ r holds inf X ≥ inf Y.

(98) If A ≠ ∅ and B ≠ ∅, then ρ(A, B) ≥ 0.

(99) ρ(A, B) = ρ(B, A).

(100) If A ≠ ∅ and B ≠ ∅, then ρ(x, A) + ρ(x, B) ≥ ρ(A, B).

(101) If A ∩ B ≠ ∅, then ρ(A, B) = 0.

Let us consider n. The open subsets of ^{n} constitute a family of subsets of n defined by:

(Def.22) the open subsets of ^{n} = {A : A is open }, where A is an element of 2^{^{n}} .

The following proposition is true

(102) For every element A of 2^{^{n}} holds A ∈ the open subsets of ^{n} if and only

if A is open. Let us consider n. The n -dimensional complex space is a topological space

defined by: (Def.23) the n -dimensional complex space = ⟨ ^{n}, the open subsets of ^{n}⟩.

We now state two propositions:

(103) The topology of the n -dimensional complex space = the open subsets of ^{n}.

(104) The carrier of the n -dimensional complex space = ^{n}.

In the sequel p denotes a point of the n -dimensional complex space and V denotes a subset of the n -dimensional complex space. Next we state several propositions:

(105) p is an element of ^{n}.

(106) V is a subset of ^{n}.

(107) For every subset A of ^{n} holds A is a subset of the n -dimensional complex space.

(108) For every subset A of ^{n} such that A = V holds A is open if and only if V is open.

(109) For every subset A of ^{n} holds A is closed if and only if A^{c} is open.

(110) For every subset A of ^{n} such that A = V holds A is closed if and only if V is closed.

(111) The n -dimensional complex space is a T_{2} space.

158 Czesław Byliński and Andrzej Trybulec

(112) The n -dimensional complex space is a T_{3} space.

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. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.

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

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

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

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

[9] Czesław Byliński. Semigroup operations on finite subsets. Formalized Mathematics, 1(4):651–656, 1990.

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

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

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

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

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

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

[16] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

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

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

[19] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

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

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

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

Several Properties of Fields. Field Theory

Józef Białas University of Łódź

Summary. The article includes a continuation of the paper [2]. Some simple theorems concerning basic properties of a field are proved.

MML Identifier: REALSET3.

The articles [8], [7], [5], [6], [3], [1], [2], and [4] provide the terminology and notation for this paper. The following propositions are true:

(1) For every field F holds −_{F} (0_{F}) = 0_{F}.

(2) For every field F holds {−1 \atop F}(1_{F}) = 1_{F}.

(3) For every field F and for all elements a, b of the support of F holds −_{F} (+_{F} (⟨⟨a, −_{F} (b)⟩⟩)) = +_{F} (⟨⟨b, −_{F} (a)⟩⟩).

(4) For every field F and for all elements a, b of the support of F \single(0_{F}) holds {−1 \atop F}(·_{F} (⟨⟨a, {−1 \atop F}(b)⟩⟩)) = ·_{F} (⟨⟨b, {−1 \atop F}(a)⟩⟩).

(5) For every field F and for all elements a, b of the support of F holds −_{F} (+_{F} (⟨⟨a, b⟩⟩)) = +_{F} (⟨⟨−_{F} (a), −_{F} (b)⟩⟩).

(6) For every field F and for all elements a, b of the support of F \single(0_{F}) holds {−1 \atop F}(·_{F} (⟨⟨a, b⟩⟩)) = ·_{F} (⟨⟨{−1 \atop F}(a), {−1 \atop F}(b)⟩⟩).

(7) For every field F and for all elements a, b, c, d of the support of F holds +_{F} (⟨⟨a, −_{F} (b)⟩⟩) = +_{F} (⟨⟨c, −_{F} (d)⟩⟩) if and only if +_{F} (⟨⟨a, d⟩⟩) = +_{F} (⟨⟨b, c⟩⟩).

(8) Let F be a field. Then for all elements a, c of the support of F and for all elements b, d of the support of F \ single(0_{F}) holds ·_{F} (⟨⟨a, {−1 \atop F}(b)⟩⟩) = ·_{F} (⟨⟨c, {−1 \atop F}(d)⟩⟩) if and only if ·_{F} (⟨⟨a, d⟩⟩) = ·_{F} (⟨⟨b, c⟩⟩).

(9) For every field F and for all elements a, b of the support of F holds ·_{F} (⟨⟨a, b⟩⟩) = 0_{F} if and only if a = 0_{F} or b = 0_{F}.

(10) Let F be a field. Let a, b be elements of the support of F. Let c, d be elements of the support of F \ single(0_{F}). Then ·_{F} (⟨⟨·_{F} (⟨⟨a, {−1 \atop F}(c)⟩⟩), ·_{F} (⟨⟨b, {−1 \atop F}(d)⟩⟩)⟩⟩) = ·_{F} (⟨⟨·_{F} (⟨⟨a, b⟩⟩), {−1 \atop F}(·_{F} (⟨⟨c, d⟩⟩))⟩⟩).

(11) Let F be a field. Let a, b be elements of the support of F. Let c, d be elements of the support of F \ single(0_{F}).

Then +_{F} (⟨⟨·_{F} (⟨⟨a, {−1 \atop F}(c)⟩⟩), ·_{F} (⟨⟨b, {−1 \atop F}(d)⟩⟩)⟩⟩) = ·_{F} (⟨⟨+_{F} (⟨⟨·_{F} (⟨⟨a, d⟩⟩), ·_{F} (⟨⟨b, c⟩⟩)⟩⟩), {−1 \atop F}(·_{F} (⟨⟨c, d⟩⟩))⟩⟩).

Let F be a field. The functor osf F yielding a binary operation of the support of F is defined as follows: (Def.1) for all elements x, y of the support of F holds

(osf F)(⟨⟨x, y⟩⟩) = +_{F} (⟨⟨x, −_{F} (y)⟩⟩).

The following propositions are true:

(12) For every field F and for every binary operation S of the support of F holds S = osf F if and only if for all elements x, y of the support of F holds S(⟨⟨x, y⟩⟩) = +_{F} (⟨⟨x, −_{F} (y)⟩⟩).

(13) For every field F and for all elements x, y of the support of F holds osf F (⟨⟨x, y⟩⟩) = +_{F} (⟨⟨x, −_{F} (y)⟩⟩).

(14) For every field F and for every element x of the support of F holds osf F (⟨⟨x, x⟩⟩) = 0_{F}.

(15) For every field F and for all elements a, b, c of the support of F holds ·_{F} (⟨⟨a, osf F (⟨⟨b, c⟩⟩)⟩⟩) = osf F (⟨⟨·_{F} (⟨⟨a, b⟩⟩), ·_{F} (⟨⟨a, c⟩⟩)⟩⟩).

(16) For every field F and for all elements a, b of the support of F holds osf F (⟨⟨a, b⟩⟩) is an element of the support of F.

(17) For every field F and for all elements a, b, c of the support of F holds ·_{F} (⟨⟨ osf F (⟨⟨a, b⟩⟩), c⟩⟩) = osf F (⟨⟨·_{F} (⟨⟨a, c⟩⟩), ·_{F} (⟨⟨b, c⟩⟩)⟩⟩).

(18) For every field F and for all elements a, b of the support of F holds osf F (⟨⟨a, b⟩⟩) = −_{F} (osf F (⟨⟨b, a⟩⟩)).

(19) For every field F and for all elements a, b of the support of F holds osf F (⟨⟨−_{F} (a), b⟩⟩) = −_{F} (+_{F} (⟨⟨a, b⟩⟩)).

(20) For every field F and for all elements a, b, c, d of the support of F holds osf F (⟨⟨a, b⟩⟩) = osf F (⟨⟨c, d⟩⟩) if and only if +_{F} (⟨⟨a, d⟩⟩) = +_{F} (⟨⟨b, c⟩⟩).

(21) For every field F and for every element a of the support of F holds osf F (⟨⟨0_{F} , a⟩⟩) = −_{F} (a).

(22) For every field F and for every element a of the support of F holds osf F (⟨⟨a, 0_{F} ⟩⟩) = a.

(23) For every field F and for all elements a, b, c of the support of F holds +_{F} (⟨⟨a, b⟩⟩) = c if and only if osf F (⟨⟨c, a⟩⟩) = b.

(24) For every field F and for all elements a, b, c of the support of F holds +_{F} (⟨⟨a, b⟩⟩) = c if and only if osf F (⟨⟨c, b⟩⟩) = a.

(25) For every field F and for all elements a, b, c of the support of F holds osf F (⟨⟨a, osf F (⟨⟨b, c⟩⟩)⟩⟩) = +_{F} (⟨⟨ osf F (⟨⟨a, b⟩⟩), c⟩⟩).

