Preface

As was stated in [3] we publish mathematical papers which are abstracts of Mizar articles to be found in the Main Mizar Library (MML). An article includes certain elements which are transferred to the data base, such as theorems or definitions. This has been due to the fact that the material published there was at first intended to help the Mizar users to handle the data base. Thus the works published there describe the present state of MML and are, in a sense, a report on the expansion of that library. Next to them there are also new mathematical papers because the new method of formalization is not trivial even though it refers to simple mathematical facts.

It must be explained at this point that both the PC-Mizar verifier and MML are being systematically developed. In the case of PC-Mizar it is mainly the Mizar language which is enriched, which makes it more convenient to write articles; the same might be said of proof-checker, which enables one to write shorter proofs and articles.

The development of MML consists in continuous revisions of articles accepted for publication, for instance in the removal of self-evident or repeated theorems (while the numbering of successive theorems in a given article is preserved). We then have the information in a footnote such as ”The proposition (5) has been removed” (see [1], page 450). Previously such a comment was, e.g., ”The proposition (9) was either repeated or obvious” (see [2], page 14).

Please note also that in the articles we use atypical symbolism for the Cartesian product [: :] , and that is no paranthesis in the case of grouping to the left. We also use overloading. For instance, see [1], page 469: ”(Def.1) F (f) = F (f)”. In the latter case, on the right side of the equality symbol we have the old functor, while on the left side we have the new functor, which differs from the old one only by the type of the result.

Our periodical appears five times a year, which is to say every two months except for the summer holidays period. The present issue, although dated September-October, also includes items contributed in November. They have been included because the editors received them before sending the issue 2(4) to the press.

Roman Matuszewski

References

[1] Formalized Mathematics: a computer assisted approach. Volume 2(4), Université Catholique de Louvain, September-October 1991.

[2] Formalized Mathematics: a computer assisted approach. Volume 2(1), Université Catholique de Louvain, January-February 1991.

[3] Andrzej Trybulec. Introduction. Formalized Mathematics, 1(1):7–8, January 1990.

448

FORMALIZED MATHEMATICS Vol.2, No.4, September–October 1991 Université Catholique de Louvain

Serieses ^{1}

Konrad Raczkowski Warsaw University Białystok Andrzej Ne‘dzusiak Warsaw University Białystok

Summary. The article contains definitions and properties of convergent serieses.

MML Identifier: SERIES 1.

The articles [12], [2], [10], [1], [7], [6], [4], [3], [5], [11], [8], and [9] provide the notation and terminology for this paper. We follow the rules: n, m will denote natural numbers, a, p, r will denote real numbers, and s, s_{1}, s_{2} will denote sequences of real numbers. We now state three propositions:

(1) If 0 < a and a < 1 and for every n holds s(n) = a^{n+1}, then s is convergent and lim s = 0.

(2) If a ≠ 0, then |a|^{n} = |a^{n}|.

(3) If |a| < 1 and for every n holds s(n) = a^{n+1}, then s is convergent and lim s = 0.

Let us consider s. The functor (∑_{α=0}^{κ} s(α))_{κ∈} yielding a sequence of real numbers is defined by: (Def.1) (∑_{α=0}^{κ} s(α))_{κ∈} (0) = s(0) and for every n holds (∑_{α=0}^{κ} s(α))_{κ∈} (n +

1) = (∑_{α=0}^{κ} s(α))_{κ∈} (n) + s(n + 1). The following proposition is true

(4) For all s, s_{1} holds s_{1} = (∑_{α=0}^{κ} s(α))_{κ∈} if and only if s_{1}(0) = s(0) and for every n holds s_{1}(n + 1) = s_{1}(n) + s(n + 1).

Let us consider s. We say that s is summable if and only if: (Def.2) (∑_{α=0}^{κ} s(α))_{κ∈} is convergent.

Let us consider s. Let us assume that s is summable. The functor ∑ s yields a real number and is defined as follows: (Def.3) ∑ s = lim((∑_{α=0}^{κ} s(α))_{κ∈} ).

1Supported by RPBP.III-24.C8

450 konrad raczkowski and andrzej ne‘dzusiak

The following propositions are true: (6)^{2} For all s, r such that s is summable holds r = ∑ s if and only if

r = lim((∑_{α=0}^{κ} s(α))_{κ∈} ).

(7) If s is summable, then s is convergent and lim s = 0.

(8) (∑_{α=0}^{κ} s_{1}(α))_{κ∈} + (∑_{α=0}^{κ} s_{2}(α))_{κ∈} = (∑_{α=0}^{κ}(s_{1} + s_{2})(α))_{κ∈} .

(9) (∑_{α=0}^{κ} s_{1}(α))_{κ∈} − (∑_{α=0}^{κ} s_{2}(α))_{κ∈} = (∑_{α=0}^{κ}(s_{1} − s_{2})(α))_{κ∈} .

(10) If s_{1} is summable and s_{2} is summable, then s_{1} + s_{2} is summable and ∑(s_{1} + s_{2}) = ∑ s_{1} + ∑ s_{2}.

(11) If s_{1} is summable and s_{2} is summable, then s_{1} − s_{2} is summable and ∑(s_{1} − s_{2}) = ∑ s_{1} − ∑ s_{2}.

(12) (∑_{α=0}^{κ}(rs)(α))_{κ∈} = r(∑_{α=0}^{κ} s(α))_{κ∈} .

(13) If s is summable, then rs is summable and ∑(rs) = r · ∑ s.

(14) For all s, s_{1} such that for every n holds s_{1}(n) = s(0) holds (∑_{α=0}^{κ}(s ↑ 1)(α))_{κ∈} = (∑_{α=0}^{κ} s(α))_{κ∈} ↑ 1 − s_{1}.

(15) If s is summable, then for every n holds s ↑ n is summable.

(16) If there exists n such that s ↑ n is summable, then s is summable.

(17) If for every n holds s_{1}(n) ≤ s_{2}(n), then for every n holds (∑_{α=0}^{κ} s_{1}(α))_{κ∈} (n) ≤ (∑_{α=0}^{κ} s_{2}(α))_{κ∈} (n).

(18) If s is summable, then for every n holds ∑ s = (∑_{α=0}^{κ} s(α))_{κ∈} (n) + ∑(s ↑ (n + 1)).

(19) If for every n holds 0 ≤ s(n), then (∑_{α=0}^{κ} s(α))_{κ∈} is non-decreasing.

(20) If for every n holds 0 ≤ s(n), then (∑_{α=0}^{κ} s(α))_{κ∈} is upper bounded if and only if s is summable.

(21) If s is summable and for every n holds 0 ≤ s(n), then 0 ≤ ∑ s.

(22) If for every n holds 0 ≤ s_{2}(n) and s_{1} is summable and there exists m such that for every n such that m ≤ n holds s_{2}(n) ≤ s_{1}(n), then s_{2} is summable.

(23) If for every n holds 0 ≤ s_{2}(n) and s_{2} is not summable and there exists m such that for every n such that m ≤ n holds s_{2}(n) ≤ s_{1}(n), then s_{1} is not summable.

(24) If for every n holds 0 ≤ s_{1}(n) and s_{1}(n) ≤ s_{2}(n) and s_{2} is summable, then s_{1} is summable and ∑ s_{1} ≤ ∑ s_{2}.

(25) s is summable if and only if for every r such that 0 < r there exists n such that for every m such that n ≤ m holds |(∑_{α=0}^{κ} s(α))_{κ∈} (m) − (∑_{α=0}^{κ} s(α))_{κ∈} (n)| < r.

(26) If a ≠ 1, then (∑_{α=0}^{κ}((a^{κ})_{κ∈} )(α))_{κ∈} (n) = \frac{1−a^{n+1}}{1−a}.

(27) If a ≠ 1 and for every n holds s(n + 1) = a · s(n), then for every n holds (∑_{α=0}^{κ} s(α))_{κ∈} (n) = \frac{s(0)·(1−a^{n+1})}{1−a}.

(28) If |a| < 1, then (a^{κ})_{κ∈} is summable and ∑((a^{κ})_{κ∈} ) = \frac{1}{1−a}.

2The proposition (5) has been removed.

(29) If |a| < 1 and for every n holds s(n + 1) = a · s(n), then s is summable and ∑ s = \frac{s(0)}{1−a}.

(30) If for every n holds s(n) > 0 and s_{1}(n) = \frac{s(n+1)}{s(n)} and s_{1} is convergent and lim s_{1} < 1, then s is summable.

(31) If for every n holds s(n) > 0 and there exists m such that for every n such that n ≥ m holds \frac{s(n+1)}{s(n)} ≥ 1, then s is not summable.

(32) If for every n holds s(n) ≥ 0 and s_{1}(n) = ^{n} \sqrt{s(n)} and s_{1} is convergent and lim s_{1} < 1, then s is summable.

(33) If for every n holds s(n) ≥ 0 and s_{1}(n) = ^{n} \sqrt{s(n)} and there exists m such that for every n such that m ≤ n holds s_{1}(n) ≥ 1, then s is not summable.

(34) If for every n holds s(n) ≥ 0 and s_{1}(n) = ^{n} \sqrt{s(n)} and s_{1} is convergent

and lim s_{1} > 1, then s is not summable. Let us consider n. The n-th power of 2 yields a natural number and is defined

as follows: (Def.4) the n-th power of 2= 2^{n}.

One can prove the following three propositions:

(35) If s is non-increasing and for every n holds s(n) ≥ 0 and s_{1}(n) = 2^{n}·s(the n-th power of 2), then s is summable if and only if s_{1} is summable.

(36) If p > 1 and for every n such that n ≥ 1 holds s(n) = \frac{1}{n^{p}}, then s is summable.

(37) If p ≤ 1 and for every n such that n ≥ 1 holds s(n) = \frac{1}{n^{p}}, then s is not

summable. Let us consider s. We say that s is absolutely summable if and only if:

(Def.5) |s| is summable.

We now state several propositions: (39)^{3} For all n, m such that n ≤ m holds |(∑_{α=0}^{κ} s(α))_{κ∈} (m)−

(∑_{α=0}^{κ} s(α))_{κ∈} (n)| ≤ |(∑_{α=0}^{κ} |s|(α))_{κ∈} (m) − (∑_{α=0}^{κ} |s|(α))_{κ∈} (n)|.

(40) If s is absolutely summable, then s is summable.

(41) If for every n holds 0 ≤ s(n) and s is summable, then s is absolutely summable.

(42) If for every n holds s(n) ≠ 0 and s_{1}(n) = \frac{|s|(n+1)}{|s|(n)} and s_{1} is convergent and lim s_{1} < 1, then s is absolutely summable.

(43) If r > 0 and there exists m such that for every n such that n ≥ m holds |s(n)| ≥ r, then s is not convergent or lim s ≠ 0.

(44) If for every n holds s(n) ≠ 0 and there exists m such that for every n such that n ≥ m holds \frac{|s|(n+1)}{|s|(n)} ≥ 1, then s is not summable.

3The proposition (38) has been removed.

452 konrad raczkowski and andrzej ne‘dzusiak

(45) If for every n holds s_{1}(n) = ^{n} \sqrt{|s|(n)} and s_{1} is convergent and lim s_{1} < 1, then s is absolutely summable.

(46) If for every n holds s_{1}(n) = ^{n} \sqrt{|s|(n)} and there exists m such that for every n such that m ≤ n holds s_{1}(n) ≥ 1, then s is not summable.

(47) If for every n holds s_{1}(n) = ^{n} \sqrt{|s|(n)} and s_{1} is convergent and lim s_{1} > 1, then s is not summable.

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

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

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

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

[8] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125– 130, 1991.

[9] Konrad Raczkowski and Andrzej Ne‘dzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

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

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

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

The Lattice of Natural Numbers and The Sublattice of it. The Set of Prime Numbers

Marek Chmur Warsaw University Białystok

Summary. Basic properties of the least common multiple and the greatest common divisor. The lattice of natural numbers (L_{}) and the lattice of natural numbers greater than zero (L_{^{+}} ) are constructed. The notion of the sublattice of the lattice of natural numbers is given. Some fact about it are proved. The last part of the article deals with some properties of prime numbers and with the notions of the set of prime numbers and the n-th prime number. It is proved that the set of prime numbers is infinite.

MML Identifier: NAT LAT.

The papers [15], [6], [18], [14], [7], [17], [9], [1], [11], [2], [16], [12], [5], [4], [8], [13], [10], and [3] provide the terminology and notation for this paper. In the sequel n, m, l, k, j will be natural numbers. We now state two propositions:

(1) For all natural numbers m, n holds m | m · n and n | m · n.

(2) For all k, l such that l ≥ 1 holds k · l ≥ k. Let us consider n. Then n! is a natural number. The following propositions are true:

(3) For all n, k, l such that l ≥ 1 holds if n ≥ k · l, then n ≥ k.

(4) k = 0 or k ≥ 1.

(5) For every l such that l ≠ 0 holds l | l!.

(6) k ≠ k + 1. (8)^{1} For every n such that n ≠ 0 holds \frac{n+1}{n} > 1.

(9) \frac{k}{k+1} < 1.

1The proposition (7) has been removed.

(10) For every l holds l! ≥ l. (12)^{2} For all m, n such that m ≠ 1 holds if m | n, then m n + 1.

(13) j | l and j | l + 1 if and only if j = 1.

(14) For every l there exists j such that j | l!.

(15) For all k, j such that j ≠ 0 holds j | (j + k)!.

(16) If j ≤ l and j ≠ 0, then j | l!.

(17) For all l, j such that j ≠ 1 and j ≠ 0 holds if j | l! + 1, then j > l.

(18) For all natural numbers m, n holds lcm(m, n) = lcm(n, m).

(19) For all natural numbers m, n, k holds lcm(m, lcm(n, k)) = lcm(lcm(m, n), k).

(20) For all natural numbers m, n holds m | n if and only if lcm(m, n) = n.

(21) m | lcm(m, n) and n | lcm(m, n).

(22) lcm(m, m) = m.

(23) n | m and k | m if and only if lcm(n, k) | m.

(24) lcm(m, n) | 0.

(25) 1 | lcm(m, n).

(26) lcm(m, 1) = m.

(27) lcm(m, n) | m · n.

(28) For all natural numbers m, n, k holds gcd(m, gcd(n, k)) = gcd(gcd(m, n), k).

(29) gcd(m, n) | m and gcd(m, n) | n.

(30) For all natural numbers m, n such that n | m holds gcd(n, m) = n.

(31) gcd(m, m) = m.

(32) m | n and m | k if and only if m | gcd(n, k).

(33) gcd(m, n) | 0.

The following propositions are true:

(34) 1 | gcd(m, n).

(35) gcd(m, 1) = 1.

(36) gcd(m, 0) = m.

(37) For all natural numbers m, n holds lcm(gcd(m, n), n) = n.

(38) For all natural numbers m, n holds gcd(m, lcm(m, n)) = m.

(39) For all natural numbers m, n holds gcd(m, lcm(m, n)) = lcm(gcd(n, m), m).

(40) If m | n, then gcd(m, k) | gcd(n, k).

(41) If m | n, then gcd(k, m) | gcd(k, n).

(42) For every m such that m > 0 holds gcd(0, m) > 0.

(43) For all m, n such that m > 0 and n > 0 holds gcd(n, m) > 0.

(44) For all m, n such that m > 0 and n > 0 holds lcm(m, n) > 0.

2The proposition (11) has been removed.

(45) lcm(gcd(n, m), gcd(n, k)) | gcd(n, lcm(m, k)).

(46) For all m, n, l such that m | l holds lcm(m, gcd(n, l)) | gcd(lcm(m, n), l).

(47) gcd(n, m) | lcm(n, m).

Let m be an element of qua a non-empty set. The functor ^{@}m yielding a natural number is defined by: (Def.1) ^{@}m = m.

Let m be a natural number. The functor ^{@}m yielding an element of qua a non-empty set is defined as follows: (Def.2) ^{@}m = m.

We now define two new functors. The binary operation hcf_{}on is defined by: (Def.3) hcf_{}(m, n) = gcd(m, n). The binary operation lcm_{}on is defined by: (Def.4) lcm_{}(m, n) = lcm(m, n).

In the sequel p, q denote elements of the carrier of ⟨ , lcm_{}, hcf_{}⟩. Let m be an element of the carrier of ⟨ , lcm_{}, hcf_{}⟩. The functor ^{@}m yielding a natural number is defined as follows: (Def.5) ^{@}m = m.

We now state several propositions:

(48) p ⊔ q = lcm(^{@}p,^{@}q).

(49) p ⊓ q = gcd(^{@}p,^{@}q).

(50) lcm_{}(p, q) = p ⊔ q.

(51) hcf_{}(p, q) = p ⊓ q.

(52) For all elements a, b of the carrier of ⟨ , lcm_{}, hcf_{}⟩ such that a ⊑ b

holds ^{@}a | ^{@}b. The element 0_{_{}}of the carrier of ⟨ , lcm_{}, hcf_{}⟩ is defined as follows:

(Def.6) 0_{_{}}= 1.

The element 1_{_{}}of the carrier of ⟨ , lcm_{}, hcf_{}⟩ is defined by: (Def.7) 1_{_{}}= 0.

We now state three propositions: (55)^{3} ^{@}(0_{_{}}) = 1.

(56) For every element a of the carrier of ⟨ , lcm_{}, hcf_{}⟩ holds 0_{_{}}⊓a = 0_{_{}}.

(57) There exists an element z of the carrier of ⟨ , lcm_{}, hcf_{}⟩ such that for

every element x of the carrier of ⟨ , lcm_{}, hcf_{}⟩ holds z ⊓ x = z. The lattice _{}is defined by:

(Def.8) _{}= ⟨ , lcm_{}, hcf_{}⟩.

The following proposition is true

(58) _{}= ⟨ , lcm_{}, hcf_{}⟩.

3The propositions (53) and (54) have been removed.

In the sequel p, q, r will denote elements of the carrier of _{}. One can prove the following propositions: (60)^{4} _{}is a lower bound lattice.

(61) lcm_{}(p, q) = lcm_{}(q, p).

(62) hcf_{}(q, p) = hcf_{}(p, q).

(63) lcm_{}(p, lcm_{}(q, r)) = lcm_{}(lcm_{}(p, q), r).

(64) (i) lcm_{}(p, lcm_{}(q, r)) = lcm_{}(lcm_{}(q, p), r), (ii) lcm_{}(p, lcm_{}(q, r)) = lcm_{}(lcm_{}(p, r), q), (iii) lcm_{}(p, lcm_{}(q, r)) = lcm_{}(lcm_{}(r, q), p), (iv) lcm_{}(p, lcm_{}(q, r)) = lcm_{}(lcm_{}(r, p), q).

(65) hcf_{}(p, hcf_{}(q, r)) = hcf_{}(hcf_{}(p, q), r).

(66) (i) hcf_{}(p, hcf_{}(q, r)) = hcf_{}(hcf_{}(q, p), r), (ii) hcf_{}(p, hcf_{}(q, r)) = hcf_{}(hcf_{}(p, r), q), (iii) hcf_{}(p, hcf_{}(q, r)) = hcf_{}(hcf_{}(r, q), p), (iv) hcf_{}(p, hcf_{}(q, r)) = hcf_{}(hcf_{}(r, p), q).

(67) hcf_{}(q, lcm_{}(q, p)) = q and hcf_{}(lcm_{}(p, q), q) = q and hcf_{}(q, lcm_{}(p, q)) = q and hcf_{}(lcm_{}(q, p), q) = q.

(68) lcm_{}(q, hcf_{}(q, p)) = q and lcm_{}(hcf_{}(p, q), q) = q and lcm_{}(q, hcf_{}(p,

q)) = q and lcm_{}(hcf_{}(q, p), q) = q. The subset ^{+} of is defined by:

(Def.9) for every natural number n holds n ∈ ^{+} if and only if 0 < n.

Let D be a non-empty set, and let S be a non-empty subset of D, and let N be a non-empty subset of S. We see that the element of N is an element of S.

A positive natural number is an element of ^{+}.

Let k be a natural number satisfying the condition: k > 0. The functor ^{@}k yields an element of ^{+} qua a non-empty set and is defined by:

(Def.10) ^{@}k = k.

Let k be an element of ^{+} qua a non-empty set. The functor ^{@}k yields a positive natural number and is defined as follows:

(Def.11) ^{@}k = k.

In the sequel m, n denote positive natural numbers. We now define two new functors. The binary operation hcf_{^{+}} on ^{+} is defined by:

(Def.12) hcf_{^{+}} (m, n) = gcd(m, n).

The binary operation lcm_{^{+}} on ^{+} is defined as follows: (Def.13) lcm_{^{+}} (m, n) = lcm(m, n).

In the sequel p, q will denote elements of the carrier of ⟨ ^{+}, lcm_{^{+}} , hcf_{^{+}} ⟩. Let m be an element of the carrier of ⟨ ^{+}, lcm_{^{+}} , hcf_{^{+}} ⟩. The functor^{@}m yields a positive natural number and is defined as follows:

(Def.14) ^{@}m = m.

4The proposition (59) has been removed.

One can prove the following four propositions:

(69) p ⊔ q = lcm(^{@}p,^{@}q).

(70) p ⊓ q = gcd(^{@}p,^{@}q).

(71) lcm_{^{+}} (p, q) = p ⊔ q.

(72) hcf_{^{+}} (p, q) = p ⊓ q.

The lattice _{^{+}} is defined by: (Def.15) _{^{+}} = ⟨ ^{+}, lcm_{^{+}} , hcf_{^{+}} ⟩.

Next we state the proposition

(73) _{^{+}} = ⟨ ^{+}, lcm_{^{+}} , hcf_{^{+}} ⟩.

Let L be a lattice. A lattice is said to be a sublattice of L if: (Def.16) the carrier of it ⊆ the carrier of L and the join operation of it = (the

join operation of L) [: the carrier of it, the carrier of it :] and the meet operation of it = (the meet operation of L) [: the carrier of it, the carrier of it :].

The following two propositions are true: (75)^{5} For every lattice L holds L is a sublattice of L.

(76) _{^{+}} is a sublattice of _{}.

In the sequel n, i, k, k_{1}, k_{2}, m, l will denote natural numbers. The set Prime of natural numbers is defined as follows:

(Def.17) for every natural number n holds n ∈ Prime if and only if n is prime.

A natural number is said to be a prime number if: (Def.18) it ∈ Prime.

In the sequel p, q denote prime numbers and f denotes a prime number. Let us consider p. The functor Prime(p) yields sets of natural numbers and is defined by:

(Def.19) for every natural number q holds q ∈ Prime(p) if and only if q < p and

q is prime. Next we state a number of propositions:

(77) Prime(p) ⊆ Prime.

(78) For every prime number q such that p < q holds Prime(p) ⊆ Prime(q).

(79) Prime(p) ⊆ Seg p.

(80) Prime(p) is finite.

(81) For every l there exists p such that p is prime and p > l.

(82) For every q such that q is prime there exists p such that p is prime and p > q.

(83) Prime ⊆ .

(84) Prime ≠ ∅.

(85) {k : k < 2 ∧ k is prime} = ∅.

5The proposition (74) has been removed.

458 marek chmur

(86) For every p holds {k : k < p ∧ k is prime} ⊆ .

(87) For every m holds {k : k < m ∧ k is prime} ⊆ Seg m.

(88) For every m holds {k : k < m ∧ k is prime} is finite.

(89) For every prime number f holds f ∉ {k : k < f ∧ k is prime}.

(90) For every f holds {k : k < f ∧ k is prime} ∪ {f} is finite.

(91) For all prime numbers f, g such that f < g holds {k_{1} : k_{1} < f ∧ k_{1} is prime} ∪ {f} ⊆ {k_{2} : k_{2} < g ∧ k_{2} is prime}.

(92) For every k such that k > m holds k ∉ {k_{1} : k_{1} < m ∧ k_{1} is prime}. Let us consider n. The functor pr(n) yielding a prime number is defined as

follows: (Def.20) n = card{k : k < pr(n) ∧ k is prime}.

One can prove the following two propositions:

(93) Prime(p) = {k : k < p ∧ k is prime}.

(94) Prime is not finite.

The following proposition is true

(95) For every i such that i is prime for all m, n such that i | m · n holds i | m or i | n.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

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

[3] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

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

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

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

[7] Marek Chmur. The lattice of real numbers. The lattice of real functions. Formalized Mathematics, 1(4):681–684, 1990.

[8] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

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

[12] Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97–104, 1991.

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

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

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

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

the lattice of natural numbers and the … 459

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

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

222, 1990.

Received April 26, 1991 460

Commutator and Center of a Group

Wojciech A. Trybulec Warsaw University

Summary. We introduce the notions of commutators of element, subgroups of a group, commutator of a group and center of a group. We prove P.Hall identity. The article is based on [6].

MML Identifier: GROUP 5.

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

Preliminaries

The scheme SubsetFD3 concerns a non-empty set A, a non-empty set B, a non-empty set C, a ternary functor F yielding an element of B, and a ternary predicate P, and states that:

{F(c, d, e) : P[c, d, e]}, where c ranges over elements of A, and d ranges over elements of B, and e ranges over elements of C, is a subset of B for all values of the parameters.

For simplicity we adopt the following rules: x will be arbitrary, k, n will denote natural numbers, i will denote an integer, G will denote a group, a, b, c, d will denote elements of G, A, B, C, D will denote subsets of G, H, H_{1}, H_{2}, H_{3}, H_{4} will denote subgroups of G, N, N_{1}, N_{2}, N_{3} will denote normal subgroups of G, F, F_{1}, F_{2} will denote finite sequences of elements of the carrier of G, and I will denote a finite sequence of elements of . Next we state several propositions:

(1) x ∈ {1}_{G} if and only if x = 1_{G}.

(2) If a ∈ H and b ∈ H, then a^{b} ∈ H.

(3) If a ∈ N, then a^{b} ∈ N.

(4) x ∈ H_{1} · H_{2} if and only if there exist a, b such that x = a · b and a ∈ H_{1} and b ∈ H_{2}.

(5) If H_{1} · H_{2} = H_{2} · H_{1}, then x ∈ H_{1} ⊔ H_{2} if and only if there exist a, b such that x = a · b and a ∈ H_{1} and b ∈ H_{2}.

(6) If G is an Abelian group, then x ∈ H_{1} ⊔ H_{2} if and only if there exist a, b such that x = a · b and a ∈ H_{1} and b ∈ H_{2}.

(7) x ∈ N_{1} ⊔ N_{2} if and only if there exist a, b such that x = a · b and a ∈ N_{1} and b ∈ N_{2}.

(8) H · N = N · H.

Let us consider G, F, a. The functor F^{a} yielding a finite sequence of elements of the carrier of G is defined by: (Def.1) len(F^{a}) = len F and for every k such that k ∈ Seg len F holds F^{a}(k) =

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

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

(10) len(F^{a}) = len F.

(11) For every k such that k ∈ Seg len F holds F^{a}(k) = (π_{k}F)^{a}.

(12) (F_{1}^{a})^{}F_{2}^{a} = (F_{1} ^{}F_{2})^{a}.

(13) ε_{(the carrier of G)}^{a} = ε.

(14) ⟨a⟩^{b} = ⟨a^{b}⟩.

(15) ⟨a, b⟩^{c} = ⟨a^{c}, b^{c}⟩.

(16) ⟨a, b, c⟩^{d} = ⟨a^{d}, b^{d}, c^{d}⟩.

(17) ∏(F^{a}) = (∏ F)^{a}.

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

Commutators

Let us consider G, a, b. The functor [a, b] yields an element of G and is defined by: (Def.2) [a, b] = a^{−1} · b^{−1} · a · b.

One can prove the following propositions:

(19) (i) [a, b] = a^{−1} · b^{−1} · a · b, (ii) [a, b] = a^{−1} · (b^{−1} · a) · b, (iii) [a, b] = a^{−1} · (b^{−1} · a · b), (iv) [a, b] = a^{−1} · (b^{−1} · (a · b)),

(v) [a, b] = a^{−1} · b^{−1} · (a · b).

(20) [a, b] = (b · a)^{−1} · (a · b).

(21) [a, b] = (b^{−1})^{a} · b and [a, b] = a^{−1} · a^{b}.

(22) [1_{G}, a] = 1_{G} and [a, 1_{G}] = 1_{G}.

(23) [a, a] = 1_{G}.

(24) [a, a^{−1}] = 1_{G} and [a^{−1}, a] = 1_{G}.

(25) [a, b]^{−1} = [b, a].

(26) [a, b]^{c} = [a^{c}, b^{c}].

(27) [a, b] = (a^{−1})^{2} · (a · b^{−1})^{2} · b^{2}.

(28) [a · b, c] = [a, c]^{b} · [b, c].

(29) [a, b · c] = [a, c] · [a, b]^{c}.

(30) [a^{−1}, b] = [b, a]^{a^{−1}}.

(31) [a, b^{−1}] = [b, a]^{b^{−1}}.

(32) [a^{−1}, b^{−1}] = [a, b]^{(a·b)^{−1}} and [a^{−1}, b^{−1}] = [a, b]^{(b·a)^{−1}}.

(33) [a, b^{a^{−1}}] = [b, a^{−1}].

(34) [a^{b^{−1}}, b] = [b^{−1}, a].

(35) [a^{n}, b] = a^{−n} · (a^{b})^{n}.

(36) [a, b^{n}] = (b^{a})^{−n} · b^{n}.

(37) [a^{i}, b] = a^{−i} · (a^{b})^{i}.

(38) [a, b^{i}] = (b^{a})^{−i} · b^{i}.

(39) [a, b] = 1_{G} if and only if a · b = b · a.

(40) G is an Abelian group if and only if for all a, b holds [a, b] = 1_{G}.

(41) If a ∈ H and b ∈ H, then [a, b] ∈ H.

Let us consider G, a, b, c. The functor [a, b, c] yielding an element of G is defined by: (Def.3) [a, b, c] = [[a, b], c].

One can prove the following propositions:

(42) [a, b, c] = [[a, b], c].

(43) [a, b, 1_{G}] = 1_{G} and [a, 1_{G}, b] = 1_{G} and [1_{G}, a, b] = 1_{G}.

(44) [a, a, b] = 1_{G}.

(45) [a, b, a] = [a^{b}, a].

(46) [b, a, a] = ([b, a^{−1}] · [b, a])^{a}.

(47) [a, b, b^{a}] = [b, [b, a]].

(48) [a · b, c] = [a, c] · [a, c, b] · [b, c].

(49) [a, b · c] = [a, c] · [a, b] · [a, b, c].

(50) [a, b^{−1}, c]^{b} · [b, c^{−1}, a]^{c} · [c, a^{−1}, b]^{a} = 1_{G}.

Let us consider G, A, B. The commutators of A & B yielding a subset of G is defined as follows: (Def.4) the commutators of A & B = {[a, b] : a ∈ A ∧ b ∈ B}.

We now state several propositions:

(51) The commutators of A & B = {[a, b] : a ∈ A ∧ b ∈ B}.

(52) x ∈ the commutators of A & B if and only if there exist a, b such that x = [a, b] and a ∈ A and b ∈ B.

(53) The commutators of ∅_{the carrier of G} & A = ∅ and the commutators of A & ∅_{the carrier of G} = ∅.

(54) The commutators of {a} & {b} = {[a, b]}.