(26) For every field F and for all elements a, b, c of the support of F holds

osf F (⟨⟨a, +_{F} (⟨⟨b, c⟩⟩)⟩⟩) = osf F (⟨⟨ osf F (⟨⟨a, b⟩⟩), c⟩⟩). Let F be a field. The functor ovf F yields a function from the support of F #(the support of F \ single(0_{F})) into the support of F and is defined as follows:

(Def.2) for every element x of the support of F and for every element y of the support of F \ single(0_{F}) holds (ovf F)(⟨⟨x, y⟩⟩) = ·_{F} (⟨⟨x, {−1 \atop F}(y)⟩⟩).

Next we state a number of propositions:

(27) Let F be a field. Then for every function D from the support of F #(the support of F \ single(0_{F})) into the support of F holds D = ovf F if and only if for every element x of the support of F and for every element y of the support of F \ single(0_{F}) holds D(⟨⟨x, y⟩⟩) = ·_{F} (⟨⟨x, {−1 \atop F}(y)⟩⟩).

(28) For every field F and for every element x of the support of F and for every element y of the support of F \ single(0_{F}) holds ovf F (⟨⟨x, y⟩⟩) = ·_{F} (⟨⟨x, {−1 \atop F}(y)⟩⟩).

(29) For every field F and for every element x of the support of F \single(0_{F}) holds ovf F (⟨⟨x, x⟩⟩) = 1_{F}.

(30) For every field F and for every element a of the support of F and for every element b of the support of F \ single(0_{F}) holds ovf F (⟨⟨a, b⟩⟩) is an element of the support of F.

(31) For every field F and for all elements a, b of the support of F and for every element c of the support of F \single(0_{F}) holds ·_{F} (⟨⟨a, ovf F (⟨⟨b, c⟩⟩)⟩⟩) = ovf F (⟨⟨·_{F} (⟨⟨a, b⟩⟩), c⟩⟩).

(32) For every field F and for every element a of the support of F \single(0_{F}) holds ·_{F} (⟨⟨a, ovf F (⟨⟨1_{F} , a⟩⟩)⟩⟩) = 1_{F} and ·_{F} (⟨⟨ ovf F (⟨⟨1_{F} , a⟩⟩), a⟩⟩) = 1_{F}.

(34)^{1} For every field F and for all elements a, b of the support of F \single(0_{F})

holds ·_{F} (⟨⟨a, {−1 \atop F}(b)⟩⟩) = {−1 \atop F}(·_{F} (⟨⟨b, {−1 \atop F}(a)⟩⟩)).

(35) For every field F and for all elements a, b of the support of F \single(0_{F}) holds ovf F (⟨⟨a, b⟩⟩) = {−1 \atop F}(ovf F (⟨⟨b, a⟩⟩)).

(36) For every field F and for all elements a, b of the support of F \single(0_{F}) holds ovf F (⟨⟨{−1 \atop F}(a), b⟩⟩) = {−1 \atop F}(·_{F} (⟨⟨a, b⟩⟩)).

(37) For every field F and for all elements a, c of the support of F and for all elements b, d of the support of F \ single(0_{F}) holds ovf F (⟨⟨a, b⟩⟩) = ovf F (⟨⟨c, d⟩⟩) if and only if ·_{F} (⟨⟨a, d⟩⟩) = ·_{F} (⟨⟨b, c⟩⟩).

(38) For every field F and for every element a of the support of F \single(0_{F}) holds ovf F (⟨⟨1_{F} , a⟩⟩) = {−1 \atop F}(a).

(39) For every field F and for every element a of the support of F holds ovf F (⟨⟨a, 1_{F} ⟩⟩) = a.

(40) For every field F and for every element a of the support of F \single(0_{F}) and for all elements b, c of the support of F holds ·_{F} (⟨⟨a, b⟩⟩) = c if and only if ovf F (⟨⟨c, a⟩⟩) = b.

(41) For every field F and for all elements a, c of the support of F and for every element b of the support of F \ single(0_{F}) holds ·_{F} (⟨⟨a, b⟩⟩) = c if and only if ovf F (⟨⟨c, b⟩⟩) = a.

1The proposition (33) was either repeated or obvious.

162 Józef Białas

(42) For every field F and for every element a of the support of F and for all elements b, c of the support of F \ single(0_{F}) holds ovf F (⟨⟨a, ovf F (⟨⟨b, c⟩⟩)⟩⟩) = ·_{F} (⟨⟨ ovf F (⟨⟨a, b⟩⟩), c⟩⟩).

(43) For every field F and for every element a of the support of F and for all elements b, c of the support of F \ single(0_{F}) holds ovf F (⟨⟨a, ·_{F} (⟨⟨b, c⟩⟩)⟩⟩) = ovf F (⟨⟨ ovf F (⟨⟨a, b⟩⟩), c⟩⟩).

References

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

[2] Józef Białas. Properties of fields. Formalized Mathematics, 1(5):807–812, 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. 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] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

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

Infimum and Supremum of the Set of Real Numbers. Measure Theory

Józef Białas University of Łódź

Summary. We introduce some properties of the least upper bound and the greatest lower bound of the subdomain of \overline{}numbers, where \overline{} denotes the enlarged set of real numbers, \overline{}= ∪ \{−∞, +∞\}. The paper contains definitions of majorant and minorant elements, bounded from above, bounded from below and bounded sets, sup and inf of set, for nonempty subset of \overline{}. We prove theorems describing the basic relationships among those definitions. The work is the first part of the series of articles concerning the Lebesgue measure theory.

MML Identifier: SUPINF 1.

The terminology and notation used here have been introduced in the following articles: [3], [1], and [2]. The constant +∞ is defined by: (Def.1) +∞ = .

The following propositions are true:

(1) +∞ = .

(2) +∞ ∉ . A positive infinite number is defined as follows:

(Def.2) it = +∞. One can prove the following proposition (4)^{1} +∞ is a positive infinite number. The constant −∞ is defined as follows:

(Def.3) −∞ = { }. The following propositions are true:

(5) −∞ = { }.

(6) −∞ ∉ .

1The proposition (3) was either repeated or obvious.

A negative infinite number is defined as follows: (Def.4) it = −∞.

One can prove the following proposition (8)^{2} −∞ is a negative infinite number. A Real number is defined as follows:

(Def.5) it ∈ ∪ {−∞, +∞}.

One can prove the following propositions: (10)^{3} For every real number x holds x is a Real number.

(11) For an arbitrary x such that x = −∞ or x = +∞ holds x is a Real

number. Let us note that it makes sense to consider the following constant. Then +∞

is a Real number.

Let us note that it makes sense to consider the following constant. Then −∞ is a Real number.

Next we state the proposition (14)^{4} −∞ ≠ +∞.

Let x, y be Real numbers. The predicate x ≤ y is defined by: (Def.6) there exist real numbers p, q such that p = x and q = y and p ≤ q or

there exists a positive infinite number q such that q = y or there exists a negative infinite number p such that p = x.

Next we state several propositions: (16)^{5} For all Real numbers x, y such that x is a real number and y is a real

number holds x ≤ y if and only if there exist real numbers p, q such that p = x and q = y and p ≤ q.

(17) For every Real number x such that x ∈ holds x −∞.

(18) For every Real number x such that x ∈ holds +∞ x.

(19) +∞ −∞.

(20) For every Real number x holds x ≤ +∞.

(21) For every Real number x holds −∞ ≤ x.

(22) For all Real numbers x, y such that x ≤ y and y ≤ x holds x = y.

(23) For every Real number x such that x ≤ −∞ holds x = −∞.

(24) For every Real number x such that +∞ ≤ x holds x = +∞. The scheme SepR eal concerns a unary predicate P, and states that: there exists a subset X of ∪ {−∞, +∞} such that for every Real number x

holds x ∈ X if and only if P[x] for all values of the parameter.

2The proposition (7) was either repeated or obvious.

3The proposition (9) was either repeated or obvious.

4The propositions (12)–(13) were either repeated or obvious.

5The proposition (15) was either repeated or obvious.

The set \overline{}is defined as follows: (Def.7) \overline{}= ∪ \{−∞, +∞\}.

We now state several propositions:

(25) \overline{}= ∪ \{−∞, +∞\}.

(26) \overline{}is a non-empty set.

(27) For an arbitrary x holds x is a Real number if and only if x ∈ \overline{}.

(28) For every Real number x holds x ≤ x.

(29) For all Real numbers x, y, z such that x ≤ y and y ≤ z holds x ≤ z. Let us note that it makes sense to consider the following constant. Then \overline{}

is a non-empty set.

Let x, y be Real numbers. The predicate x < y is defined by: (Def.8) x ≤ y and x ≠ y.

The following proposition is true (31)^{6} For every Real number x such that x ∈ holds −∞ < x and x < +∞.

Let X be a non-empty subset of \overline{}. A Real number is said to be a majorant of X if: (Def.9) for every Real number x such that x ∈ X holds x ≤ it.

We now state two propositions: (33)^{7} For every non-empty subset X of \overline{}holds +∞ is a majorant of X.

(34) For all non-empty subsets X, Y of \overline{}such that X ⊆ Y for every Real

number U_{1} such that U_{1} is a majorant of Y holds U_{1} is a majorant of X. Let X be a non-empty subset of \overline{}. A Real number is said to be a minorant

of X if: (Def.10) for every Real number x such that x ∈ X holds it ≤ x.

We now state four propositions: (36)^{8} For every non-empty subset X of \overline{}holds −∞ is a minorant of X.

(37) For every non-empty subset X of \overline{}such that X = \overline{}holds +∞ is a majorant of X.

(38) For every non-empty subset X of \overline{}such that X = \overline{}holds −∞ is a minorant of X.

(39) For all non-empty subsets X, Y of \overline{}such that X ⊆ Y for every Real

number L_{1} such that L_{1} is a minorant of Y holds L_{1} is a minorant of X. Let us note that it makes sense to consider the following constant. Then

is a non-empty subset of \overline{}.

One can prove the following propositions: (41)^{9} +∞ is a majorant of .

6The proposition (30) was either repeated or obvious.

7The proposition (32) was either repeated or obvious.

8The proposition (35) was either repeated or obvious.

9The proposition (40) was either repeated or obvious.

(42) −∞ is a minorant of .

Let X be a non-empty subset of \overline{}. We say that X is upper bounded if and only if:

(Def.11) there exists a majorant U_{1} of X such that U_{1} ∈ .

The following two propositions are true: (44)^{10} For all non-empty subsets X, Y of \overline{}such that X ⊆ Y holds if Y is

upper bounded, then X is upper bounded.

(45) is not upper bounded.

Let X be a non-empty subset of \overline{}. We say that X is lower bounded if and only if:

(Def.12) there exists a minorant L_{1} of X such that L_{1} ∈ .

The following two propositions are true: (47)^{11} For all non-empty subsets X, Y of \overline{}such that X ⊆ Y holds if Y is

lower bounded, then X is lower bounded.

(48) is not lower bounded.

Let X be a non-empty subset of \overline{}. We say that X is bounded if and only if: (Def.13) X is upper bounded and X is lower bounded.

The following two propositions are true: (50)^{12} For all non-empty subsets X, Y of \overline{}such that X ⊆ Y holds if Y is

bounded, then X is bounded.

(51) For every non-empty subset X of \overline{}there exists a non-empty subset Y of \overline{}such that for every Real number x holds x ∈ Y if and only if x is a majorant of X.

Let X be a non-empty subset of \overline{}. The functor \overline{X} yields a non-empty subset of \overline{}and is defined as follows:

(Def.14) for every Real number x holds x ∈ \overline{X} if and only if x is a majorant of X.

One can prove the following four propositions:

(52) For every non-empty subset X of \overline{}and for every non-empty subset Y of \overline{}holds Y = \overline{X} if and only if for every Real number x holds x ∈ Y if and only if x is a majorant of X.

(53) For every non-empty subset X of \overline{}and for every Real number x holds x ∈ \overline{X} if and only if x is a majorant of X.

(54) For all non-empty subsets X, Y of \overline{}such that X ⊆ Y for every Real number x such that x ∈ \overline{Y} holds x ∈ \overline{X}.

(55) For every non-empty subset X of \overline{}there exists a non-empty subset Y of \overline{}such that for every Real number x holds x ∈ Y if and only if x is a minorant of X.

10The proposition (43) was either repeated or obvious.

11The proposition (46) was either repeated or obvious.

12The proposition (49) was either repeated or obvious.

Let X be a non-empty subset of \overline{}. The functor \underline{X} yields a non-empty subset of \overline{}and is defined by:

(Def.15) for every Real number x holds x ∈ \underline{X} if and only if x is a minorant of X.

We now state a number of propositions:

(56) For every non-empty subset X of \overline{}and for every non-empty subset Y of \overline{}holds Y = \underline{X} if and only if for every Real number x holds x ∈ Y if and only if x is a minorant of X.

(57) For every non-empty subset X of \overline{}and for every Real number x holds x ∈ \underline{X} if and only if x is a minorant of X.

(58) For all non-empty subsets X, Y of \overline{}such that X ⊆ Y for every Real number x such that x ∈ \underline{Y} holds x ∈ \underline{X}.

(59) For every non-empty subset X of \overline{}such that X is upper bounded and X ≠ {−∞} there exists a real number x such that x ∈ X and x ≠ −∞.

(60) For every non-empty subset X of \overline{}such that X is lower bounded and X ≠ {+∞} there exists a real number x such that x ∈ X and x ≠ +∞.

(62)^{13} For every non-empty subset X of \overline{}such that X is upper bounded and X ≠ {−∞} there exists a Real number U_{1} such that U_{1} is a majorant of X and for every Real number y such that y is a majorant of X holds U_{1} ≤ y.

(63) For every non-empty subset X of \overline{}such that X is lower bounded and X ≠ {+∞} there exists a Real number L_{1} such that L_{1} is a minorant of X and for every Real number y such that y is a minorant of X holds y ≤ L_{1}.

(64) For every non-empty subset X of \overline{}such that X = \{−∞\} holds X is upper bounded.

(65) For every non-empty subset X of \overline{}such that X = \{+∞\} holds X is lower bounded.

(66) For every non-empty subset X of \overline{}such that X = \{−∞\} there exists a Real number U_{1} such that U_{1} is a majorant of X and for every Real number y such that y is a majorant of X holds U_{1} ≤ y.

(67) For every non-empty subset X of \overline{}such that X = \{+∞\} there exists a Real number L_{1} such that L_{1} is a minorant of X and for every Real number y such that y is a minorant of X holds y ≤ L_{1}.

(68) For every non-empty subset X of \overline{}such that X is upper bounded there exists a Real number U_{1} such that U_{1} is a majorant of X and for every Real number y such that y is a majorant of X holds U_{1} ≤ y.

(69) For every non-empty subset X of \overline{}such that X is lower bounded there exists a Real number L_{1} such that L_{1} is a minorant of X and for every Real number y such that y is a minorant of X holds y ≤ L_{1}.

13The proposition (61) was either repeated or obvious.

(70) For every non-empty subset X of \overline{}such that X is not upper bounded for every Real number y such that y is a majorant of X holds y = +∞.

(71) For every non-empty subset X of \overline{}such that X is not lower bounded for every Real number y such that y is a minorant of X holds y = −∞.

(72) For every non-empty subset X of \overline{}there exists a Real number U_{1} such that U_{1} is a majorant of X and for every Real number y such that y is a majorant of X holds U_{1} ≤ y.

(73) For every non-empty subset X of \overline{}there exists a Real number L_{1} such that L_{1} is a minorant of X and for every Real number y such that y is a minorant of X holds y ≤ L_{1}.

Let X be a non-empty subset of \overline{}. The functor sup X yields a Real number and is defined as follows:

(Def.16) sup X is a majorant of X and for every Real number y such that y is a majorant of X holds sup X ≤ y.

The following propositions are true:

(74) For every non-empty subset X of \overline{}and for every Real number S holds S = sup X if and only if S is a majorant of X and for every Real number y such that y is a majorant of X holds S ≤ y.

(75) For every non-empty subset X of \overline{}holds sup X is a majorant of X and for every Real number y such that y is a majorant of X holds sup X ≤ y.

(76) For every non-empty subset X of \overline{}and for every Real number x such that x ∈ X holds x ≤ sup X.

Let X be a non-empty subset of \overline{}. The functor inf X yields a Real number and is defined by:

(Def.17) inf X is a minorant of X and for every Real number y such that y is a minorant of X holds y ≤ inf X.

The following propositions are true:

(77) For every non-empty subset X of \overline{}and for every Real number S holds S = inf X if and only if S is a minorant of X and for every Real number y such that y is a minorant of X holds y ≤ S.

(78) For every non-empty subset X of \overline{}holds inf X is a minorant of X and for every Real number y such that y is a minorant of X holds y ≤ inf X.

(79) For every non-empty subset X of \overline{}and for every Real number x such that x ∈ X holds inf X ≤ x.

(80) For every non-empty subset X of \overline{}and for every majorant x of X such that x ∈ X holds x = sup X.

(81) For every non-empty subset X of \overline{}and for every minorant x of X such that x ∈ X holds x = inf X.

(82) For every non-empty subset X of \overline{}holds sup X = inf \overline{X} and inf X = sup \underline{X}.

(83) For every non-empty subset X of \overline{}such that X is upper bounded and X ≠ {−∞} holds sup X ∈ .

(84) For every non-empty subset X of \overline{}such that X is lower bounded and

X ≠ {+∞} holds inf X ∈ . Let x be a Real number. Then \{x\} is a non-empty subset of \overline{}. Let x, y be Real numbers. Then \{x, y\} is a non-empty subset of \overline{}. We now state a number of propositions:

(85) For every Real number x holds sup{x} = x.

(86) For every Real number x holds inf{x} = x.

(87) sup{−∞} = −∞.

(88) sup{+∞} = +∞.

(89) inf{−∞} = −∞.

(90) inf{+∞} = +∞.

(91) For all non-empty subsets X, Y of \overline{}such that X ⊆ Y holds sup X ≤ sup Y.