(55) If A ⊆ B and C ⊆ D, then the commutators of A & C ⊆ the commutators of B & D.

(56) G is an Abelian group if and only if for all A, B such that A ≠ ∅ and

B ≠ ∅ holds the commutators of A & B = {1_{G}}. Let us consider G, H_{1}, H_{2}. The commutators of H_{1} & H_{2} yields a subset of

G and is defined by: (Def.5) the commutators of H_{1} & H_{2} = the commutators of \overline{H_{1}} & \overline{H_{2}}.

Next we state several propositions:

(57) The commutators of H_{1} & H_{2} = the commutators of \overline{H_{1}} & \overline{H_{2}}.

(58) x ∈ the commutators of H_{1} & H_{2} if and only if there exist a, b such that x = [a, b] and a ∈ H_{1} and b ∈ H_{2}.

(59) 1_{G} ∈ the commutators of H_{1} & H_{2}.

(60) The commutators of {1}_{G} & H = {1_{G}} and the commutators of H & {1}_{G} = {1_{G}}.

(61) The commutators of H & N ⊆ \overline{N} and the commutators of N & H ⊆ \overline{N}.

(62) If H_{1} is a subgroup of H_{2} and H_{3} is a subgroup of H_{4}, then the commutators of H_{1} & H_{3} ⊆ the commutators of H_{2} & H_{4}.

(63) G is an Abelian group if and only if for all H_{1}, H_{2} holds the commutators

of H_{1} & H_{2} = {1_{G}}. Let us consider G. The commutators of G yielding a subset of G is defined

by: (Def.6) the commutators of G = the commutators of Ω_{G} & Ω_{G}.

Next we state three propositions:

(64) The commutators of G = the commutators of Ω_{G} & Ω_{G}.

(65) x ∈ the commutators of G if and only if there exist a, b such that x = [a, b].

(66) G is an Abelian group if and only if the commutators of G = {1_{G}}. Let us consider G, A, B. The functor [A, B] yielding a subgroup of G is

defined as follows: (Def.7) [A, B] = gr(the commutators of A & B).

Next we state four propositions:

(67) [A, B] = gr(the commutators of A & B).

(68) If a ∈ A and b ∈ B, then [a, b] ∈ [A, B].

(69) x ∈ [A, B] if and only if there exist F, I such that len F = len I and rng F ⊆ the commutators of A & B and x = ∏(F^{I}).

(70) If A ⊆ C and B ⊆ D, then [A, B] is a subgroup of [C, D].

Let us consider G, H_{1}, H_{2}. The functor [H_{1}, H_{2}] yielding a subgroup of G is defined by: (Def.8) [H_{1}, H_{2}] = [\overline{H_{1}}, \overline{H_{2}}].

Next we state a number of propositions:

(71) [H_{1}, H_{2}] = [\overline{H_{1}}, \overline{H_{2}}].

(72) [H_{1}, H_{2}] = gr(the commutators of H_{1} & H_{2}).

(73) x ∈ [H_{1}, H_{2}] if and only if there exist F, I such that len F = len I and rng F ⊆ the commutators of H_{1} & H_{2} and x = ∏(F^{I}).

(74) If a ∈ H_{1} and b ∈ H_{2}, then [a, b] ∈ [H_{1}, H_{2}].

(75) If H_{1} is a subgroup of H_{2} and H_{3} is a subgroup of H_{4}, then [H_{1}, H_{3}] is a subgroup of [H_{2}, H_{4}].

(76) [N, H] is a subgroup of N and [H, N] is a subgroup of N.

(77) [N_{1}, N_{2}] is a normal subgroup of G.

(78) [N_{1}, N_{2}] = [N_{2}, N_{1}].

(79) [N_{1} ⊔ N_{2}, N_{3}] = [N_{1}, N_{3}] ⊔ [N_{2}, N_{3}].

(80) [N_{1}, N_{2} ⊔ N_{3}] = [N_{1}, N_{2}] ⊔ [N_{1}, N_{3}].

Let us consider G. The functor G^{c} yields a normal subgroup of G and is defined by: (Def.9) G^{c} = [Ω_{G}, Ω_{G}].

Next we state several propositions:

(81) G^{c} = [Ω_{G}, Ω_{G}].

(82) G^{c} = gr(the commutators of G).

(83) x ∈ G^{c} if and only if there exist F, I such that len F = len I and rng F ⊆ the commutators of G and x = ∏(F^{I}).

(84) [a, b] ∈ G^{c}.

(85) G is an Abelian group if and only if G^{c} = {1}_{G}.

(86) If the left cosets of H is finite and |• : H|_{}= 2, then G^{c} is a subgroup of H.

Center of a Group

Let us consider G. The functor Z(G) yielding a subgroup of G is defined as follows:

(Def.10) the carrier of Z(G) = {a : ⋀_{b} a · b = b · a}.

We now state several propositions:

(87) If the carrier of H = {a : ⋀_{b} a · b = b · a}, then H = Z(G).

(88) The carrier of Z(G) = {a : ⋀_{b} a · b = b · a}.

(89) a ∈ Z(G) if and only if for every b holds a · b = b · a.

(90) Z(G) is a normal subgroup of G.

(91) If H is a subgroup of Z(G), then H is a normal subgroup of G.

(92) Z(G) is an Abelian group.

(93) a ∈ Z(G) if and only if a^{•} = {a}.

(94) G is an Abelian group if and only if Z(G) = G.

466 wojciech a. trybulec

Auxiliary theorems

In the sequel E will be a non-empty set and p, q will be finite sequences of elements of E. The following propositions are true:

(95) If k ∈ dom p or k ∈ Seg len p, then π_{k}(p^{}q) = π_{k}p.

(96) If k ∈ dom q or k ∈ Seg len q, then π_{len p+k}(p^{}q) = π_{k}q.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

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

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

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

[5] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 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] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

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

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

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

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

[13] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991.

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

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

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

Received May 15, 1991

Natural Transformations. Discrete Categories

Andrzej Trybulec Warsaw University Białystok

Summary. We present well known concepts of category theory: natural transofmations and functor categories, and prove propositions related to. Because of the formalization it proved to be convenient to introduce some auxiliary notions, for instance: transformations. We mean by a transformation of a functor F to a functor G, both covariant functors from A to B, a function mapping the objects of A to the morphisms of B and assigning to an object a of A an element of Hom(F (a), G(a)). The material included roughly corresponds to that presented on pages 18,129–130,137–138 of the monography ([10]). We also introduce discrete categories and prove some propositions to illustrate the concepts introduced.

MML Identifier: NATTRA 1.

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

Preliminaries

For simplicity we follow a convention: A_{1}, A_{2}, B_{1}, B_{2} are non-empty sets, f is a function from A_{1} into B_{1}, g is a function from A_{2} into B_{2}, Y_{1} is a non-empty subset of A_{1}, and Y_{2} is a non-empty subset of A_{2}. Let A_{1}, A_{2} be non-empty sets, and let Y_{1} be a non-empty subset of A_{1}, and let Y_{2} be a non-empty subset of A_{2}. Then [: Y_{1}, Y_{2} :] is a non-empty subset of [: A_{1}, A_{2} :].

Let us consider A_{1}, B_{1}, f, Y_{1}. Then f Y_{1} is a function from Y_{1} into B_{1}. We now state the proposition

(1) [: f, g:] [: Y_{1}, Y_{2} :] = [: f Y_{1}, g Y_{2} :].

Let A, B be non-empty sets, and let A_{1} be a non-empty subset of A, and let B_{1} be a non-empty subset of B, and let f be a partial function from [: A_{1}, A_{1} :] to A_{1}, and let g be a partial function from [: B_{1}, B_{1} :] to B_{1}. Then |:f, g:| is a partial function from [: [: A_{1}, B_{1} :], [: A_{1}, B_{1} :] :] to [: A_{1}, B_{1} :].

One can prove the following proposition

(2) Let f be a partial function from [: A_{1}, A_{1} :] to A_{1}. Let g be a partial function from [: A_{2}, A_{2} :] to A_{2}. Then for every partial function F from [: Y_{1}, Y_{1} :] to Y_{1} such that F = f [: Y_{1}, Y_{1} :] for every partial function G from [: Y_{2}, Y_{2} :] to Y_{2} such that G = g [: Y_{2}, Y_{2} :] holds |:F, G:| = |:f, g:| [: [: Y_{1}, Y_{2} :], [: Y_{1}, Y_{2} :] :].

We adopt the following convention: A, B, C will be categories, F, F_{1}, F_{2}, F_{3} will be functors from A to B, and G will be a functor from B to C. In this article we present several logical schemes. The scheme M Choice deals with a set A, a set B, and a unary functor F yielding a set and states that:

there exists a function t from A into B such that for every element a of A holds t(a) ∈ F(a) provided the following requirement is met:

• for every element a of A holds B meets F(a). The scheme LambdaT concerns a 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 following requirement is met:

• for every element x of A holds F(x) ∈ B. We now state the proposition

(3) For every object a of A and for every morphism m from a to a holds m ∈ hom(a, a).

In the sequel m, o will be arbitrary. One can prove the following propositions:

(4) For all morphisms f, g of ˙ (o, m) holds f = g.

(5) For every object a of A holds ⟨⟨⟨⟨ id_{a}, id_{a} ⟩⟩, id_{a} ⟩⟩ ∈ the composition of A.

(6) The composition of ˙ (o, m) = {⟨⟨⟨⟨m, m⟩⟩, m⟩⟩}.

(7) For every object a of A holds ˙ (a, id_{a}) is a subcategory of A.

(8) For every subcategory C of A holds the dom-map of C = (the dom-map of A) the morphisms of C and the cod-map of C = (the cod-map of A) the morphisms of C and the composition of C = (the composition of A) [: the morphisms of C, the morphisms of C:] and the id-map of C = (the id-map of A) the objects of C.

(9) Let O be a non-empty subset of the objects of A. Let M be a non-empty subset of the morphisms of A. Let D_{1}, C_{1} be functions from M into O. Suppose D_{1} = (the dom-map of A) M and C_{1} = (the cod-map of A) M. Then for every partial function C_{2} from [: M, M qua a non-empty set :] to M such that C_{2} = (the composition of A) [: M, M :] for every function I_{1} from O into M such that I_{1} = (the id-map of A) O holds ⟨O, M, D_{1}, C_{1}, C_{2}, I_{1}⟩ is a subcategory of A.

(10) For every subcategory A of C such that the objects of A = the objects of C and the morphisms of A = the morphisms of C holds A = C.

Application of a functor to a morphism

Let us consider A, B, F, and let a, b be objects of A satisfying the condition: hom(a, b) ≠ ∅. Let f be a morphism from a to b. The functor F (f) yields a morphism from F (a) to F (b) and is defined by: (Def.1) F (f) = F (f).

One can prove the following propositions:

(11) For all objects a, b of A such that hom(a, b) ≠ ∅ for every morphism f from a to b holds (G · F)(f) = G(F (f)).

(12) For all functors F_{1}, F_{2} from A to B such that for all objects a, b of A such that hom(a, b) ≠ ∅ for every morphism f from a to b holds F_{1}(f) = F_{2}(f) holds F_{1} = F_{2}.

(13) For all objects a, b, c of A such that hom(a, b) ≠ ∅ and hom(b, c) ≠ ∅ for every morphism f from a to b and for every morphism g from b to c holds F (g · f) = F (g) · F (f).

(14) For every object c of A and for every object d of B such that F (id_{c}) = id_{d} holds F (c) = d.

(15) For every object a of A holds F (id_{a}) = id_{F (a)}.

(16) For all objects a, b of A such that hom(a, b) ≠ ∅ for every morphism f from a to b holds id_{A}(f) = f.

(17) For all objects a, b, c, d of A such that hom(a, b) meets hom(c, d) holds a = c and b = d.

Transformations

Let us consider A, B, F_{1}, F_{2}. We say that F_{1} is transformable to F_{2} if and only if: (Def.2) for every object a of A holds hom(F_{1}(a), F_{2}(a)) ≠ ∅.

One can prove the following propositions:

(18) F is transformable to F.

(19) If F is transformable to F_{1} and F_{1} is transformable to F_{2}, then F is

transformable to F_{2}. Let us consider A, B, F_{1}, F_{2}. Let us assume that F_{1} is transformable to

F_{2}. A function from the objects of A into the morphisms of B is said to be a transformation from F_{1} to F_{2} if: (Def.3) for every object a of A holds it(a) is a morphism from F_{1}(a) to F_{2}(a).

Let us consider A, B, and let F be a functor from A to B. The functor id_{F} yields a transformation from F to F and is defined as follows: (Def.4) for every object a of A holds id_{F} (a) = id_{F (a)}.

Let us consider A, B, F_{1}, F_{2}. Let us assume that F_{1} is transformable to F_{2}. Let t be a transformation from F_{1} to F_{2}, and let a be an object of A. The functor t(a) yields a morphism from F_{1}(a) to F_{2}(a) and is defined by: (Def.5) t(a) = t(a).

Let us consider A, B, F, F_{1}, F_{2}. Let us assume that F is transformable to F_{1} and F_{1} is transformable to F_{2}. Let t_{1} be a transformation from F to F_{1}, and let t_{2} be a transformation from F_{1} to F_{2}. The functor t_{2}^{∘}t_{1} yields a transformation from F to F_{2} and is defined by: (Def.6) for every object a of A holds (t_{2}^{∘}t_{1})(a) = t_{2}(a) · t_{1}(a).

The following propositions are true:

(20) If F_{1} is transformable to F_{2}, then for all transformations t_{1}, t_{2} from F_{1} to F_{2} such that for every object a of A holds t_{1}(a) = t_{2}(a) holds t_{1} = t_{2}.

(21) For every object a of A holds id_{F} (a) = id_{F (a)}.

(22) If F_{1} is transformable to F_{2}, then for every transformation t from F_{1} to F_{2} holds id_{F_{2}} ^{∘}t = t and t^{∘} id_{F_{1}} = t.

(23) If F is transformable to F_{1} and F_{1} is transformable to F_{2} and F_{2} is transformable to F_{3}, then for every transformation t_{1} from F to F_{1} and for every transformation t_{2} from F_{1} to F_{2} and for every transformation t_{3} from F_{2} to F_{3} holds t_{3}^{∘}t_{2}^{∘}t_{1} = t_{3}^{∘}(t_{2}^{∘}t_{1}).

Natural transformations

Let us consider A, B, F_{1}, F_{2}. We say that F_{1} is naturally transformable to F_{2} if and only if: (Def.7) F_{1} is transformable to F_{2} and there exists a transformation t from F_{1}

to F_{2} such that for all objects a, b of A such that hom(a, b) ≠ ∅ for every morphism f from a to b holds t(b) · F_{1}(f) = F_{2}(f) · t(a).

Next we state two propositions:

(24) F is naturally transformable to F.

(25) If F is naturally transformable to F_{1} and F_{1} is naturally transformable

to F_{2}, then F is naturally transformable to F_{2}. Let us consider A, B, F_{1}, F_{2}. Let us assume that F_{1} is naturally trans-

formable to F_{2}. A transformation from F_{1} to F_{2} is called a natural transformation from F_{1} to F_{2} if: (Def.8) for all objects a, b of A such that hom(a, b) ≠ ∅ for every morphism f

from a to b holds it(b) · F_{1}(f) = F_{2}(f) · it(a).

Let us consider A, B, F. Then id_{F} is a natural transformation from F to F. Let us consider A, B, F, F_{1}, F_{2}. satisfying the conditions: F is naturally

transformable to F_{1} and F_{1} is naturally transformable to F_{2}. Let t_{1} be a natural transformation from F to F_{1}, and let t_{2} be a natural transformation from F_{1} to F_{2}. The functor t_{2}^{∘}t_{1} yields a natural transformation from F to F_{2} and is defined by:

(Def.9) t_{2}^{∘}t_{1} = t_{2}^{∘}t_{1}.