(92) For all Real numbers x, y and for every Real number a such that x ≤ a and y ≤ a holds sup{x, y} ≤ a.

(93) For all Real numbers x, y holds if x ≤ y, then sup{x, y} = y but if y ≤ x, then sup{x, y} = x.

(94) For all non-empty subsets X, Y of \overline{}such that X ⊆ Y holds inf Y ≤ inf X.

(95) For all Real numbers x, y and for every Real number a such that a ≤ x and a ≤ y holds a ≤ inf{x, y}.

(96) For all Real numbers x, y holds if x ≤ y, then inf{x, y} = x but if y ≤ x, then inf{x, y} = y.

(97) For every non-empty subset X of \overline{}and for every Real number x such that there exists a Real number y such that y ∈ X and x ≤ y holds x ≤ sup X.

(98) For every non-empty subset X of \overline{}and for every Real number x such that there exists a Real number y such that y ∈ X and y ≤ x holds inf X ≤ x.

(99) For all non-empty subsets X, Y of \overline{}such that for every Real number x such that x ∈ X there exists a Real number y such that y ∈ Y and x ≤ y holds sup X ≤ sup Y.

(100) For all non-empty subsets X, Y of \overline{}such that for every Real number y such that y ∈ Y there exists a Real number x such that x ∈ X and x ≤ y holds inf X ≤ inf Y.

Let X, Y be non-empty subsets of \overline{}. Then X ∪ Y is a non-empty subset of \overline{}.

One can prove the following propositions:

(101) For all non-empty subsets X, Y of \overline{}and for every majorant U_{2} of X and for every majorant U_{3} of Y holds sup{U_{2}, U_{3}} is a majorant of X ∪Y.

(102) For all non-empty subsets X, Y of \overline{}and for every minorant L_{2} of X and for every minorant L_{3} of Y holds inf{L_{2}, L_{3}} is a minorant of X ∪ Y.

(103) For all non-empty subsets X, Y, S of \overline{}and for every majorant U_{2} of X and for every majorant U_{3} of Y such that S = X ∩ Y holds inf{U_{2}, U_{3}} is a majorant of S.

(104) For all non-empty subsets X, Y, S of \overline{}and for every minorant L_{2} of X and for every minorant L_{3} of Y such that S = X ∩ Y holds sup{L_{2}, L_{3}} is a minorant of S.

(105) For all non-empty subsets X, Y of \overline{}holds sup(X ∪ Y) = sup{sup X, sup Y }.

(106) For all non-empty subsets X, Y of \overline{}holds inf(X∪Y) = inf\{inf X, inf Y \}.

(107) For all non-empty subsets X, Y, S of \overline{}such that S = X ∩ Y holds sup S ≤ inf{sup X, sup Y }.

(108) For all non-empty subsets X, Y, S of \overline{}such that S = X ∩ Y holds

sup{inf X, inf Y } ≤ inf S. Let X be a non-empty set. A set is called a non-empty set of non-empty

subsets of X if:

(Def.18) it is a non-empty subset of 2^{X} and for every set A such that A ∈ it holds A is a non-empty set.

Let F be a non-empty set of non-empty subsets of \overline{}. The functor sup\overline{}F yielding a non-empty subset of \overline{}is defined as follows:

(Def.19) for every Real number a holds a ∈ sup\overline{}F if and only if there exists a non-empty subset A of \overline{}such that A ∈ F and a = sup A.

We now state several propositions: (110)^{14} For every non-empty set F of non-empty subsets of \overline{}and for every

non-empty subset S of \overline{}holds S = sup\overline{}F if and only if for every Real number a holds a ∈ S if and only if there exists a non-empty subset A of \overline{}such that A ∈ F and a = sup A.

(111) For every non-empty set F of non-empty subsets of \overline{}and for every Real number a holds a ∈ sup\overline{}F if and only if there exists a non-empty subset A of \overline{}such that A ∈ F and a = sup A.

(112) For every non-empty set F of non-empty subsets of \overline{}and for every non-empty subset S of \overline{}such that S = ⋃ F holds sup S is a majorant of sup\overline{}F.

(113) For every non-empty set F of non-empty subsets of \overline{}and for every nonempty subset S of \overline{}such that S = ⋃ F holds sup(sup\overline{}F) is a majorant of S.

14The proposition (109) was either repeated or obvious.

(114) For every non-empty set F of non-empty subsets of \overline{}and for every

non-empty subset S of \overline{}such that S = ⋃ F holds sup S = sup(sup\overline{}F). Let F be a non-empty set of non-empty subsets of \overline{}. The functor inf\overline{}F

yields a non-empty subset of \overline{}and is defined as follows: (Def.20) for every Real number a holds a ∈ inf\overline{}F if and only if there exists a

non-empty subset A of \overline{}such that A ∈ F and a = inf A. We now state several propositions:

(115) For every non-empty set F of non-empty subsets of \overline{}and for every non-empty subset S of \overline{}holds S = inf\overline{}F if and only if for every Real number a holds a ∈ S if and only if there exists a non-empty subset A of \overline{}such that A ∈ F and a = inf A.

(116) For every non-empty set F of non-empty subsets of \overline{}and for every Real number a holds a ∈ inf\overline{}F if and only if there exists a non-empty subset A of \overline{}such that A ∈ F and a = inf A.

(117) For every non-empty set F of non-empty subsets of \overline{}and for every non-empty subset S of \overline{}such that S = ⋃ F holds inf S is a minorant of inf\overline{}F.

(118) For every non-empty set F of non-empty subsets of \overline{}and for every nonempty subset S of \overline{}such that S = ⋃ F holds inf(inf\overline{}F) is a minorant of S.

(119) For every non-empty set F of non-empty subsets of \overline{}and for every non-empty subset S of \overline{}such that S = ⋃ F holds inf S = inf(inf\overline{}F).

References

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

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

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

172

Series of Positive Real Numbers. Measure Theory

Józef Białas University of Łódź

Summary. We introduce properties of a series of nonnegative \overline{} numbers, where \overline{}denotes the enlarged set of real numbers, \overline{}= ∪ {−∞, +∞}. The paper contains definitions of sup F and inf F, for F being function, and a definition of a sumable subset of \overline{}. We prove the basic theorems regarding the definitions mentioned above. The work is the second part of a series of articles concerning the Lebesgue measure theory.

MML Identifier: SUPINF 2.

The notation and terminology used here are introduced in the following articles: [6], [5], [2], [3], [4], and [1]. Let x, y be Real numbers. Let us assume that neither x = +∞ and y = −∞ nor x = −∞ and y = +∞. The functor x + y yielding a Real number is defined by: (Def.1) there exist real numbers a, b such that x = a and y = b and x+y = a+b

or x = +∞ and x + y = +∞ or y = +∞ and x + y = +∞ or x = −∞ and x + y = −∞ or y = −∞ and x + y = −∞.

Next we state four propositions:

(1) Let x, y be Real numbers. Suppose neither x = +∞ and y = −∞ nor x = −∞ and y = +∞. Then

(i) there exist real numbers a, b such that x = a and y = b and x+y = a+b, or

(ii) x = +∞ and x + y = +∞, or (iii) y = +∞ and x + y = +∞, or (iv) x = −∞ and x + y = −∞, or

(v) y = −∞ and x + y = −∞.

(2) For all Real numbers x, y and for all real numbers a, b such that x = a and y = b holds x + y = a + b.

(3) For every Real number x such that x ≠ −∞ holds +∞ + x = +∞ and x + +∞ = +∞.

(4) For every Real number x such that x ≠ +∞ holds −∞ + x = −∞ and x + −∞ = −∞.

Let x, y be Real numbers. Let us assume that neither x = +∞ and y = +∞ nor x = −∞ and y = −∞. The functor x − y yielding a Real number is defined by: (Def.2) there exist real numbers a, b such that x = a and y = b and x−y = a−b

or x = +∞ and x − y = +∞ or y = +∞ and x − y = −∞ or x = −∞ and x − y = −∞ or y = −∞ and x − y = +∞.

We now state a number of propositions:

(5) Let x, y be Real numbers. Suppose neither x = +∞ and y = +∞ nor x = −∞ and y = −∞. Then

(i) there exist real numbers a, b such that x = a and y = b and x−y = a−b, or

(ii) x = +∞ and x − y = +∞, or (iii) y = +∞ and x − y = −∞, or (iv) x = −∞ and x − y = −∞, or

(v) y = −∞ and x − y = +∞.

(6) For all Real numbers x, y and for all real numbers a, b such that x = a and y = b holds x − y = a − b.

(7) For every Real number x such that x ≠ +∞ holds +∞ − x = +∞ and x − +∞ = −∞.

(8) For every Real number x such that x ≠ −∞ holds −∞ − x = −∞ and x − −∞ = +∞.

(9) For all Real numbers x, s such that x + s = +∞ holds x = +∞ or s = +∞.

(10) For all Real numbers x, s such that x + s = −∞ holds x = −∞ or s = −∞.

(11) For all Real numbers x, s such that x − s = +∞ holds x = +∞ or s = −∞.

(12) For all Real numbers x, s such that x − s = −∞ holds x = −∞ or s = +∞.

(13) For all Real numbers x, s such that neither x = +∞ and s = −∞ nor x = −∞ and s = +∞ and x + s ∈ holds x ∈ and s ∈ .

(14) For all Real numbers x, s such that neither x = +∞ and s = +∞ nor x = −∞ and s = −∞ and x − s ∈ holds x ∈ and s ∈ .

(15) Let x, y, s, t be Real numbers. Then if neither x = +∞ and s = −∞ nor x = −∞ and s = +∞ and neither y = +∞ and t = −∞ nor y = −∞ and t = +∞ and x ≤ y and s ≤ t, then x + s ≤ y + t.

(16) Let x, y, s, t be Real numbers. Then if neither x = +∞ and t = +∞ nor x = −∞ and t = −∞ and neither y = +∞ and s = +∞ nor y = −∞

and s = −∞ and x ≤ y and s ≤ t, then x − t ≤ y − s. Let x be a Real number. The functor −x yields a Real number and is defined

by: (Def.3) there exists a real number a such that x = a and −x = −a or x = +∞

and −x = −∞ or x = −∞ and −x = +∞. We now state several propositions:

(17) For every Real number x and for every Real number z holds z = −x if and only if there exists a real number a such that x = a and z = −a or x = +∞ and z = −∞ or x = −∞ and z = +∞.

(18) For every Real number x holds there exists a real number a such that x = a and −x = −a or x = +∞ and −x = −∞ or x = −∞ and −x = +∞.

(19) For every Real number x and for every real number a such that x = a holds −x = −a.

(20) For every Real number x holds if x = +∞, then −x = −∞ but if x = −∞, then −x = +∞.

(21) For every Real number x holds −(−x) = x.

(22) For all Real numbers x, y holds x ≤ y if and only if −y ≤ −x.

(23) For all Real numbers x, y holds x < y if and only if −y < −x.

(24) For all Real numbers x, y such that x = y holds x ≤ y.

The Real number 0\overline{}is defined by: (Def.4) 0\overline{}= 0.

We now state several propositions:

(25) 0\overline{}= 0.

(26) For every Real number x holds x + 0\overline{}= x and 0\overline{}+ x = x.

(27) −∞ < 0\overline{}and 0\overline{}< +∞.

(28) For all Real numbers x, y, z such that 0\overline{}≤ z and 0\overline{}≤ x and y = x + z holds x ≤ y.

(29) For every real number x such that x ∈ holds 0 ≤ x.

(30) For every Real number x such that x ∈ holds 0\overline{}≤ x.

Let X, Y be non-empty subsets of \overline{}. Let us assume that neither −∞ ∈ X and +∞ ∈ Y nor +∞ ∈ X and −∞ ∈ Y. The functor X + Y yielding a non-empty subset of \overline{}is defined as follows: (Def.5) for every Real number z holds z ∈ X + Y if and only if there exist Real

numbers x, y such that x ∈ X and y ∈ Y and z = x + y. We now state two propositions:

(31) For all non-empty subsets X, Y of \overline{}such that neither −∞ ∈ X and +∞ ∈ Y nor +∞ ∈ X and −∞ ∈ Y for every Real number z holds z ∈ X + Y if and only if there exist Real numbers x, y such that x ∈ X and y ∈ Y and z = x + y.

(32) Let X, Y, Z be non-empty subsets of \overline{}. Then if neither −∞ ∈ X and +∞ ∈ Y nor +∞ ∈ X and −∞ ∈ Y, then Z = X + Y if and only if for every Real number z holds z ∈ Z if and only if there exist Real numbers x, y such that x ∈ X and y ∈ Y and z = x + y.

Let X be a non-empty subset of \overline{}. The functor −X yielding a non-empty subset of \overline{}is defined as follows: (Def.6) for every Real number z holds z ∈ −X if and only if there exists a Real

number x such that x ∈ X and z = −x.

Next we state a number of propositions:

(33) For every non-empty subset X of \overline{}and for every Real number z holds z ∈ −X if and only if there exists a Real number x such that x ∈ X and z = −x.

(34) For all non-empty subsets X, Z of \overline{}holds Z = −X if and only if for every Real number z holds z ∈ Z if and only if there exists a Real number x such that x ∈ X and z = −x.

(35) For every non-empty subset X of \overline{}holds −(−X) = X.

(36) For every non-empty subset X of \overline{}and for every Real number z holds z ∈ X if and only if −z ∈ −X.

(37) For all non-empty subsets X, Y of \overline{}holds X ⊆ Y if and only if −X ⊆ −Y.

(38) For every Real number z holds z ∈ if and only if −z ∈ .

(39) Let X, Y be non-empty subsets of \overline{}. Then if neither −∞ ∈ X and +∞ ∈ Y nor +∞ ∈ X and −∞ ∈ Y and neither sup X = +∞ and sup Y = −∞ nor sup X = −∞ and sup Y = +∞, then sup(X + Y) ≤ sup X + sup Y.

(40) Let X, Y be non-empty subsets of \overline{}. Then if neither −∞ ∈ X and +∞ ∈ Y nor +∞ ∈ X and −∞ ∈ Y and neither inf X = +∞ and inf Y = −∞ nor inf X = −∞ and inf Y = +∞, then inf X + inf Y ≤ inf(X + Y).

(41) For all non-empty subsets X, Y of \overline{}such that X is upper bounded and Y is upper bounded holds sup(X + Y) ≤ sup X + sup Y.

(42) For all non-empty subsets X, Y of \overline{}such that X is lower bounded and Y is lower bounded holds inf X + inf Y ≤ inf(X + Y).

(43) For every non-empty subset X of \overline{}and for every Real number a holds a is a majorant of X if and only if −a is a minorant of −X.

(44) For every non-empty subset X of \overline{}and for every Real number a holds a is a minorant of X if and only if −a is a majorant of −X.

(45) For every non-empty subset X of \overline{}holds inf(−X) = − sup X.

(46) For every non-empty subset X of \overline{}holds sup(−X) = − inf X. Let X be a non-empty set, and let Y be a non-empty subset of \overline{}, and let F

be a function from X into Y. Then rng F is a non-empty subset of \overline{}.

Let X be a non-empty set, and let Y be a non-empty subset of \overline{}, and let F be a function from X into Y. The functor sup F yielding a Real number is defined by: (Def.7) sup F = sup(rng F).

The following proposition is true

(47) For every non-empty set X and for every non-empty subset Y of \overline{}and

for every function F from X into Y holds sup F = sup(rng F). Let X be a non-empty set, and let Y be a non-empty subset of \overline{}, and let

F be a function from X into Y. The functor inf F yields a Real number and is defined by: (Def.8) inf F = inf(rng F).

Next we state the proposition

(48) For every non-empty set X and for every non-empty subset Y of \overline{}and

for every function F from X into Y holds inf F = inf(rng F). Let X be a non-empty set, and let Y be a non-empty subset of \overline{}, and let

F be a function from X into Y, and let x be an element of X. Then F (x) is a Real number.

The scheme FunctR ealEx concerns a non-empty set A, a set B, and a unary functor F and states that:

there exists a function f from A into B such that for every element x of A holds f(x) = F(x) provided the parameters have the following property:

• for every element x of A holds F(x) ∈ B. Let X be a non-empty set, and let Y, Z be non-empty subsets of \overline{}, and let

F be a function from X into Y, and let G be a function from X into Z. Let us assume that neither −∞ ∈ Y and +∞ ∈ Z nor +∞ ∈ Y and −∞ ∈ Z. The functor F + G yields a function from X into Y + Z and is defined by: (Def.9) for every element x of X holds (F + G)(x) = F (x) + G(x).

Next we state several propositions:

(49) Let X be a non-empty set. Let Y, Z be non-empty subsets of \overline{}. Suppose neither −∞ ∈ Y and +∞ ∈ Z nor +∞ ∈ Y and −∞ ∈ Z. Then for every function F from X into Y and for every function G from X into Z and for every function H from X into Y + Z holds H = F + G if and only if for every element x of X holds H(x) = F (x) + G(x).

(50) Let X be a non-empty set. Then for all non-empty subsets Y, Z of \overline{} such that neither −∞ ∈ Y and +∞ ∈ Z nor +∞ ∈ Y and −∞ ∈ Z for every function F from X into Y and for every function G from X into Z and for every element x of X holds (F + G)(x) = F (x) + G(x).

(51) For every non-empty set X and for all non-empty subsets Y, Z of \overline{} such that neither −∞ ∈ Y and +∞ ∈ Z nor +∞ ∈ Y and −∞ ∈ Z for every function F from X into Y and for every function G from X into Z holds rng(F + G) ⊆ rng F + rng G.