One can prove the following proposition

(26) If F_{1} is naturally transformable to F_{2}, then for every natural transfor-

mation t from F_{1} to F_{2} holds id_{F_{2}} ^{∘}t = t and t^{∘} id_{F_{1}} = t. In the sequel t denotes a natural transformation from F to F_{1} and t_{1} denotes

a natural transformation from F_{1} to F_{2}. Next we state two propositions:

(27) If F is naturally transformable to F_{1} and F_{1} is naturally transformable to F_{2}, then for every natural transformation t_{1} from F to F_{1} and for every natural transformation t_{2} from F_{1} to F_{2} and for every object a of A holds (t_{2}^{∘}t_{1})(a) = t_{2}(a) · t_{1}(a).

(28) If F is naturally transformable to F_{1} and F_{1} is naturally transformable to F_{2} and F_{2} is naturally transformable to F_{3}, then for every natural transformation t_{3} from F_{2} to F_{3} holds t_{3}^{∘}t_{1}^{∘}t = t_{3}^{∘}(t_{1}^{∘}t).

Let us consider A, B, F_{1}, F_{2}. A transformation from F_{1} to F_{2} is invertible if:

(Def.10) for every object a of A holds it(a) is invertible.

We now define two new predicates. Let us consider A, B, F_{1}, F_{2}. We say that F_{1}, F_{2} are naturally equivalent if and only if:

(Def.11) F_{1} is naturally transformable to F_{2} and there exists a natural transfor-

mation t from F_{1} to F_{2} such that t is invertible. We write F_{1} ≅ F_{2} if and only if F_{1}, F_{2} are naturally equivalent.

One can prove the following proposition

(29) F ≅ F.

Let us consider A, B, F_{1}, F_{2}. satisfying the condition: F_{1} is transformable to F_{2}. Let t_{1} be a transformation from F_{1} to F_{2} satisfying the condition: t_{1} is invertible. The functor t_{1}^{−1} yielding a transformation from F_{2} to F_{1} is defined as follows:

(Def.12) for every object a of A holds t_{1}^{−1}(a) = t_{1}(a)^{−1}.

Let us consider A, B, F_{1}, F_{2}, t_{1}. satisfying the conditions: F_{1} is naturally transformable to F_{2} and t_{1} is invertible. The functor t_{1}^{−1} yielding a natural transformation from F_{2} to F_{1} is defined by:

(Def.13) t_{1}^{−1} = (t_{1} qua a transformation from F_{1} to F_{2})^{−1}.

Next we state three propositions:

(30) For all A, B, F_{1}, F_{2}, t_{1} such that F_{1} is naturally transformable to F_{2} and t_{1} is invertible for every object a of A holds t_{1}^{−1}(a) = t_{1}(a)^{−1}.

(31) If F_{1} ≅ F_{2}, then F_{2} ≅ F_{1}.

(32) If F_{1} ≅ F_{2} and F_{2} ≅ F_{3}, then F_{1} ≅ F_{3}.

Let us consider A, B, F_{1}, F_{2}. Let us assume that F_{1}, F_{2} are naturally equivalent. A natural transformation from F_{1} to F_{2} is called a natural equivalence of F_{1} and F_{2} if:

(Def.14) it is invertible.

We now state two propositions:

(33) id_{F} is a natural equivalence of F and F.

(34) If F_{1} ≅ F_{2} and F_{2} ≅ F_{3}, then for every natural equivalence t of F_{1} and F_{2} and for every natural equivalence t′ of F_{2} and F_{3} holds t′^{∘}t is a natural equivalence of F_{1} and F_{3}.

Functor category

Let us consider A, B. A non-empty set is called a set of natural transformations from A to B if:

(Def.15) for an arbitrary x such that x ∈ it there exist functors F_{1}, F_{2} from A to B and there exists a natural transformation t from F_{1} to F_{2} such that x = ⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t⟩⟩ and F_{1} is naturally transformable to F_{2}.

Let us consider A, B. The functor NatTrans(A, B) yielding a set of natural transformations from A to B is defined as follows:

(Def.16) for an arbitrary x holds x ∈ NatTrans(A, B) if and only if there exist functors F_{1}, F_{2} from A to B and there exists a natural transformation t from F_{1} to F_{2} such that x = ⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t⟩⟩ and F_{1} is naturally transformable to F_{2}.

Let A_{1}, B_{1}, A_{2}, B_{2} be non-empty sets, and let f_{1} be a function from A_{1} into B_{1}, and let f_{2} be a function from A_{2} into B_{2}. Let us note that one can characterize the predicate f_{1} = f_{2} by the following (equivalent) condition:

(Def.17) A_{1} = A_{2} and for every element a of A_{1} holds f_{1}(a) = f_{2}(a).

The following two propositions are true:

(35) F_{1} is naturally transformable to F_{2} if and only if ⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t_{1}⟩⟩ ∈ NatTrans(A, B).

(36) ⟨⟨⟨⟨F, F ⟩⟩, id_{F} ⟩⟩ ∈ NatTrans(A, B).

Let us consider A, B. The functor B^{A} yielding a category is defined by the conditions (Def.18).

(Def.18) (i) The objects of B^{A} = Funct(A, B), (ii) the morphisms of B^{A} = NatTrans(A, B), (iii) for every morphism f of B^{A} holds dom f = (f_{1})_{1} and cod f = (f_{1})_{2}, (iv) for all morphisms f, g of B^{A} such that dom g = cod f holds ⟨⟨g, f⟩⟩ ∈

dom (the composition of B^{A}),

(v) for all morphisms f, g of B^{A} such that ⟨⟨g, f⟩⟩ ∈ dom (the composition of B^{A}) there exist F, F_{1}, F_{2}, t, t_{1} such that f = ⟨⟨⟨⟨F, F_{1}⟩⟩, t⟩⟩ and g = ⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t_{1}⟩⟩ and (the composition of B^{A})(⟨⟨g, f⟩⟩) = ⟨⟨⟨⟨F, F_{2}⟩⟩, t_{1}^{∘}t⟩⟩,

(vi) for every object a of B^{A} and for every F such that F = a holds id_{a} = ⟨⟨⟨⟨F, F ⟩⟩, id_{F} ⟩⟩.

We now state several propositions:

(37) The objects of B^{A} = Funct(A, B).

(38) The morphisms of B^{A} = NatTrans(A, B).

(39) For every morphism f of B^{A} such that f = ⟨⟨⟨⟨F, F_{1}⟩⟩, t⟩⟩ holds dom f = F and cod f = F_{1}.

(40) For all objects a, b of B^{A} and for every morphism f from a to b such that hom(a, b) ≠ ∅ there exist F, F_{1}, t such that a = F and b = F_{1} and f = ⟨⟨⟨⟨F, F_{1}⟩⟩, t⟩⟩.

(41) For every natural transformation t′ from F_{2} to F_{3} and for all morphisms f, g of B^{A} such that f = ⟨⟨⟨⟨F, F_{1}⟩⟩, t⟩⟩ and g = ⟨⟨⟨⟨F_{2}, F_{3}⟩⟩, t′⟩⟩ holds ⟨⟨g, f⟩⟩ ∈ dom (the composition of B^{A}) if and only if F_{1} = F_{2}.

(42) For all morphisms f, g of B^{A} such that f = ⟨⟨⟨⟨F, F_{1}⟩⟩, t⟩⟩ and g = ⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t_{1}⟩⟩ holds g · f = ⟨⟨⟨⟨F, F_{2}⟩⟩, t_{1}^{∘}t⟩⟩.

(43) For every object a of B^{A} and for every F such that F = a holds id_{a} = ⟨⟨⟨⟨F, F ⟩⟩, id_{F} ⟩⟩.

Discrete categories

A category is discrete if: (Def.19) for every morphism f of it there exists an object a of it such that

f = id_{a}. One can prove the following propositions:

(44) For every discrete category A and for every object a of A holds hom(a, a) = {id_{a}}.

(45) A is discrete if and only if for every object a of A holds hom(a, a) is finite and card hom(a, a) = 1 and for every object b of A such that a ≠ b holds hom(a, b) = ∅.

(46) ˙ (o, m) is discrete.

(47) For every discrete category A and for every subcategory C of A holds C is discrete.

(48) If A is discrete and B is discrete, then [: A, B:] is discrete.

(49) For every discrete category A and for every category B and for all functors F_{1}, F_{2} from B to A such that F_{1} is transformable to F_{2} holds F_{1} = F_{2}.

(50) For every discrete category A and for every category B and for every functor F from B to A and for every transformation t from F to F holds t = id_{F}.

(51) If A is discrete, then A^{B} is discrete.

Let us consider C. The functor IdCat C yields a discrete subcategory of C and is defined as follows:

(Def.20) the objects of IdCat C = the objects of C and the morphisms of IdCat C = {id_{a}}, where a ranges over objects of C.

Next we state four propositions:

(52) If C is discrete, then IdCat C = C.

474 andrzej trybulec

(53) IdCat IdCat C = IdCat C.

(54) IdCat ˙ (o, m) = ˙ (o, m).

(55) IdCat[: A, B:] = [: IdCat A, IdCat B:].

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

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

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

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

[5] Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1(2):409–420, 1990.

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

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

[8] Czesław Byliński. Subcategories and products of categories. Formalized Mathematics, 1(4):725–732, 1990.

[9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[10] Zbigniew Semadeni and Antoni Wiweger. Wste‘p do teorii kategorii i funktorów. Volume 45 of Biblioteka Matematyczna, PWN, Warszawa, 1978.

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

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

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

Received May 15, 1991

Matrices. Abelian Group of Matrices

Katarzyna Jankowska Warsaw University Białystok

Summary. The basic conceptions of matrix algebra are introduced. The matrix is introduced as the finite sequence of sequences with the same length, i.e. as a sequence of lines. There are considered matrices over a field, and the fact that these matrices with addition form an Abelian group is proved.

MML Identifier: MATRIX 1.

The notation and terminology used here have been introduced in the following papers: [9], [5], [6], [1], [8], [4], [2], [3], and [7]. For simplicity we adopt the following rules: x will be arbitrary, i, j, n, m will be natural numbers, D will be a non-empty set, K will be a field structure, s will be a finite sequence, a, a_{1}, a_{2}, b_{1}, b_{2}, d will be elements of D, p, p_{1}, p_{2} will be finite sequences of elements of D, and F will be a field. A finite sequence is tabular if: (Def.1) there exists a natural number n such that for every x such that x ∈ rng it

there exists s such that s = x and len s = n. The following propositions are true:

(1) ⟨⟨d⟩⟩ is tabular.

(2) m ⟼ (n ⟼ x) is tabular.

(3) For every s holds ⟨s⟩ is tabular.

(4) For all finite sequences s_{1}, s_{2} such that len s_{1} = n and len s_{2} = n holds ⟨s_{1}, s_{2}⟩ is tabular.

(5) ε is tabular.

(6) ⟨ε, ε⟩ is tabular.

(7) ⟨⟨a_{1}⟩, ⟨a_{2}⟩⟩ is tabular.

(8) ⟨⟨a_{1}, a_{2}⟩, ⟨b_{1}, b_{2}⟩⟩ is tabular. A tabular finite sequence is non-trivial if:

(Def.2) there exists s such that s ∈ rng it and len s > 0.

Let D be a non-empty set.

Let D be a non-empty set. A matrix over D is a tabular finite sequence of elements of D^{∗}.

We now state the proposition

(9) s is a matrix over D if and only if there exists n such that for every x such that x ∈ rng s there exists p such that x = p and len p = n.

Let us consider D, m, n. A matrix over D is said to be a matrix over D of dimension m × n if: (Def.3) len it = m and for every p such that p ∈ rng it holds len p = n.

Let us consider D, n. A matrix over D of dimension n is a matrix over D of dimension n × n.

We now define three new modes. Let us consider K. A matrix over K is a matrix over the carrier of K.

Let us consider n. A matrix over K of dimension n is a matrix over the carrier of K of dimension n × n.

Let us consider m. A matrix over K of dimension n × m is a matrix over the carrier of K of dimension n × m.

We now state a number of propositions:

(10) m ⟼ (n ⟼ a) is a matrix over D of dimension m × n.

(11) For every finite sequence p of elements of D holds ⟨p⟩ is a matrix over D of dimension 1 × len p.

(12) For all p_{1}, p_{2} such that len p_{1} = n and len p_{2} = n holds ⟨p_{1}, p_{2}⟩ is a matrix over D of dimension 2 × n.

(13) ε is a matrix over D of dimension 0 × m.

(14) ⟨ε⟩ is a matrix over D of dimension 1 × 0.

(15) ⟨⟨a⟩⟩ is a matrix over D of dimension 1.

(16) ⟨ε, ε⟩ is a matrix over D of dimension 2 × 0.

(17) ⟨⟨a_{1}, a_{2}⟩⟩ is a matrix over D of dimension 1 × 2.

(18) ⟨⟨a_{1}⟩, ⟨a_{2}⟩⟩ is a matrix over D of dimension 2 × 1.

(19) ⟨⟨a_{1}, a_{2}⟩, ⟨b_{1}, b_{2}⟩⟩ is a matrix over D of dimension 2.

In the sequel M, M_{1}, M_{2} will be matrices over D. Let M be a tabular finite sequence. The functor width M yields a natural number and is defined as follows: (Def.4) (i) there exists s such that s ∈ rng M and len s = width M if len M > 0,

(ii) width M = 0, otherwise.

Next we state the proposition

(20) If len M > 0, then for every n holds M is a matrix over D of dimension

len M × n if and only if n = width M. Let M be a tabular finite sequence. The indices of M yielding a set is defined

by: (Def.5) the indices of M = [: Seg len M, Seg width M :].

Let us consider D, and let M be a matrix over D, and let us consider i, j. Let us assume that ⟨⟨i, j⟩⟩ ∈ the indices of M. The functor M_{i,j} yielding an element of D is defined as follows: (Def.6) there exists p such that p = M(i) and M_{i,j} = p(j).

The following proposition is true

(21) If len M_{1} = len M_{2} and width M_{1} = width M_{2} and for all i, j such that

⟨⟨i, j⟩⟩ ∈ the indices of M_{1} holds M_{1}_{i,j} = M_{2}_{i,j}, then M_{1} = M_{2}. In this article we present several logical schemes. The scheme MatrixLambda

deals with a non-empty set A, a natural number B, a natural number C, and a binary functor F yielding an element of A and states that:

there exists a matrix M over A of dimension B × C such that for all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of M holds M_{i,j} = F(i, j) for all values of the parameters.

The scheme MatrixEx concerns a non-empty set A, a natural number B, a natural number C, and a ternary predicate P, and states that:

there exists a matrix M over A of dimension B × C such that for all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of M holds P[i, j, M_{i,j}] provided the parameters have the following properties:

• for all i, j such that ⟨⟨i, j⟩⟩ ∈ [: Seg B, Seg C:] for all elements x_{1}, x_{2} of A such that P[i, j, x_{1}] and P[i, j, x_{2}] holds x_{1} = x_{2},

• for all i, j such that ⟨⟨i, j⟩⟩ ∈ [: Seg B, Seg C:] there exists an element x of A such that P[i, j, x].

The scheme SeqDLambda concerns a non-empty set A, a natural number B, and a unary functor F yielding an element of A and states that:

there exists a finite sequence p of elements of A such that len p = B and for every i such that i ∈ Seg B holds p(i) = F(i) for all values of the parameters.

We now state several propositions:

(22) For every matrix M over D of dimension n × m such that len M = 0 holds width M = 0.

(23) For every matrix M over D of dimension 0 × m holds len M = 0 and width M = 0 and the indices of M = ∅.