(52) Let X be a non-empty set. Let Y, Z be non-empty subsets of \overline{}. Suppose neither −∞ ∈ Y and +∞ ∈ Z nor +∞ ∈ Y and −∞ ∈ Z. Then for every function F from X into Y and for every function G from X into Z such that neither sup F = +∞ and sup G = −∞ nor sup F = −∞ and sup G = +∞ holds sup(F + G) ≤ sup F + sup G.

(53) Let X be a non-empty set. Let Y, Z be non-empty subsets of \overline{}. Suppose neither −∞ ∈ Y and +∞ ∈ Z nor +∞ ∈ Y and −∞ ∈ Z. Then for every function F from X into Y and for every function G from X into Z such that neither inf F = +∞ and inf G = −∞ nor inf F = −∞ and inf G = +∞ holds inf F + inf G ≤ inf(F + G).

Let X be a non-empty set, and let Y be a non-empty subset of \overline{}, and let F be a function from X into Y. The functor −F yielding a function from X into −Y is defined by:

(Def.10) for every element x of X holds (−F)(x) = −F (x).

One can prove the following three propositions:

(54) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y and for every function G from X into −Y holds G = −F if and only if for every element x of X holds G(x) = −F (x).

(55) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y holds rng(−F) = − rng F.

(56) For every non-empty set X and for every non-empty subset Y of \overline{} and for every function F from X into Y holds inf(−F) = − sup F and sup(−F) = − inf F.

Let X be a non-empty set, and let Y be a non-empty subset of \overline{}, and let F be a function from X into Y. We say that F is upper bounded if and only if:

(Def.11) sup F < +∞.

Let X be a non-empty set, and let Y be a non-empty subset of \overline{}, and let F be a function from X into Y. We say that F is lower bounded if and only if:

(Def.12) −∞ < inf F.

Let X be a non-empty set, and let Y be a non-empty subset of \overline{}, and let F be a function from X into Y. We say that F is bounded if and only if:

(Def.13) F is upper bounded and F is lower bounded.

We now state a number of propositions: (60)^{1} For every non-empty set X and for every non-empty subset Y of \overline{}and

for every function F from X into Y holds F is bounded if and only if sup F < +∞ and −∞ < inf F.

(61) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y holds F is upper bounded if and only if −F is lower bounded.

1The propositions (57)–(59) were either repeated or obvious.

(62) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y holds F is lower bounded if and only if −F is upper bounded.

(63) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y holds F is bounded if and only if −F is bounded.

(64) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y and for every element x of X holds −∞ ≤ F (x) and F (x) ≤ +∞.

(65) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y and for every element x of X such that Y ⊆ holds −∞ < F (x) and F (x) < +∞.

(66) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y and for every element x of X holds inf F ≤ F (x) and F (x) ≤ sup F.

(67) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y such that Y ⊆ holds F is upper bounded if and only if sup F ∈ .

(68) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y such that Y ⊆ holds F is lower bounded if and only if inf F ∈ .

(69) For every non-empty set X and for every non-empty subset Y of \overline{}and for every function F from X into Y such that Y ⊆ holds F is bounded if and only if inf F ∈ and sup F ∈ .

(70) For every non-empty set X and for all non-empty subsets Y, Z of \overline{} such that Y ⊆ and Z ⊆ for every function F_{1} from X into Y and for every function F_{2} from X into Z such that F_{1} is upper bounded and F_{2} is upper bounded holds F_{1} + F_{2} is upper bounded.

(71) For every non-empty set X and for all non-empty subsets Y, Z of \overline{} such that Y ⊆ and Z ⊆ for every function F_{1} from X into Y and for every function F_{2} from X into Z such that F_{1} is lower bounded and F_{2} is lower bounded holds F_{1} + F_{2} is lower bounded.

(72) For every non-empty set X and for all non-empty subsets Y, Z of \overline{} such that Y ⊆ and Z ⊆ for every function F_{1} from X into Y and for every function F_{2} from X into Z such that F_{1} is bounded and F_{2} is bounded holds F_{1} + F_{2} is bounded.

(73) There exists a function F from into \overline{}such that F is one-to-one and = rng F and rng F is a non-empty subset of \overline{}.

A non-empty subset of \overline{}is called a denumerable set of larged real if:

(Def.14) there exists a function F from into \overline{}such that it = rng F.

Next we state the proposition (75)^{2} is a denumerable set of larged real.

A denumerable set of larged real is said to be a denumerable set of positive larged real if:

(Def.15) for every Real number x such that x ∈ it holds 0\overline{}≤ x.

Let D be a denumerable set of larged real. A function from into \overline{}is said to be a numeration of D if:

(Def.16) D = rng it.

One can prove the following proposition (78)^{3} For every denumerable set D of positive larged real and for every func-

tion F from into \overline{}holds F is a numeration of D if and only if D = rng F. Let N be a function from into \overline{}, and let n be a natural number. Then

N(n) is a Real number.

We see that the Real number is an element of \overline{}. The scheme RecFuncExR eal concerns a Real number A and a binary functor

F yielding a Real number and states that:

there exists a function F from into \overline{}such that F (0) = A and for every natural number n and for every Real number x such that x = F (n) holds F (n + 1) = F(n, x) for all values of the parameters.

We now state the proposition

(79) For every denumerable set D of larged real and for every numeration N of D there exists a function F from into \overline{}such that F (0) = N(0) and for every natural number n and for every Real number y such that y = F (n) holds F (n + 1) = y + N(n + 1).

Let D be a denumerable set of larged real, and let N be a numeration of D. The functor Ser(D, N) yields a function from into \overline{}and is defined by:

(Def.17) Ser(D, N)(0) = N(0) and for every natural number n and for every Real number y such that y = Ser(D, N)(n) holds Ser(D, N)(n + 1) = y + N(n + 1).

The following propositions are true:

(80) Let D be a denumerable set of larged real. Then for every numeration N of D and for every function F from into \overline{}holds F = Ser(D, N) if and only if F (0) = N(0) and for every natural number n and for every Real number y such that y = F (n) holds F (n + 1) = y + N(n + 1).

(81) For every denumerable set D of larged real and for every numeration N of D holds Ser(D, N)(0) = N(0) and for every natural number n and for every Real number y such that y = Ser(D, N)(n) holds Ser(D, N)(n+1) = y + N(n + 1).

(82) For every denumerable set D of positive larged real and for every numeration N of D and for every natural number n holds 0\overline{}≤ N(n).

2The proposition (74) was either repeated or obvious.

3The propositions (76)–(77) were either repeated or obvious.

(83) For every denumerable set D of positive larged real and for every numeration N of D and for every natural number n holds Ser(D, N)(n) ≤ Ser(D, N)(n + 1) and 0\overline{}≤ Ser(D, N)(n).

(84) For every denumerable set D of positive larged real and for every numeration N of D and for all natural numbers n, m holds Ser(D, N)(n) ≤ Ser(D, N)(n + m).

Let D be a denumerable set of larged real. A non-empty subset of \overline{}is called a set of series of D if:

(Def.18) there exists a numeration N of D such that it = rng Ser(D, N).

Let F be a function from into \overline{}. Then rng F is a non-empty subset of \overline{}. Let D be a denumerable set of positive larged real, and let N be a numeration

of D. The functor ∑_{D} N yields a Real number and is defined as follows: (Def.19) ∑_{D} N = sup(rng Ser(D, N)).

One can prove the following propositions: (86)^{4} For every denumerable set D of positive larged real and for every nu-

meration N of D and for every Real number s holds s = ∑_{D} N if and only if s = sup(rng Ser(D, N)).

(87) For every denumerable set D of positive larged real and for every nu-

meration N of D holds ∑_{D} N = sup(rng Ser(D, N)). Let D be a denumerable set of positive larged real, and let N be a numeration

of D. We say that D is N sumable if and only if: (Def.20) ∑_{D} N ∈ .

One can prove the following proposition (89)^{5} For every function F from into \overline{}holds rng F is a denumerable set of

larged real. Let F be a function from into \overline{}. Then rng F is a denumerable set of larged

real.

Next we state the proposition

(90) For every function F from into \overline{}holds F is a numeration of rng F. Let F be a function from into \overline{}. The functor Ser F yields a function from

into \overline{}and is defined by: (Def.21) for every numeration N of rng F such that N = F holds Ser F =

Ser(rng F, N).

We now state the proposition

(91) For every function F from into \overline{}and for every numeration N of

rng F such that N = F holds Ser F = Ser(rng F, N). Let F be a function from into \overline{}. We say that F is non-negative if and

only if:

4The proposition (85) was either repeated or obvious.

5The proposition (88) was either repeated or obvious.

(Def.22) rng F is a denumerable set of positive larged real.

Let F be a function from into \overline{}. Let us assume that F is non-negative. The functor ∑ F yields a Real number and is defined by:

(Def.23) ∑ F = sup(rng Ser F).

The following propositions are true: (93)^{6} For every function F from into \overline{}such that F is non-negative holds

∑ F = sup(rng Ser F).

(94) For every function F from into \overline{}holds F is non-negative if and only if for every natural number n holds 0\overline{}≤ F (n).

(95) For every function F from into \overline{}and for every natural number n such that F is non-negative holds Ser F (n) ≤ Ser F (n + 1) and 0\overline{}≤ Ser F (n).

(96) For every function F from into \overline{}such that F is non-negative for all natural numbers n, m holds Ser F (n) ≤ Ser F (n + m).

(97) For all functions F_{1}, F_{2} from into \overline{}such that F_{1} is non-negative holds if for every natural number n holds F_{1}(n) ≤ F_{2}(n), then for every natural number n holds Ser F_{1}(n) ≤ Ser F_{2}(n).

(98) For all functions F_{1}, F_{2} from into \overline{}such that F_{1} is non-negative holds if for every natural number n holds F_{1}(n) ≤ F_{2}(n), then ∑ F_{1} ≤ ∑ F_{2}.

(99) For every function F from into \overline{}holds Ser F (0) = F (0) and for every natural number n and for every Real number y such that y = Ser F (n) holds Ser F (n + 1) = y + F (n + 1).

(100) For every function F from into \overline{}such that F is non-negative holds if

there exists a natural number n such that F (n) = +∞, then ∑ F = +∞. Let F be a function from into \overline{}. Let us assume that F is non-negative.

We say that F is sumable if and only if: (Def.24) ∑ F ∈ .

One can prove the following propositions: (102)^{7} For every function F from into \overline{}such that F is non-negative holds

if there exists a natural number n such that F (n) = +∞, then F is not sumable.

(103) For all functions F_{1}, F_{2} from into \overline{}such that F_{1} is non-negative holds if for every natural number n holds F_{1}(n) ≤ F_{2}(n), then if F_{2} is sumable, then F_{1} is sumable.

(104) For all functions F_{1}, F_{2} from into \overline{}such that F_{1} is non-negative holds if for every natural number n holds F_{1}(n) ≤ F_{2}(n), then if F_{1} is not sumable, then F_{2} is not sumable.

(105) For every function F from into \overline{}such that F is non-negative for every natural number n such that for every natural number r such that n ≤ r holds F (r) = 0\overline{}holds ∑ F = Ser F (n).

6The proposition (92) was either repeated or obvious.

7The proposition (101) was either repeated or obvious.

Series of Positive Real Numbers. Measure … 183

(106) For every function F from into \overline{}such that for every natural number n holds F (n) ∈ for every natural number n holds Ser F (n) ∈ .

(107) For every function F from into \overline{}such that F is non-negative holds if there exists a natural number n such that for every natural number k such that n ≤ k holds F (k) = 0\overline{}and for every natural number k such that k ≤ n holds F (k) ≠ +∞, then F is sumable.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.

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

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

184

From Double Loops to Fields

Wojciech Skaba Toruń University Michał Muzalewski^{1} Warsaw University Białystok

Summary. This paper contains the second part of the set of articles concerning the theory of algebraic structures, based on the [9], pp. 9-12 (pages 4-6 of the English edition).

First the basic structure ⟨F, +, ·, 1, 0⟩ is defined. Following it the consecutive structures are contemplated in detail, including double loop, left quasi-field, right quasi-field, double sided quasi-field, skew field and field. These structures are created by gradually augmenting the basic structure with new axioms of commutativity, associativity, distributivity etc. Each part of the article begins with the set of auxiliary theorems related to the structure under consideration that are necessary for the existence proof of each defined mode. Next the mode and proof of its existence is included. If the current set of axioms may be replaced with a different and equivalent one, the appropriate proof is performed following the definition of that mode. With the introduction of double loop the ”−” function is defined. Some interesting features of this function are also included.

MML Identifier: ALGSTR 2.

The terminology and notation used here have been introduced in the following articles: [11], [10], [3], [4], [1], [2], [6], [5], [7], and [8]. We consider double loop structures which are systems

⟨a carrier, an addition, a multiplication, a unity, a zero⟩, where the carrier is a non-empty set, the addition is a binary operation on the carrier, the multiplication is a binary operation on the carrier, the unity is an element of the carrier, and the zero is an element of the carrier.

In the sequel G_{1} will be a double loop structure and L will be a double loop structure. Let us consider G_{1}. An element of G_{1} is an element of the carrier of G_{1}.

In the sequel a, b will denote elements of G_{1}. Let us consider G_{1}, a, b. The functor a + b yields an element of G_{1} and is defined by:

1Supported by RPBP.III-24.B5

(Def.1) a + b = (the addition of G_{1})(a, b).

Let us consider G_{1}, a, b. The functor a · b yields an element of G_{1} and is defined by: (Def.2) a · b = (the multiplication of G_{1})(a, b).

One can prove the following propositions:

(1) a + b = (the addition of G_{1})(a, b).

(2) a · b = (the multiplication of G_{1})(a, b). Let us consider G_{1}. The functor 0_{G_{1}} yielding an element of G_{1} is defined as

follows: (Def.3) 0_{G_{1}} = the zero of G_{1}.

Let us consider G_{1}. The functor 1_{G_{1}} yields an element of G_{1} and is defined as follows: (Def.4) 1_{G_{1}} = the unity of G_{1}.

The following two propositions are true:

(3) 0_{G_{1}} = the zero of G_{1}.

(4) 1_{G_{1}} = the unity of G_{1}. The double loop structure loop_{}is defined by:

(Def.5) loop_{}= ⟨ , +_{}, ·_{}, 1, 0⟩.

One can prove the following three propositions:

(5) loop_{}= ⟨ , +_{}, ·_{}, 1, 0⟩.

(6) For every real numbers q, p there exists a real number y such that p = q + y.

(7) For every real numbers q, p there exists a real number y such that p = y + q.

A double loop structure is said to be a double loop if: (Def.6) (i) for every element a of it holds a + 0_{it} = a,

(ii) for every element a of it holds 0_{it} + a = a, (iii) for every elements a, b of it there exists an element x of it such that

a + x = b, (iv) for every elements a, b of it there exists an element x of it such that

x + a = b,

(v) for all elements a, x, y of it such that a + x = a + y holds x = y, (vi) for all elements a, x, y of it such that x + a = y + a holds x = y, (vii) 0_{it} ≠ 1_{it}, (viii) for every element a of it holds a · (1_{it}) = a, (ix) for every element a of it holds (1_{it}) · a = a,

(x) for all elements a, b of it such that a ≠ 0_{it} there exists an element x of it such that a · x = b,

(xi) for all elements a, b of it such that a ≠ 0_{it} there exists an element x of it such that x · a = b, (xii) for all elements a, x, y of it such that a ≠ 0_{it} holds if a · x = a · y, then

x = y, (xiii) for all elements a, x, y of it such that a ≠ 0_{it} holds if x · a = y · a, then

x = y, (xiv) for every element a of it holds a · 0_{it} = 0_{it}, (xv) for every element a of it holds 0_{it} · a = 0_{it}.

Let us note that it makes sense to consider the following constant. Then loop_{}is a double loop.

Let L be a double loop, and let a be an element of L. The functor −a yielding an element of L is defined as follows: (Def.7) a + (−a) = 0_{L}.

Next we state the proposition (9)^{2} For every double loop L and for every element a of L holds a + (−a) =

0_{L}. Let L be a double loop, and let a, b be elements of L. The functor a − b

yielding an element of L is defined by: (Def.8) a − b = a + (−b).

We now state the proposition

(10) For every double loop L and for all elements a, b of L holds a − b =

a + (−b). A double loop is said to be a left quasi-field if:

(Def.9) (i) for all elements a, b, c of it holds (a + b) + c = a + (b + c), (ii) for all elements a, b of it holds a + b = b + a, (iii) for all elements a, b, c of it holds a · (b + c) = a · b + a · c.

In the sequel a, b, c, x, y are elements of L. The following proposition is true (12)^{3} L is a left quasi-field if and only if the following conditions are satisfied:

(i) for every a holds a + 0_{L} = a, (ii) for every a there exists x such that a + x = 0_{L}, (iii) for all a, b, c holds (a + b) + c = a + (b + c), (iv) for all a, b holds a + b = b + a,

(v) 0_{L} ≠ 1_{L}, (vi) for every a holds a · (1_{L}) = a, (vii) for every a holds (1_{L}) · a = a, (viii) for all a, b such that a ≠ 0_{L} there exists x such that a · x = b, (ix) for all a, b such that a ≠ 0_{L} there exists x such that x · a = b,

(x) for all a, x, y such that a ≠ 0_{L} holds if a · x = a · y, then x = y, (xi) for all a, x, y such that a ≠ 0_{L} holds if x · a = y · a, then x = y, (xii) for every a holds a · 0_{L} = 0_{L}, (xiii) for every a holds 0_{L} · a = 0_{L}, (xiv) for all a, b, c holds a · (b + c) = a · b + a · c.

2The proposition (8) was either repeated or obvious.

3The proposition (11) was either repeated or obvious.