(24) If n > 0, then for every matrix M over D of dimension n × m holds len M = n and width M = m and the indices of M = [: Seg n, Seg m:].

(25) For every matrix M over D of dimension n holds len M = n and width M = n and the indices of M = [: Seg n, Seg n:].

(26) For every matrix M over D of dimension n × m holds len M = n and the indices of M = [: Seg n, Seg width M :].

(27) For all matrices M_{1}, M_{2} over D of dimension n × m holds the indices of M_{1} = the indices of M_{2}.

(28) For all matrices M_{1}, M_{2} over D of dimension n×m such that for all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of M_{1} holds M_{1}_{i,j} = M_{2}_{i,j} holds M_{1} = M_{2}.

(29) For every matrix M_{1} over D of dimension n and for all i, j such that

⟨⟨i, j⟩⟩ ∈ the indices of M_{1} holds ⟨⟨j, i⟩⟩ ∈ the indices of M_{1}. Let us consider D, and let M be a matrix over D. The functor M^{T} yielding

a matrix over D is defined as follows:

(Def.7) len(M^{T}) = width M and for all i, j holds ⟨⟨i, j⟩⟩ ∈ the indices of M^{T} if and only if ⟨⟨j, i⟩⟩ ∈ the indices of M and for all i, j such that ⟨⟨j, i⟩⟩ ∈ the indices of M holds M_{i,j}^{T} = M_{j,i}.

We now define two new functors. Let us consider D, M, i. The functor Line(M, i) yields a finite sequence of elements of D and is defined by:

(Def.8) len Line(M, i) = width M and for every j such that j ∈ Seg width M

holds Line(M, i)(j) = M_{i,j}. The functor M_{,i} yields a finite sequence of elements of D and is defined as follows:

(Def.9) len(M_{,i} ) = len M and for every j such that j ∈ Seg len M holds M_{,i} (j) = M_{j,i}.

Let us consider D, and let M be a matrix over D, and let us consider i. Then Line(M, i) is an element of D^{width M}. Then M_{,i} is an element of D^{len M}.

In the sequel A, B are matrices over K of dimension n. We now define five new functors. Let us consider K, n. The functor K^{n×n} yields a non-empty set and is defined as follows:

(Def.10) K^{n×n} = ( (the carrier of K)^{n})^{n}.

The functor ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{n×n} yielding a matrix over K of dimension n is de-

fined as follows:

(Def.11) ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{n×n} = n ⟼ (n ⟼ 0_{K}).

The functor ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} yielding a matrix over K of dimension n is de-

fined as follows:

(Def.12) for every i such that ⟨⟨i, i⟩⟩ ∈ the indices of ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} holds

(( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n})_{i,i} = 1_{K} and for all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices

of ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n} and i ≠ j holds (( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )_{K}^{n×n})_{i,j} = 0_{K}.

Let us consider A. The functor −A yielding a matrix over K of dimension n is defined as follows:

(Def.13) for all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of A holds (−A)_{i,j} = −A_{i,j}. Let us consider B. The functor A + B yielding a matrix over K of dimension n is defined by:

(Def.14) for all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of A holds (A+B)_{i,j} = A_{i,j}+B_{i,j}.

The following two propositions are true:

(30) For all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{n×n} holds

(( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{K}^{n×n})_{i,j} = 0_{K}.

(31) For every x holds x is an element of K^{n×n} if and only if x is a matrix

over K of dimension n. Let us consider K, n. A matrix over K of dimension n is called a diagonal

n-dimensional matrix over K if: (Def.15) for all i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of it and it_{i,j} ≠ 0_{K} holds i = j.

In the sequel A, B, C will denote matrices over F of dimension n. One can prove the following four propositions:

(32) A + B = B + A.

(33) A + B + C = A + (B + C).

(34) A + ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{F}^{n×n} = A.

(35) A + −A = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{F}^{n×n}.

Let us consider F, n. The functor F_{G}^{n×n} yielding an Abelian group is defined by:

(Def.16) the carrier of F_{G}^{n×n} = F^{n×n} and for all A, B holds (the addition of F_{G}^{n×n})(A, B) = A + B and for every A holds (the reverse-map of

F_{G}^{n×n})(A) = −A and the zero of F_{G}^{n×n} = ( \begin{matrix}0 & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & 0 \end{matrix} )_{F}^{n×n}.

480 katarzyna jankowska

Acknowledgements

I would like to thank Grzegorz Bancerek for his useful sugestions and valuable comments.

References

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

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

[3] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.

[4] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 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] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.

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

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

Received June 8, 1990

Paracompact and Metrizable Spaces

Leszek Borys Warsaw University Białystok

Summary. We give an example of a compact space. Next we define a locally finite subset family of topological spaces and paracompact topological spaces. An open sets family of a metric space is defined next and it has been shown that the metric space with any open sets family is a topological space. Next we define metrizable space.

MML Identifier: PCOMPS 1.

The papers [15], [5], [6], [11], [10], [12], [13], [18], [8], [17], [9], [7], [16], [3], [2], [1], [4], and [14] provide the terminology and notation for this paper. In the sequel P_{1} denotes a metric space, x denotes an element of the carrier of P_{1}, and r, p denote real numbers. Next we state the proposition

(1) If r ≤ p and r > 0, then Ball(x, r) ⊆ Ball(x, p).

For simplicity we adopt the following convention: T will be a topological space, x will be a point of T, W, A will be subsets of T, and F_{1} will be a family of subsets of T. One can prove the following four propositions:

(2) \overline{A} ≠ ∅ if and only if A ≠ ∅.

(3) If \overline{A} = ∅, then A = ∅.

(4) \overline{A} is closed.

(5) If F_{1} is a cover of T, then for every x there exists W such that x ∈ W and W ∈ F_{1}.

Let X be arbitrary. Then {X} is a non-empty set. Then 2^{X} is a non-empty family of subsets of X.

Let a be arbitrary. The functor {a}_{top} yields a topological space and is defined by: (Def.1) {a}_{top} = ⟨{a}, 2^{{a}}⟩.

In the sequel a is arbitrary. We now state four propositions:

(6) {a}_{top} = ⟨{a}, 2^{{a}}⟩.

(7) The topology of {a}_{top} = 2^{{a}}.

(8) The carrier of {a}_{top} = {a}.

(9) {a}_{top} is compact.

Let us consider T, x. Then {x} is a subset of T.

We now state the proposition

(10) If T is a T_{2} space, then {x} is closed.

For simplicity we follow the rules: T will be a topological space, x will be a point of T, Z, V, W, Y, A, B will be subsets of T, and F_{1}, G_{1} will be families of subsets of T. Let us consider T. A family of subsets of T is locally finite if: (Def.2) for every x there exists W such that x ∈ W and W is open and {V :

V ∈ it ∧ V ∩ W ≠ ∅} is finite. Next we state three propositions:

(11) For every W holds {V : V ∈ F_{1} ∧ V ∩ W ≠ ∅} ⊆ F_{1}.

(12) If F_{1} ⊆ G_{1} and G_{1} is locally finite, then F_{1} is locally finite.

(13) If F_{1} is finite, then F_{1} is locally finite.

Let us consider T, F_{1}. The functor clf F_{1} yielding a family of subsets of T is defined by: (Def.3) Z ∈ clf F_{1} if and only if there exists W such that Z = \overline{W} and W ∈ F_{1}.

Next we state several propositions:

(14) clf F_{1} is closed.

(15) If F_{1} = ∅, then clf F_{1} = ∅.

(16) If F_{1} = \{V \}, then clf F_{1} = \{\overline{V} \}.

(17) If F_{1} ⊆ G_{1}, then clf F_{1} ⊆ clf G_{1}.

(18) clf(F_{1} ∪ G_{1}) = clf F_{1} ∪ clf G_{1}.

Next we state two propositions:

(19) If F_{1} is finite, then \overline{⋃ F_{1}} = ⋃ clf F_{1}.

(20) F_{1} is finer than clf F_{1}.

The scheme Lambda1top deals with a topological space A, a family B of subsets of A, a family C of subsets of A, and a unary functor F yielding a subset of A and states that:

there exists a function f from B into C such that for every subset Z of A such that Z ∈ B holds f(Z) = F(Z) provided the following condition is satisfied:

• for every subset Z of A such that Z ∈ B holds F(Z) ∈ C.

Next we state four propositions:

(21) If F_{1} is locally finite, then clf F_{1} is locally finite.

(22) ⋃ F_{1} ⊆ ⋃ clf F_{1}.

(23) If F_{1} is locally finite, then \overline{⋃ F_{1}} = ⋃ clf F_{1}.

(24) If F_{1} is locally finite and F_{1} is closed, then ⋃ F_{1} is closed. A topological space is paracompact if:

(Def.4) for every family F_{1} of subsets of it such that F_{1} is a cover of it and F_{1} is open there exists a family G_{1} of subsets of it such that G_{1} is open and G_{1} is a cover of it and G_{1} is finer than F_{1} and G_{1} is locally finite.

The following propositions are true:

(25) If T is compact, then T is paracompact.

(26) Suppose T is paracompact and A is closed and B is closed and A misses B and for every x such that x ∈ B there exist V, W such that V is open and W is open and A ⊆ V and x ∈ W and V misses W. Then there exist Y, Z such that Y is open and Z is open and A ⊆ Y and B ⊆ Z and Y misses Z.

(27) If T is a T_{2} space and T is paracompact, then T is a T_{3} space.

(28) If T is a T_{2} space and T is paracompact, then T is a T_{4} space. For simplicity we follow a convention: P_{1} will denote a metric space, x, y, z

will denote elements of the carrier of P_{1}, r, p, q will denote real numbers, and V, W will denote subsets of the carrier of P_{1}. Let us consider P_{1}. The open set family of P_{1} yielding a family of subsets of the carrier of P_{1} is defined as follows: (Def.5) for every V holds V ∈the open set family of P_{1} if and only if for every

x such that x ∈ V there exists r such that r > 0 and Ball(x, r) ⊆ V.

One can prove the following propositions:

(29) For every x there exists r such that r > 0 and Ball(x, r) ⊆ the carrier of P_{1}.

(30) If y ∈ Ball(x, r), then there exists p such that p > 0 and Ball(y, p) ⊆ Ball(x, r).

(31) If y ∈ Ball(x, r) ∩ Ball(z, p), then there exists q such that Ball(y, q) ⊆ Ball(x, r) and Ball(y, q) ⊆ Ball(z, p).

(32) For every V holds V ∈ the open set family of P_{1} if and only if for every x such that x ∈ V there exists r such that r > 0 and Ball(x, r) ⊆ V.

(33) For all x, r holds Ball(x, r) ∈ the open set family of P_{1}.

(34) The carrier of P_{1} ∈ the open set family of P_{1}.

(35) For all V, W such that V ∈ the open set family of P_{1} and W ∈ the open set family of P_{1} holds V ∩ W ∈ the open set family of P_{1}.

(36) For every family A of subsets of the carrier of P_{1} such that A ⊆ the open set family of P_{1} holds ⋃ A ∈ the open set family of P_{1}.

(37) ⟨The carrier of P_{1},the open set family of P_{1}⟩ is a topological space. Let us consider P_{1}. The functor P_{1}_{top} yielding a topological space is defined

as follows: (Def.6) P_{1}_{top} = ⟨ the carrier of P_{1},the open set family of P_{1}⟩.

We now state the proposition

(38) P_{1}_{top} is a T_{2} space.

Let D be a non-empty set, and let f be a function from [: D, D:] into . We say that f is a metric of D if and only if:

484 leszek borys

(Def.7) for all elements a, b, c of D holds f(a, b) = 0 if and only if a = b but f(a, b) = f(b, a) and f(a, c) ≤ f(a, b) + f(b, c).

We now state two propositions:

(39) For every non-empty set D and for every function f from [: D, D:] into holds f is a metric of D if and only if ⟨D, f⟩ is a metric space.

(40) For every metric space M_{1} holds the distance of M_{1} is a metric of the carrier of M_{1}.

Let D be a non-empty set, and let f be a function from [: D, D:] into . Let us assume that f is a metric of D. The functor MetrSp(D, f) yielding a metric space is defined by: (Def.8) MetrSp(D, f) = ⟨D, f⟩.

A topological space is metrizable if: (Def.9) there exists a function f from [: the carrier of it, the carrier of it :] into

such that f is a metric of the carrier of it and the open set family of MetrSp((the carrier of it), f) = the topology of it.

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

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

[9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

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

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

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

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

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

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

[18] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231–237, 1990.

paracompact and metrizable spaces 485 Received June 8, 1990 486

Atlas of Midpoint Algebra

Michał Muzalewski Warsaw University Białystok

Summary. This article is a continuation of [4]. We have established a one-to-one correspondence between midpoint algebras and groups with the operator \frac{1}{2} . In general we shall say that a given midpoint algebra M and a group V are w-assotiated iff w is an atlas from M to V. At the beginning of the paper a few facts which rather belong to [3], [5] are proved.

MML Identifier: MIDSP 2.

The terminology and notation used here have been introduced in the following articles: [2], [1], [3], [4], and [5]. In the sequel G is a group structure and x is an element of G. Let us consider G, x. The functor 2x yielding an element of G is defined by: (Def.1) 2x = x + x.

In the sequel M is a midpoint algebra structure. Let us consider M. A point of M is an element of the points of M.

In the sequel p, q, r will be points of M and w will be a function from [: the points of M, the points of M :] into the carrier of G. Let us consider M, G, w. We say that M, G are associated w.r.t. w if and only if: (Def.2) p ⊕ q = r if and only if w(p, r) = w(r, q).

The following proposition is true

(1) If M, G are associated w.r.t. w, then p ⊕ p = p.

We follow the rules: S will be a non-empty set, a, b, b′, c, c′, d will be elements of S, and w will be a function from [: S, S:] into the carrier of G. Let us consider S, G, w. We say that w is an atlas of S, G if and only if: (Def.3) for every a, x there exists b such that w(a, b) = x and for all a, b, c such

that w(a, b) = w(a, c) holds b = c and for all a, b, c holds w(a, b) + w(b,

c) = w(a, c).

Let us consider S, G, w, a, x. Let us assume that w is an atlas of S, G. The functor (a, x).w yielding an element of S is defined by: (Def.4) w(a, (a, x).w) = x.

In the sequel G denotes a group, x, y denote elements of G, and w denotes a function from [: S, S:] into the carrier of G. One can prove the following propositions:

(2) 2(0_{G}) = 0_{G}.

(3) If x + y = x, then y = 0_{G}.

(4) If w is an atlas of S, G, then w(a, a) = 0_{G}.

(5) If w is an atlas of S, G and w(a, b) = 0_{G}, then a = b.

(6) If w is an atlas of S, G, then w(a, b) = −w(b, a).

(7) If w is an atlas of S, G and w(a, b) = w(c, d), then w(b, a) = w(d, c).

(8) If w is an atlas of S, G, then for every b, x there exists a such that w(a, b) = x.

(9) If w is an atlas of S, G and w(b, a) = w(c, a), then b = c.

(10) For every function w from [: the points of M, the points of M :] into the carrier of G such that w is an atlas of the points of M, G and M, G are associated w.r.t. w holds p ⊕ q = q ⊕ p.

(11) For every function w from [: the points of M, the points of M :] into the carrier of G such that w is an atlas of the points of M, G and M, G are associated w.r.t. w there exists r such that r ⊕ p = q.

We adopt the following rules: G will denote an Abelian group and x, y, z, t will denote elements of G. The following propositions are true:

(12) −(x + y) = −x + −y.

(13) x + y + (z + t) = x + z + (y + t).

(14) 2(x + y) = 2x + 2y.

(15) 2(−x) = −2x.