We follow the rules: G will be a left quasi-field and a, b, x, y will be elements of G. We now state several propositions:

(13) a + (−a) = 0_{G} and (−a) + a = 0_{G}.

(14) a · (−b) = −a · b.

(15) −(−a) = a.

(16) (−1_{G}) · (−1_{G}) = 1_{G}.

(17) a · (x − y) = a · x − a · y.

A double loop is called a right quasi-field if: (Def.10) (i) for all elements a, b, c of it holds (a + b) + c = a + (b + c),

(ii) for all elements a, b of it holds a + b = b + a, (iii) for all elements a, b, c of it holds (b + c) · a = b · a + c · a. In the sequel a, b, c, x, y are elements of L. One can prove the following

proposition (19)^{4} L is a right quasi-field if and only if the following conditions are satisfied:

(i) for every a holds a + 0_{L} = a, (ii) for every a there exists x such that a + x = 0_{L}, (iii) for all a, b, c holds (a + b) + c = a + (b + c), (iv) for all a, b holds a + b = b + a,

(v) 0_{L} ≠ 1_{L}, (vi) for every a holds a · (1_{L}) = a, (vii) for every a holds (1_{L}) · a = a, (viii) for all a, b such that a ≠ 0_{L} there exists x such that a · x = b, (ix) for all a, b such that a ≠ 0_{L} there exists x such that x · a = b,

(x) for all a, x, y such that a ≠ 0_{L} holds if a · x = a · y, then x = y, (xi) for all a, x, y such that a ≠ 0_{L} holds if x · a = y · a, then x = y, (xii) for every a holds a · 0_{L} = 0_{L}, (xiii) for every a holds 0_{L} · a = 0_{L}, (xiv) for all a, b, c holds (b + c) · a = b · a + c · a. We adopt the following rules: G will be a right quasi-field and a, b, x, y will

be elements of G. We now state several propositions:

(20) a + (−a) = 0_{G} and (−a) + a = 0_{G}.

(21) (−b) · a = −b · a.

(22) −(−a) = a.

(23) (−1_{G}) · (−1_{G}) = 1_{G}.

(24) (x − y) · a = x · a − y · a.

In the sequel a, b, c, x, y will denote elements of L. A double loop is called a double sided quasi-field if:

(Def.11) (i) for all elements a, b, c of it holds (a + b) + c = a + (b + c), (ii) for all elements a, b of it holds a + b = b + a, (iii) for all elements a, b, c of it holds a · (b + c) = a · b + a · c, (iv) for all elements a, b, c of it holds (b + c) · a = b · a + c · a.

4The proposition (18) was either repeated or obvious.

Let us note that it makes sense to consider the following constant. Then loop_{}is a double sided quasi-field.

The following propositions are true: (26)^{5} L is a double sided quasi-field if and only if the following conditions are

satisfied:

(i) for every a holds a + 0_{L} = a, (ii) for every a there exists x such that a + x = 0_{L}, (iii) for all a, b, c holds (a + b) + c = a + (b + c), (iv) for all a, b holds a + b = b + a,

(v) 0_{L} ≠ 1_{L}, (vi) for every a holds a · (1_{L}) = a, (vii) for every a holds (1_{L}) · a = a, (viii) for all a, b such that a ≠ 0_{L} there exists x such that a · x = b, (ix) for all a, b such that a ≠ 0_{L} there exists x such that x · a = b,

(x) for all a, x, y such that a ≠ 0_{L} holds if a · x = a · y, then x = y, (xi) for all a, x, y such that a ≠ 0_{L} holds if x · a = y · a, then x = y, (xii) for every a holds a · 0_{L} = 0_{L}, (xiii) for every a holds 0_{L} · a = 0_{L}, (xiv) for all a, b, c holds a · (b + c) = a · b + a · c, (xv) for all a, b, c holds (b + c) · a = b · a + c · a.

(27) For every double sided quasi-field L holds L is a left quasi-field.

(28) For every double sided quasi-field L holds L is a right quasi-field. We adopt the following rules: G will be a double sided quasi-field and a, b,

x, y will be elements of G. Next we state two propositions:

(29) a · (−b) = −a · b and (−b) · a = −b · a.

(30) a · (x − y) = a · x − a · y and (x − y) · a = x · a − y · a.

We see that the double sided quasi-field is a left quasi-field. In the sequel a, b, c, x will be elements of L. A double sided quasi-field is

called a skew field if: (Def.12) for all elements a, b, c of it holds (a · b) · c = a · (b · c).

Let us note that it makes sense to consider the following constant. Then loop_{}is a skew field.

The following proposition is true (32)^{6} L is a skew field if and only if the following conditions are satisfied:

(i) for every a holds a + 0_{L} = a, (ii) for every a there exists x such that a + x = 0_{L}, (iii) for all a, b, c holds (a + b) + c = a + (b + c), (iv) for all a, b holds a + b = b + a,

(v) 0_{L} ≠ 1_{L}, (vi) for every a holds a · (1_{L}) = a, 190 Wojciech Skaba and Michał Muzalewski

5The proposition (25) was either repeated or obvious.

6The proposition (31) was either repeated or obvious.

(vii) for every a such that a ≠ 0_{L} there exists x such that a · x = 1_{L}, (viii) for every a holds a · 0_{L} = 0_{L}, (ix) for every a holds 0_{L} · a = 0_{L},

(x) for all a, b, c holds (a · b) · c = a · (b · c), (xi) for all a, b, c holds a · (b + c) = a · b + a · c, (xii) for all a, b, c holds (b + c) · a = b · a + c · a. A skew field is said to be a field if:

(Def.13) for all elements a, b of it holds a · b = b · a.

Let us note that it makes sense to consider the following constant. Then loop_{}is a field.

The following proposition is true (34)^{7} L is a field if and only if the following conditions are satisfied:

(i) for every a holds a + 0_{L} = a, (ii) for every a there exists x such that a + x = 0_{L}, (iii) for all a, b, c holds (a + b) + c = a + (b + c), (iv) for all a, b holds a + b = b + a,

(v) 0_{L} ≠ 1_{L}, (vi) for every a holds a · (1_{L}) = a, (vii) for every a such that a ≠ 0_{L} there exists x such that a · x = 1_{L}, (viii) for every a holds a · 0_{L} = 0_{L}, (ix) for all a, b, c holds (a · b) · c = a · (b · c),

(x) for all a, b, c holds a · (b + c) = a · b + a · c, (xi) for all a, b holds a · b = b · a.

References

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

[2] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

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

[7] Michał Muzalewski. Midpoint algebras. Formalized Mathematics, 1(3):483–488, 1990.

[8] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833–840, 1990.

[9] Wanda Szmielew. From Affine to Euclidean Geometry. Volume 27, PWN – D.Reidel Publ. Co., Warszawa – Dordrecht, 1983.

[10] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

7The proposition (33) was either repeated or obvious.

From Double Loops to Fields 191

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

1990.

Received September 27, 1990

Index of MML Identifiers

AFVECT0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 ALGSEQ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 ALGSTR 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 CARD 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 COMPLSP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 CONNSP 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 GROUP 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 HOMOTHET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 LIMFUNC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 LIMFUNC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 LIMFUNC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 LIMFUNC4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 LUKASI 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 NORMSP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 PREPOWER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 PROJDES1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 REALSET3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 REAL 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 SCHEME1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 SUPINF 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 SUPINF 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 TOLER 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 VECTSP 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 ZFMODEL2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Contents Form. Math. 2 (1)

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Construction of Rings and Left-, Right-, and Bi-Modules over a Ring

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Desargues Theorem In Projective 3-Space

By Eugeniusz Kusak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

The Limit of a Real Function at Infinity

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

One-Side Limits of a Real Function at a Point

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Lattice of Subgroups of a Group. Frattini Subgroup

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Equalities and Inequalities in Real Numbers

By Andrzej Kondracki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Countable Sets and Hessenberg’s Theorem

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

The Limit of a Real Function at a Point

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

The Limit of a Composition of Real Functions

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Locally Connected Spaces

By Beata Padlewska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Construction of Finite Sequences over Ring and Left-, Right-, and Bi-Modules over a Ring

By Michał Muzalewski et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Continued on inside back cover

Relations of Tolerance

By Krzysztof Hryniewiecki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Real Normed Space

By Jan Popiołek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Schemes of Existence of some Types of Functions

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Integer and Rational Exponents

By Konrad Raczkowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Homotheties and Shears in Affine Planes

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Directed Geometrical Bundles and Their Analytical Representation

By Grzegorz Lewandowski et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Definable Functions

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Propositional Calculus

By Grzegorz Bancerek et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Complex Spaces

By Czesław Byliński and Andrzej Trybulec . . . . . . . . . . . . 151

Several Properties of Fields. Field Theory

By Józef Białas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Infimum and Supremum of the Set of Real Numbers. Measure Theory

By Józef Białas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Series of Positive Real Numbers. Measure Theory

By Józef Białas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

From Double Loops to Fields

By Wojciech Skaba and Michał Muzalewski . . . . . . . . . . . . 185

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192