(16) For every function w from [: the points of M, the points of M :] into the carrier of G such that w is an atlas of the points of M, G and M, G are associated w.r.t. w for all points a, b, c, d of M holds a ⊕ b = c ⊕ d if and only if w(a, d) = w(c, b).

In the sequel w denotes a function from [: S, S:] into the carrier of G. Next we state the proposition

(17) If w is an atlas of S, G, then for all a, b, b′, c, c′ such that w(a, b) = w(b,

c) and w(a, b′) = w(b′, c′) holds w(c, c′) = 2w(b, b′). We follow the rules: M denotes a midpoint algebra and p, q, r, s denote

points of M. Let us consider M. Then vectgroup M is an Abelian group.

The following proposition is true

(18) For an arbitrary a holds a is an element of vectgroup M if and only if a is a vector of M and 0_{vectgroup M} = I_{M} and for all elements a, b of vectgroup M and for all vectors x, y of M such that a = x and b = y holds a + b = x + y.

An Abelian group is called a group with the operator \frac{1}{2} if: (Def.5) for every element a of it there exists an element x of it such that 2x = a

and for every element a of it such that 2a = 0_{it} holds a = 0_{it}.

In the sequel G is a group with the operator \frac{1}{2} and x, y are elements of G. One can prove the following two propositions:

(19) If x = −x, then x = 0_{G}.

(20) If 2x = 2y, then x = y.

Let us consider G, x. The functor \frac{1}{2}x yielding an element of G is defined as follows: (Def.6) 2\frac{1}{2}x = x.

The following three propositions are true:

(21) \frac{1}{2}(0_{G}) = 0_{G} and \frac{1}{2}(x + y) = \frac{1}{2}x + \frac{1}{2}y but if \frac{1}{2}x = \frac{1}{2}y, then x = y and \frac{1}{2}2x = x.

(22) For every M being a midpoint algebra structure and for every function w from [: the points of M, the points of M :] into the carrier of G such that w is an atlas of the points of M, G and M, G are associated w.r.t. w for all points a, b, c, d of M holds a ⊕ b ⊕ (c ⊕ d) = a ⊕ c ⊕ (b ⊕ d).

(23) For every M being a midpoint algebra structure and for every function w from [: the points of M, the points of M :] into the carrier of G such that w is an atlas of the points of M, G and M, G are associated w.r.t. w holds M is a midpoint algebra.

Let us consider M. Then vectgroup M is a group with the operator \frac{1}{2}. Let us consider M, p, q. The functor q^{p} yields an element of vectgroup M

and is defined as follows:

(Def.7) q^{p} = \overrightarrow{[p, q]}.

Let us consider M. The functor vect M yields a function from [: the points of M, the points of M :] into the carrier of vectgroup M and is defined by:

(Def.8) (vect M)(p, q) = \overrightarrow{[p, q]}.

We now state four propositions:

(24) vect M is an atlas of the points of M, vectgroup M.

(25) \overrightarrow{[p, q]} = \overrightarrow{[r, s]} if and only if p ⊕ s = q ⊕ r.

(26) p ⊕ q = r if and only if \overrightarrow{[p, r]} = \overrightarrow{[r, q]}.

(27) M, vectgroup M are associated w.r.t. vect M.

In the sequel w will denote a function from [: S, S:] into the carrier of G. Let us consider S, G, w. Let us assume that w is an atlas of S, G. The functor ^{@}w yielding a binary operation on S is defined as follows: (Def.9) w(a, (^{@}w)(a, b)) = w((^{@}w)(a, b), b).

We now state the proposition

(28) If w is an atlas of S, G, then for all a, b, c holds (^{@}w)(a, b) = c if and

only if w(a, c) = w(c, b). In the sequel a, b, c are points of ⟨S,^{@}w⟩. We now state two propositions:

(29) (^{@}w)(a, b) = a ⊕ b.

(30) a ⊕ b = c if and only if (^{@}w)(a, b) = c.

Let us consider S, G, w. The functor Atlas w yielding a function from [: the points of ⟨S,^{@}w⟩, the points of ⟨S,^{@}w⟩:] into the carrier of G is defined as follows:

(Def.10) Atlas w = w.

Next we state two propositions:

(31) If w is an atlas of S, G, then Atlas w is an atlas of the points of ⟨S, @w⟩, G.

(32) If w is an atlas of S, G, then ⟨S,^{@}w⟩, G are associated w.r.t. Atlas w. Let us consider S, G, w. Let us assume that w is an atlas of S, G. The

functor MidSp(w) yielding a midpoint algebra is defined by: (Def.11) MidSp(w) = ⟨S,^{@}w⟩.

We follow the rules: M is a midpoint algebra structure, w is a function from [: the points of M, the points of M :] into the carrier of G, and a, b, b_{1}, b_{2}, c are points of M. The following proposition is true

(33) M is a midpoint algebra if and only if there exists G and there exists w such that w is an atlas of the points of M, G and M, G are associated w.r.t. w.

Let us consider M. We consider atlas structures over M which are systems ⟨an algebra, a function⟩,

where the algebra is a group with the operator \frac{1}{2} and the function is a function from [: the points of M, the points of M :] into the carrier of the algebra.

Let M be a midpoint algebra. An atlas structure over M is said to be an atlas of M if:

(Def.12) M, the algebra of it are associated w.r.t. the function of it and the function of it is an atlas of the points of M, the algebra of it.

Let M be a midpoint algebra, and let W be an atlas of M. A vector of W is an element of the algebra of W.

Let M be a midpoint algebra, and let W be an atlas of M, and let a, b be points of M. The functor W (a, b) yields an element of the algebra of W and is defined as follows:

(Def.13) W (a, b) = (the function of W)(a, b).

Let M be a midpoint algebra, and let W be an atlas of M, and let a be a point of M, and let x be a vector of W. The functor (a, x).W yielding a point of M is defined as follows:

(Def.14) (a, x).W = (a, x).(the function of W).

Let M be a midpoint algebra, and let W be an atlas of M. The functor 0_{W} yielding a vector of W is defined as follows:

(Def.15) 0_{W} = 0_{the algebra of W}.

We now state two propositions:

(34) If w is an atlas of the points of M, G and M, G are associated w.r.t. w, then a ⊕ c = b_{1} ⊕ b_{2} if and only if w(a, c) = w(a, b_{1}) + w(a, b_{2}).

(35) If w is an atlas of the points of M, G and M, G are associated w.r.t.

w, then a ⊕ c = b if and only if w(a, c) = 2w(a, b). For simplicity we adopt the following convention: M will be a midpoint

algebra, W will be an atlas of M, a, b, b_{1}, b_{2}, c, d will be points of M, and x will be a vector of W. One can prove the following propositions:

(36) a ⊕ c = b_{1} ⊕ b_{2} if and only if W (a, c) = W (a, b_{1}) + W (a, b_{2}).

(37) a ⊕ c = b if and only if W (a, c) = 2W (a, b).

(38) For every a, x there exists b such that W (a, b) = x and for all a, b, c such that W (a, b) = W (a, c) holds b = c and for all a, b, c holds W (a,

b) + W (b, c) = W (a, c).

(39) (i) W (a, a) = 0_{W}, (ii) if W (a, b) = 0_{W}, then a = b, (iii) W (a, b) = −W (b, a), (iv) if W (a, b) = W (c, d), then W (b, a) = W (d, c),

(v) for every b, x there exists a such that W (a, b) = x, (vi) if W (b, a) = W (c, a), then b = c, (vii) a ⊕ b = c if and only if W (a, c) = W (c, b), (viii) a ⊕ b = c ⊕ d if and only if W (a, d) = W (c, b), (ix) W (a, b) = x if and only if (a, x).W = b.

(40) (a, 0_{W}).W = a.

References

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

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

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

[4] Michał Muzalewski. Midpoint algebras. Formalized Mathematics, 1(3):483–488, 1990.

[5] Michał Muzalewski and Wojciech Skaba. Groups, rings, left- and right-modules. Formalized Mathematics, 2(2):275–278, 1991.

492

Several Properties of the σ-additive Measure

Józef Białas University of Łódź

Summary. A continuation of [5]. The paper contains the definition and basic properties of a σ-additive, nonnegative measure, with values in \overline{}, the enlarged set of real numbers, where \overline{}denotes set \overline{}= ∪ {−∞, +∞} - by R.Sikorski [12]. Some simple theorems concerning basic properties of a σ-additive measure, measurable sets, measure zero sets are proved. The work is the fourth part of the series of articles concerning the Lebesgue measure theory.

MML Identifier: MEASURE2.

The terminology and notation used here have been introduced in the following papers: [14], [13], [8], [9], [6], [7], [1], [11], [2], [10], [3], [4], and [5]. The following proposition is true

(1) For every set X and for every σ-field S of subsets of X and for every σ-measure M on S and for every function F from into S holds M · F is non-negative.

The scheme RecExFun concerns a set A, a σ-field B of subsets of A, an element C of B, and a ternary predicate P, and states that:

there exists a function f from into B such that f(0) = C and for every element n of holds P[n, f(n), f(n + 1)] provided the following conditions are satisfied:

• for every natural number n and for every element x of B there exists an element y of B such that P[n, x, y],

• for every natural number n and for all elements x, y_{1}, y_{2} of B such that P[n, x, y_{1}] and P[n, x, y_{2}] holds y_{1} = y_{2}.

Let X be a set, and let S be a σ-field of subsets of X. A denumerable family of subsets of X is called a family of measureable sets of S if: (Def.1) it ⊆ S.

One can prove the following propositions:

(2) For every set X and for every σ-field S of subsets of X and for every denumerable family T of subsets of X holds T is a family of measureable sets of S if and only if T ⊆ S.

(3) For every set X and for every σ-field S of subsets of X and for every family T of measureable sets of S holds ⋂ T ∈ S and ⋃ T ∈ S.

Let X be a set, and let S be a σ-field of subsets of X, and let T be a family of measureable sets of S. Then ⋂ T is an element of S.

Let X be a set, and let S be a σ-field of subsets of X, and let T be a family of measureable sets of S. Then ⋃ T is an element of S.

Let X be a set, and let S be a σ-field of subsets of X, and let F be a function from into S, and let n be an element of . Then F (n) is an element of S.

One can prove the following propositions:

(4) For every set X and for every σ-field S of subsets of X and for every function N from into S there exists a function F from into S such that F (0) = N(0) and for every element n of holds F (n + 1) = N(n + 1) \ N(n).

(5) For every set X and for every σ-field S of subsets of X and for every function N from into S there exists a function F from into S such that F (0) = N(0) and for every element n of holds F (n + 1) = N(n + 1) ∪ F (n).

(6) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Let F be a function from into S. Suppose F (0) = N(0) and for every element n of holds F (n + 1) = N(n + 1) ∪ F (n). Then for an arbitrary r and for every natural number n holds r ∈ F (n) if and only if there exists a natural number k such that k ≤ n and r ∈ N(k).

(7) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Then for every function F from into S such that F (0) = N(0) and for every element n of holds F (n+1) = N(n+1)∪F (n) for all natural numbers n, m such that n < m holds F (n) ⊆ F (m).

(8) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Let G be a function from into S. Let F be a function from into S. Suppose that

(i) G(0) = N(0), (ii) for every element n of holds G(n + 1) = N(n + 1) ∪ G(n), (iii) F (0) = N(0), (iv) for every element n of holds F (n + 1) = N(n + 1) \ G(n).

Then for all natural numbers n, m such that n ≤ m holds F (n) ⊆ G(m).

(9) For every set X and for every σ-field S of subsets of X and for every function N from into S and for every function G from into S there exists a function F from into S such that F (0) = N(0) and for every element n of holds F (n + 1) = N(n + 1) \ G(n).

(10) For every set X and for every σ-field S of subsets of X and for every function N from into S there exists a function F from into S such

several properties of the σ-additive … 495

that F (0) = ∅ and for every element n of holds F (n+1) = N(0)\N(n).

(11) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Let G be a function from into S. Let F be a function from into S. Suppose that

(i) G(0) = N(0), (ii) for every element n of holds G(n + 1) = N(n + 1) ∪ G(n), (iii) F (0) = N(0), (iv) for every element n of holds F (n + 1) = N(n + 1) \ G(n).

Then for all natural numbers n, m such that n < m holds F (n) ∩ F (m) = ∅.

(12) For every set X and for every σ-field S of subsets of X and for every function N from into S and for every element n of holds N(n) ∈ rng N.

(13) For every set X and for every σ-field S of subsets of X and for every σ-measure M on S and for every family T of measureable sets of S and for every function F from into S such that T = rng F holds M(⋃ T) ≤ ∑(M · F).

(14) For every set X and for every σ-field S of subsets of X and for every family T of measureable sets of S there exists a function F from into S such that T = rng F.

(15) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Let F be a function from into S. Then if F (0) = ∅ and for every element n of holds F (n + 1) = N(0) \ N(n) and N(n + 1) ⊆ N(n), then for every element n of holds F (n) ⊆ F (n + 1).

(16) For every set X and for every σ-field S of subsets of X and for every σ-measure M on S and for every family T of measureable sets of S such that for every set A such that A ∈ T holds A is a set of measure zero w.r.t. M holds ⋃ T is a set of measure zero w.r.t. M.

(17) For every set X and for every σ-field S of subsets of X and for every σ-measure M on S and for every family T of measureable sets of S such that there exists a set A such that A ∈ T and A is a set of measure zero w.r.t. M holds ⋂ T is a set of measure zero w.r.t. M.

(18) For every set X and for every σ-field S of subsets of X and for every σ-measure M on S and for every family T of measureable sets of S such that for every set A such that A ∈ T holds A is a set of measure zero w.r.t. M holds ⋂ T is a set of measure zero w.r.t. M.

Let X be a set, and let S be a σ-field of subsets of X. A family of measureable sets of S is called a family of measureable non-decrement sets of S if: (Def.2) there exists a function F from into S such that it = rng F and for

every element n of holds F (n) ⊆ F (n + 1).

We now state the proposition

(19) For every set X and for every σ-field S of subsets of X and for every family T of measureable sets of S holds T is a family of measureable nondecrement sets of S if and only if there exists a function F from into S such that T = rng F and for every element n of holds F (n) ⊆ F (n + 1).

Let X be a set, and let S be a σ-field of subsets of X. A family of measureable sets of S is called a family of measureable non-increment sets of S if:

(Def.3) there exists a function F from into S such that it = rng F and for every element n of holds F (n + 1) ⊆ F (n).

We now state several propositions:

(20) For every set X and for every σ-field S of subsets of X and for every family T of measureable sets of S holds T is a family of measureable nonincrement sets of S if and only if there exists a function F from into S such that T = rng F and for every element n of holds F (n + 1) ⊆ F (n).

(21) Let X be a set. Let S be a σ-field of subsets of X. Then for every function N from into S and for every function F from into S such that F (0) = ∅ and for every element n of holds F (n + 1) = N(0) \ N(n) and N(n + 1) ⊆ N(n) holds rng F is a family of measureable non-decrement sets of S.

(22) For every set X and for every non-empty family S of subsets of X and for every function N from into S such that for every element n of holds N(n) ⊆ N(n + 1) for all natural numbers m, n such that n < m holds N(n) ⊆ N(m).

(23) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Let F be a function from into S. Suppose F (0) = N(0) and for every element n of holds F (n + 1) = N(n + 1) \ N(n) and N(n) ⊆ N(n + 1). Then for all natural numbers n, m such that n < m holds F (n) ∩ F (m) = ∅.

(24) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Then for every function F from into S such that F (0) = N(0) and for every element n of holds F (n+1) = N(n+1)\N(n) and N(n) ⊆ N(n + 1) holds ⋃ rng F = ⋃ rng N.

(25) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Then for every function F from into S such that F (0) = N(0) and for every element n of holds F (n+1) = N(n+1)\N(n) and N(n) ⊆ N(n + 1) holds F is a sequence of separated subsets of S.

(26) Let X be a set. Let S be a σ-field of subsets of X. Let N be a function from into S. Let F be a function from into S. Suppose F (0) = N(0) and for every element n of holds F (n + 1) = N(n + 1) \ N(n) and N(n) ⊆ N(n + 1). Then N(0) = F (0) and for every element n of holds N(n + 1) = F (n + 1) ∪ N(n).

(27) For every set X and for every σ-field S of subsets of X and for every σmeasure M on S and for every function F from into S such that for every element n of holds F (n) ⊆ F (n+1) holds M(⋃ rng F) = sup rng(M ·F). several properties of the σ-additive … 497

References

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

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

[3] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163–171, 1991.

[4] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173–183, 1991.

[5] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263–270, 1991.

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

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

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

[12] R. Sikorski. Rachunek różniczkowy i całkowy - funkcje wielu zmiennych. Biblioteka Matematyczna, PWN - Warszawa, 1968.

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

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

498

Metrics in the Cartesian Product - Part II

Stanisława Kanas Technical University of Rzeszów

Adam Lecko Technical University of Rzeszów

Summary. A continuation of [9]. It deals with the method of creation of the distance in the Cartesian product of metric spaces. The distance between two points belonging to Cartesian product of metric spaces has been defined as square root of the sum of squares of distances of appriopriate coordinates (or projections) of these points. It is shown that the product of metric spaces with such a distance is a metric space. Examples of metric spaces defined in this way are given.

MML Identifier: METRIC 4.

The articles [7], [15], [4], [5], [2], [6], [1], [10], [11], [3], [8], [13], [12], [14], and [9] provide the terminology and notation for this paper. We adopt the following convention: X, Y are metric spaces, x_{1}, y_{1}, z_{1} are elements of the carrier of X, and x_{2}, y_{2}, z_{2} are elements of the carrier of Y. Let us consider X, Y. The functor ρ^{[:X,Y:]} yields a function from [: [: the carrier of X, the carrier of Y :], [: the carrier of X, the carrier of Y :] :] into and is defined by: (Def.1) for all elements x_{1}, y_{1} of the carrier of X and for all elements x_{2}, y_{2}

of the carrier of Y and for all elements x, y of [: the carrier of X, the carrier of Y :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds ρ^{[:X,Y:]}(x,

y) = \sqrt{(ρ(x_{1}, y_{1}))^{2} + (ρ(x_{2}, y_{2}))^{2}}.

Next we state the proposition

(1) Let X be a metric space. Let Y be a metric space. Let F be a function from [: [: the carrier of X, the carrier of Y :], [: the carrier of X, the carrier of Y :] :] into . Then F = ρ^{[:X,Y:]} if and only if for all elements x_{1}, y_{1} of the carrier of X and for all elements x_{2}, y_{2} of the carrier of Y and for all elements x, y of [: the carrier of X, the carrier of Y :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds F (x, y) = \sqrt{(ρ(x_{1}, y_{1}))^{2} + (ρ(x_{2}, y_{2}))^{2}}.

Next we state several propositions:

(2) For all elements a, b of such that 0 ≤ a and 0 ≤ b holds \sqrt{a + b} = 0 if and only if a = 0 and b = 0.

(3) For all elements x, y of [: the carrier of X, the carrier of Y :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds ρ^{[:X,Y:]}(x, y) = 0 if and only if x = y.

(4) For all elements x, y of [: the carrier of X, the carrier of Y :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds ρ^{[:X,Y:]}(x, y) = ρ^{[:X,Y:]}(y, x).

(5) For all elements a, b, c, d of such that 0 ≤ a and 0 ≤ b and 0 ≤ c and 0 ≤ d holds \sqrt{(a + c)^{2} + (b + d)^{2}} ≤ \sqrt{a^{2} + b^{2}} + \sqrt{c^{2} + d^{2}}.

(6) For all elements x, y, z of [: the carrier of X, the carrier of Y :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ and z = ⟨⟨z_{1}, z_{2}⟩⟩ holds ρ^{[:X,Y:]}(x,

z) ≤ ρ^{[:X,Y:]}(x, y) + ρ^{[:X,Y:]}(y, z). Let us consider X, Y, and let x, y be elements of [: the carrier of X, the

carrier of Y :]. The functor ρ^{2}(x, y) yielding a real number is defined as follows:

(Def.2) ρ^{2}(x, y) = ρ^{[:X,Y:]}(x, y).

Next we state the proposition

(7) For all elements x, y of [: the carrier of X, the carrier of Y :] holds ρ^{2}(x, y) = ρ^{[:X,Y:]}(x, y).

Let X, Y be metric spaces. The functor [:X, Y:] yielding a metric space is defined as follows:

(Def.3) [:X, Y:] = ⟨[: the carrier of X, the carrier of Y :], ρ^{[:X,Y:]}⟩.

We now state the proposition

(8) For every metric space X and for every metric space Y holds ⟨[: the carrier of X, the carrier of Y :], ρ^{[:X,Y:]}⟩ is a metric space.

In the sequel Z will be a metric space and x_{3}, y_{3}, z_{3} will be elements of the carrier of Z. Let us consider X, Y, Z. The functor ρ^{[:X,Y,Z:]} yielding a function from [: [: the carrier of X, the carrier of Y, the carrier of Z:], [: the carrier of X, the carrier of Y, the carrier of Z:] :] into is defined by the condition (Def.4).

(Def.4) Let x_{1}, y_{1} be elements of the carrier of X. Let x_{2}, y_{2} be elements of the carrier of Y. Let x_{3}, y_{3} be elements of the carrier of Z. Then for all elements x, y of [: the carrier of X, the carrier of Y, the carrier of Z:] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{[:X,Y,Z:]}(x,

y) = \sqrt{(ρ(x_{1}, y_{1}))^{2} + (ρ(x_{2}, y_{2}))^{2} + (ρ(x_{3}, y_{3}))^{2}}.

One can prove the following propositions:

(9) Let X be a metric space. Let Y be a metric space. Let Z be a metric space. Let F be a function from [: [: the carrier of X, the carrier of Y, the carrier of Z:], [: the carrier of X, the carrier of Y, the carrier of Z:] :] into . Then F = ρ^{[:X,Y,Z:]} if and only if for all elements x_{1}, y_{1} of the carrier of X and for all elements x_{2}, y_{2} of the carrier of Y and for all elements x_{3}, y_{3} of the carrier of Z and for all elements x, y of [: the carrier of X, the carrier of Y, the carrier of Z:] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds F (x, y) = \sqrt{(ρ(x_{1}, y_{1}))^{2} + (ρ(x_{2}, y_{2}))^{2} + (ρ(x_{3}, y_{3}))^{2}}.

(10) For all elements x, y of [: the carrier of X, the carrier of Y, the carrier of Z:] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{[:X,Y,Z:]}(x,

y) = 0 if and only if x = y.

(11) For all elements x, y of [: the carrier of X, the carrier of Y, the carrier of Z:] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{[:X,Y,Z:]}(x,

y) = ρ^{[:X,Y,Z:]}(y, x).

(12) For all elements a, b, c of holds (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + (2 · a · b + 2 · a · c + 2 · b · c).

(13) Let a, b, c, d, e, f be elements of . Suppose 0 ≤ a and 0 ≤ b and 0 ≤ c and 0 ≤ d and 0 ≤ e and 0 ≤ f. Then 2 · (a · d) · (c · b) + 2 · (a · f) · (e · c) + 2 · (b · f) · (e · d) ≤ (a · d)^{2} + (c · b)^{2} + (a · f)^{2} + (e · c)^{2} + (b · f)^{2} + (e · d)^{2}.

(14) Let a, b, c, d, e, f be elements of . Then a^{2} ·d^{2} +(a^{2} ·f^{2} +c^{2} ·b^{2})+e^{2} · c^{2} +b^{2} ·f^{2} +e^{2} ·d^{2} +e^{2} ·f^{2} +b^{2} ·d^{2} +a^{2} ·c^{2} = (a^{2} +b^{2} +e^{2})·(c^{2} +d^{2} +f^{2}).

(15) Let a, b, c, d, e, f be elements of . Suppose 0 ≤ a and 0 ≤ b and 0 ≤ c and 0 ≤ d and 0 ≤ e and 0 ≤ f. Then (a · c + b · d + e · f)^{2} ≤ (a^{2} + b^{2} + e^{2}) · (c^{2} + d^{2} + f^{2}).

(16) Let x, y, z be elements of [: the carrier of X, the carrier of Y, the carrier of Z:]. Then if x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ and z = ⟨⟨z_{1}, z_{2}, z_{3}⟩⟩, then ρ^{[:X,Y,Z:]}(x, z) ≤ ρ^{[:X,Y,Z:]}(x, y) + ρ^{[:X,Y,Z:]}(y, z).

Let us consider X, Y, Z, and let x, y be elements of [: the carrier of X, the carrier of Y, the carrier of Z:]. The functor ρ^{3}(x, y) yielding a real number is defined as follows:

(Def.5) ρ^{3}(x, y) = ρ^{[:X,Y,Z:]}(x, y).

One can prove the following proposition

(17) For all elements x, y of [: the carrier of X, the carrier of Y, the carrier

of Z:] holds ρ^{3}(x, y) = ρ^{[:X,Y,Z:]}(x, y). Let X, Y, Z be metric spaces. The functor [:X, Y:] yields a metric space and

is defined by:

(Def.6) [:X, Y:] = ⟨[: the carrier of X, the carrier of Y, the carrier of Z:], ρ^{[:X,Y,Z:]}⟩.

The following proposition is true

(18) For every metric space X and for every metric space Y and for every metric space Z holds ⟨[: the carrier of X, the carrier of Y, the carrier of Z:], ρ^{[:X,Y,Z:]}⟩ is a metric space.

In the sequel x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2} denote elements of . The function ρ^{[: , :]} from [: [: , :], [: , :] :] into is defined by:

(Def.7) for all elements x_{1}, y_{1}, x_{2}, y_{2} of and for all elements x, y of [: , :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds ρ^{[: , :]} (x, y) = ρ_{}(x_{1}, y_{1}) + ρ_{}(x_{2}, y_{2}).

The following propositions are true:

(19) For all elements x_{1}, x_{2}, y_{1}, y_{2} of and for all elements x, y of [: , :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds ρ^{[: , :]} (x, y) = 0 if and only if x = y.

(20) For all elements x, y of [: , :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds ρ^{[: , :]} (x, y) = ρ^{[: , :]} (y, x).

(21) For all elements x, y, z of [: , :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1},

y_{2}⟩⟩ and z = ⟨⟨z_{1}, z_{2}⟩⟩ holds ρ^{[: , :]} (x, z) ≤ ρ^{[: , :]} (x, y) + ρ^{[: , :]} (y, z). The metric space [: _{M}, _{M}:] is defined by:

(Def.8) [: _{M}, _{M}:] = ⟨[: , :], ρ^{[: , :]} ⟩.

The function ρ^{^{2}} from [: [: , :], [: , :] :] into is defined as follows: (Def.9) for all elements x_{1}, y_{1}, x_{2}, y_{2} of and for all elements x, y of [: , :]

such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds

ρ^{^{2}} (x, y) = \sqrt{ρ_{}(x_{1}, y_{1})^{2} + ρ_{}(x_{2}, y_{2})^{2}}.

We now state three propositions:

(22) For all elements x_{1}, x_{2}, y_{1}, y_{2} of and for all elements x, y of [: , :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds ρ^{^{2}} (x, y) = 0 if and only if x = y.

(23) For all elements x, y of [: , :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1}, y_{2}⟩⟩ holds ρ^{^{2}} (x, y) = ρ^{^{2}} (y, x).

(24) For all elements x, y, z of [: , :] such that x = ⟨⟨x_{1}, x_{2}⟩⟩ and y = ⟨⟨y_{1},

y_{2}⟩⟩ and z = ⟨⟨z_{1}, z_{2}⟩⟩ holds ρ^{^{2}} (x, z) ≤ ρ^{^{2}} (x, y) + ρ^{^{2}} (y, z). The Euclidean plain being a metric space is defined as follows:

(Def.10) the Euclidean plain= ⟨[: , :], ρ^{^{2}} ⟩.

In the sequel x_{3}, y_{3}, z_{3} denote elements of . The function ρ^{[: , , :]} from [: [: , , :], [: , , :] :] into is defined by the condition (Def.11).

(Def.11) Let x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3} be elements of . Then for all elements x, y of [: , , :] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{[: , , :]} (x,

y) = ρ_{}(x_{1}, y_{1}) + ρ_{}(x_{2}, y_{2}) + ρ_{}(x_{3}, y_{3}).

We now state three propositions:

(25) For all elements x_{1}, x_{2}, y_{1}, y_{2}, x_{3}, y_{3} of and for all elements x, y of [: , , :] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{[: , , :]} (x,

y) = 0 if and only if x = y.

(26) For all elements x, y of [: , , :] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{[: , , :]} (x, y) = ρ^{[: , , :]} (y, x).

(27) For all elements x, y, z of [: , , :] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ and z = ⟨⟨z_{1}, z_{2}, z_{3}⟩⟩ holds ρ^{[: , , :]} (x, z) ≤ ρ^{[: , , :]} (x,

y) + ρ^{[: , , :]} (y, z). The metric space [: _{M}, _{M}, _{M}:] is defined as follows:

(Def.12) [: _{M}, _{M}, _{M}:] = ⟨[: , , :], ρ^{[: , , :]} ⟩.

metrics in the cartesian product - part … 503

The function ρ^{^{3}} from [: [: , , :], [: , , :] :] into is defined by the condition (Def.13).

(Def.13) Let x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3} be elements of . Then for all elements x, y of [: , , :] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{^{3}} (x,

y) = \sqrt{ρ_{}(x_{1}, y_{1})^{2} + ρ_{}(x_{2}, y_{2})^{2} + ρ_{}(x_{3}, y_{3})^{2}}.

One can prove the following three propositions:

(28) For all elements x_{1}, x_{2}, y_{1}, y_{2}, x_{3}, y_{3} of and for all elements x, y of [: , , :] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{^{3}} (x,

y) = 0 if and only if x = y.

(29) For all elements x, y of [: , , :] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ holds ρ^{^{3}} (x, y) = ρ^{^{3}} (y, x).

(30) For all elements x, y, z of [: , , :] such that x = ⟨⟨x_{1}, x_{2}, x_{3}⟩⟩ and y = ⟨⟨y_{1}, y_{2}, y_{3}⟩⟩ and z = ⟨⟨z_{1}, z_{2}, z_{3}⟩⟩ holds ρ^{^{3}} (x, z) ≤ ρ^{^{3}} (x, y) + ρ^{^{3}} (y, z).

The Euclidean space being a metric space is defined as follows: (Def.14) the Euclidean space= ⟨[: , , :], ρ^{^{3}} ⟩.

References

[1] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537–541, 1990.

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

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

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

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

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

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

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

[9] Stanisława Kanas and Jan Stankiewicz. Metrics in Cartesian product. Formalized Mathematics, 2(2):193–197, 1991.

[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.

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

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

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

504 stanisława kanas and adam lecko Received July 8, 1991

Fix Point Theorem for Compact Spaces

Alicia de la Cruz Universidad Politecnica de Madrid

Summary. The Banach theorem in a compact metric spaces is proved.

MML Identifier: ALI2.

The terminology and notation used in this paper have been introduced in the following papers: [9], [15], [3], [4], [8], [11], [13], [9], [11], [5], [7], [18], [6], [17], [1], [2], [6], [4], and [5]. In the sequel M will be a metric space. Next we state the proposition

(1) For every set F such that F is finite and F ≠ ∅ and for all sets B, C such that B ∈ F and C ∈ F holds B ⊆ C or C ⊆ B there exists a set m such that m ∈ F and for every set C such that C ∈ F holds m ⊆ C.

Let M be a metric space. A function from the carrier of M into the carrier of M is said to be a contraction of M if: (Def.1) there exists a real number L such that 0 < L and L < 1 and for all

points x, y of M holds ρ(it(x), it(y)) ≤ L · ρ(x, y).

Next we state the proposition

(2) For every contraction f of M such that M_{top} is compact there exists a point c of M such that f(c) = c and for every point x of M such that f(x) = x holds x = c.

References

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

[2] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 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.

506 alicia de la cruz

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

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

[7] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

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

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

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

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

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

[15] Konrad Raczkowski and Andrzej Ne‘dzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

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

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

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

[19] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231–237, 1990.

Quadratic Inequalities

Jan Popiołek Warsaw University Białystok

Summary. Consider a quadratic trinomial of the form P (x) = ax^{2} + bx + c, where a ≠ 0. The determinat of the equation P (x) = 0 is of the form Δ(a, b, c) = b^{2} − 4ac. We prove several quadratic inequalities when Δ(a, b, c) < 0, Δ(a, b, c) = 0 and Δ(a, b, c) > 0.

MML Identifier: QUIN 1.

The articles [3], [1], [2], and [4] provide the terminology and notation for this paper. In the sequel x is a real number and a, b, c are real numbers. Let us consider a, b, c. The functor Δ(a, b, c) yielding a real number is defined as follows: (Def.1) Δ(a, b, c) = b^{2} − 4 · a · c.

The following propositions are true:

(1) If a ≠ 0, then a · x^{2} + b · x + c = a · (x + \frac{b}{2·a})^{2} − \frac{Δ(a,b,c)}{4·a}.

(2) If a > 0 and Δ(a, b, c) ≤ 0, then a · x^{2} + b · x + c ≥ 0.

(3) If a > 0 and Δ(a, b, c) < 0, then a · x^{2} + b · x + c > 0.

(4) If a < 0 and Δ(a, b, c) ≤ 0, then a · x^{2} + b · x + c ≤ 0.

(5) If a < 0 and Δ(a, b, c) < 0, then a · x^{2} + b · x + c < 0.

(6) If a > 0 and a · x^{2} + b · x + c ≥ 0, then (2 · a · x + b)^{2} − Δ(a, b, c) ≥ 0.

(7) If a > 0 and a · x^{2} + b · x + c > 0, then (2 · a · x + b)^{2} − Δ(a, b, c) > 0.

(8) If a < 0 and a · x^{2} + b · x + c ≤ 0, then (2 · a · x + b)^{2} − Δ(a, b, c) ≥ 0.

(9) If a < 0 and a · x^{2} + b · x + c < 0, then (2 · a · x + b)^{2} − Δ(a, b, c) > 0.

(10) If for every x holds a · x^{2} + b · x + c ≥ 0 and a > 0, then Δ(a, b, c) ≤ 0.

(11) If for every x holds a · x^{2} + b · x + c ≤ 0 and a < 0, then Δ(a, b, c) ≤ 0.

(12) If for every x holds a · x^{2} + b · x + c > 0 and a > 0, then Δ(a, b, c) < 0.

(13) If for every x holds a · x^{2} + b · x + c < 0 and a < 0, then Δ(a, b, c) < 0.

(14) If a ≠ 0 and a · x^{2} + b · x + c = 0, then (2 · a · x + b)^{2} − Δ(a, b, c) = 0.

(15) Suppose a ≠ 0 and Δ(a, b, c) > 0 and a · x^{2} + b · x + c = 0. Then

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

(16) Suppose a ≠ 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c = a · (x −

\frac{−b−\sqrt{Δ(a,b,c)}}{2·a}) · (x − ^{\frac{−b+\sqrt{Δ(a,b,c)}}{2·a}}).

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

(18) Suppose a < 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c > 0 if and only

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

(19) Suppose a < 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c < 0 if and only

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

(20) Suppose a < 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c ≥ 0 if and only

if ^{\frac{−b+\sqrt{Δ(a,b,c)}}{2·a}} ≤ x and x ≤ ^{\frac{−b−\sqrt{Δ(a,b,c)}}{2·a}}.

(21) Suppose a < 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c ≤ 0 if and only

if x ≤ ^{\frac{−b+\sqrt{Δ(a,b,c)}}{2·a}} or x ≥ ^{\frac{−b−\sqrt{Δ(a,b,c)}}{2·a}}.

(22) If a ≠ 0 and Δ(a, b, c) = 0 and a · x^{2} + b · x + c = 0, then x = −\frac{b}{2·a}.

(23) If a > 0 and (2 · a · x + b)^{2} − Δ(a, b, c) > 0, then a · x^{2} + b · x + c > 0.

(24) If a > 0 and Δ(a, b, c) = 0, then a · x^{2} + b · x + c > 0 if and only if x ≠ −\frac{b}{2·a}.

(25) If a < 0 and (2 · a · x + b)^{2} − Δ(a, b, c) > 0, then a · x^{2} + b · x + c < 0.

(26) If a < 0 and Δ(a, b, c) = 0, then a · x^{2} + b · x + c < 0 if and only if x ≠ −\frac{b}{2·a}.

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

(28) Suppose a > 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c < 0 if and only

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

(29) Suppose a > 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c > 0 if and only

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

(30) Suppose a > 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c ≤ 0 if and only

if ^{\frac{−b−\sqrt{Δ(a,b,c)}}{2·a}} ≤ x and x ≤ ^{\frac{−b+\sqrt{Δ(a,b,c)}}{2·a}}.

(31) Suppose a > 0 and Δ(a, b, c) > 0. Then a · x^{2} + b · x + c ≥ 0 if and only

if x ≤ ^{\frac{−b−\sqrt{Δ(a,b,c)}}{2·a}} or x ≥ ^{\frac{−b+\sqrt{Δ(a,b,c)}}{2·a}}.

References

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

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

quadratic inequalities 509

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

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

510

Introduction to Banach and Hilbert Spaces - Part I

Jan Popiołek Warsaw University Białystok

Summary. Basing on the notion of real linear space (see [15]) we introduce real unitary space. At first, we define the scalar product of two vectors and examine some of its properties. On the basis of this notion we introduce the norm and the distance in real unitary space and study the properties of these concepts. Next, proceeding from the definition of the sequence in real unitary space and basic operations on sequences we prove several theorems which will be used in our further considerations.

MML Identifier: BHSP 1.

The terminology and notation used here are introduced in the following articles: [5], [12], [16], [3], [4], [1], [6], [2], [17], [10], [11], [9], [15], [14], [13], [8], and [7]. We consider unitary space structures which are systems

⟨vectors, a scalar product⟩, where the vectors constitute a real linear space and the scalar product is a function from [: the vectors of the vectors, the vectors of the vectors :] into .

In the sequel X will denote a unitary space structure and a, b will denote 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, y will denote points of X. Let us consider X, x, y. The functor (x|y) yielding a real number is defined as follows: (Def.1) (x|y) = (the scalar product of X)(⟨⟨x, y⟩⟩).

A unitary space structure is said to be a real unitary space if it satisfies the condition (Def.2). (Def.2) Let x, y, z be points of it. Given a. Then

(i) (x|x) = 0 if and only if x = 0_{the vectors of it}, (ii) 0 ≤ (x|x), (iii) (x|y) = (y|x), (iv) ((x + y)|z) = (x|z) + (y|z),

(v) ((a · x)|y) = a · (x|y).

We follow the rules: X denotes a real unitary space and x, y, z, u, v denote points of X. We now state a number of propositions:

(1) (x|x) = 0 if and only if x = 0_{the vectors of X}.

(2) 0 ≤ (x|x).

(3) (x|y) = (y|x).

(4) ((x + y)|z) = (x|z) + (y|z).

(5) ((a · x)|y) = a · (x|y).

(6) (0_{the vectors of X}|0_{the vectors of X}) = 0.

(7) (x|(y + z)) = (x|y) + (x|z).

(8) (x|(a · y)) = a · (x|y).

(9) ((a · x)|y) = (x|(a · y)).

(10) ((a · x + b · y)|z) = a · (x|z) + b · (y|z).

(11) (x|(a · y + b · z)) = a · (x|y) + b · (x|z).

(12) ((−x)|y) = (x|−y).

(13) ((−x)|y) = −(x|y).

(14) (x|−y) = −(x|y).

(15) ((−x)|−y) = (x|y).

(16) ((x − y)|z) = (x|z) − (y|z).

(17) (x|(y − z)) = (x|y) − (x|z).

(18) ((x − y)|(u − v)) = ((x|u) − (x|v) − (y|u)) + (y|v).

(19) (0_{the vectors of X}|x) = 0.

(20) (x|0_{the vectors of X}) = 0.

(21) ((x + y)|(x + y)) = (x|x) + 2 · (x|y) + (y|y).

(22) ((x + y)|(x − y)) = (x|x) − (y|y).

(23) ((x − y)|(x − y)) = ((x|x) − 2 · (x|y)) + (y|y).

(24) |(x|y)| ≤ \sqrt{(x|x)} · \sqrt{(y|y)}.

Let us consider X, x, y. We say that x, y are ortogonal if and only if: (Def.3) (x|y) = 0.

The following propositions are true:

(25) If x, y are ortogonal, then y, x are ortogonal.

(26) If x, y are ortogonal, then x, −y are ortogonal.

(27) If x, y are ortogonal, then −x, y are ortogonal.

(28) If x, y are ortogonal, then −x, −y are ortogonal.

(29) x, 0_{the vectors of X} are ortogonal.

(30) If x, y are ortogonal, then ((x + y)|(x + y)) = (x|x) + (y|y).

(31) If x, y are ortogonal, then ((x − y)|(x − y)) = (x|x) + (y|y). Let us consider X, x. The functor ∥x∥ yielding a real number is defined by:

(Def.4) ∥x∥ = \sqrt{(x|x)}.

The following propositions are true:

(32) ∥x∥ = 0 if and only if x = 0_{the vectors of X}.

(33) ∥a · x∥ = |a| · ∥x∥.

(34) 0 ≤ ∥x∥.

(35) |(x|y)| ≤ ∥x∥ · ∥y∥.

(36) ∥x + y∥ ≤ ∥x∥ + ∥y∥.

(37) ∥−x∥ = ∥x∥.

(38) ∥x∥ − ∥y∥ ≤ ∥x − y∥.

(39) |∥x∥ − ∥y∥| ≤ ∥x − y∥.

Let us consider X, x, y. The functor ρ(x, y) yielding a real number is defined by: (Def.5) ρ(x, y) = ∥x − y∥.

One can prove the following propositions:

(40) ρ(x, y) = ρ(y, x).

(41) ρ(x, x) = 0.

(42) ρ(x, z) ≤ ρ(x, y) + ρ(y, z).

(43) x ≠ y if and only if ρ(x, y) ≠ 0.

(44) ρ(x, y) ≥ 0.

(45) x ≠ y if and only if ρ(x, y) > 0.

(46) ρ(x, y) = \sqrt{((x − y)|(x − y))}.

(47) ρ(x + y, u + v) ≤ ρ(x, u) + ρ(y, v).

(48) ρ(x − y, u − v) ≤ ρ(x, u) + ρ(y, v).

(49) ρ(x − z, y − z) = ρ(x, y).

(50) ρ(x − z, y − z) ≤ ρ(z, x) + ρ(z, y).

Let us consider X. A subset of X is a subset of the vectors of the vectors of X.

Let us consider X. A function is called a sequence of X if: (Def.6) dom it = and rng it ⊆ the vectors of the vectors of X.

For simplicity we adopt the following rules: s_{1}, s_{2}, s_{3}, s_{4}, s′_{1} denote sequences of X, k, n, m denote natural numbers, f denotes a function, and d is arbitrary. We now state four propositions:

(51) f is a sequence of X if and only if dom f = and rng f ⊆ the vectors of the vectors of X.

(52) f is a sequence of X if and only if dom f = and for every d such that d ∈ holds f(d) is a point of X.

(53) For all s_{1}, s′_{1} such that for every n holds s_{1}(n) = s′_{1}(n) holds s_{1} = s′_{1}.

(54) For every n holds s_{1}(n) is a point of X.

Let us consider X, s_{1}, n. Then s_{1}(n) is a point of X. The scheme Ex Seq in RUS concerns a real unitary space A and a unary

functor F yielding a point of A and states that:

there exists a sequence s_{1} of A such that for every n holds s_{1}(n) = F(n) for all values of the parameters.

Let us consider X, s_{2}, s_{3}. The functor s_{2} + s_{3} yielding a sequence of X is defined by: (Def.7) for every n holds (s_{2} + s_{3})(n) = s_{2}(n) + s_{3}(n).

Let us consider X, s_{2}, s_{3}. The functor s_{2} − s_{3} yielding a sequence of X is defined as follows: (Def.8) for every n holds (s_{2} − s_{3})(n) = s_{2}(n) − s_{3}(n).

Let us consider X, s_{1}, a. The functor a · s_{1} yields a sequence of X and is defined as follows: (Def.9) for every n holds (a · s_{1})(n) = a · s_{1}(n).

Let us consider X, s_{1}. The functor −s_{1} yields a sequence of X and is defined by:

(Def.10) for every n holds (−s_{1})(n) = −s_{1}(n).

Let us consider X, s_{1}. We say that s_{1} is constant if and only if: (Def.11) there exists x such that for every n holds s_{1}(n) = x.

Let us consider X, s_{1}, x. The functor s_{1} + x yielding a sequence of X is defined as follows:

(Def.12) for every n holds (s_{1} + x)(n) = s_{1}(n) + x.

Let us consider X, s_{1}, x. The functor s_{1} − x yields a sequence of X and is defined by:

(Def.13) for every n holds (s_{1} − x)(n) = s_{1}(n) − x.

We now state a number of propositions:

(55) s_{2} + s_{3} = s_{3} + s_{2}.

(56) s_{2} + (s_{3} + s_{4}) = s_{2} + s_{3} + s_{4}.

(57) If s_{2} is constant and s_{3} is constant and s_{1} = s_{2} +s_{3}, then s_{1} is constant.

(58) If s_{2} is constant and s_{3} is constant and s_{1} = s_{2} −s_{3}, then s_{1} is constant.

(59) If s_{2} is constant and s_{1} = a · s_{2}, then s_{1} is constant.

(60) For every x there exists s_{1} such that rng s_{1} = {x}.

(61) There exists s_{1} such that rng s_{1} = {0_{the vectors of X}}.

(62) If there exists x such that for every n holds s_{1}(n) = x, then there exists x such that rng s_{1} = {x}.

(63) If there exists x such that rng s_{1} = {x}, then for every n holds s_{1}(n) = s_{1}(n + 1).

(64) If for every n holds s_{1}(n) = s_{1}(n + 1), then for all n, k holds s_{1}(n) = s_{1}(n + k).

introduction to banach and hilbert … 515

(65) If for all n, k holds s_{1}(n) = s_{1}(n + k), then for all n, m holds s_{1}(n) = s_{1}(m).

(66) If for all n, m holds s_{1}(n) = s_{1}(m), then there exists x such that for every n holds s_{1}(n) = x.

(67) s_{1} is constant if and only if there exists x such that rng s_{1} = {x}.

(68) s_{1} is constant if and only if for every n holds s_{1}(n) = s_{1}(n + 1).

(69) s_{1} is constant if and only if for all n, k holds s_{1}(n) = s_{1}(n + k).

(70) s_{1} is constant if and only if for all n, m holds s_{1}(n) = s_{1}(m).

(71) s_{2} − s_{3} = s_{2} + −s_{3}.

(72) s_{1} = s_{1} + 0_{the vectors of X}.

(73) a · (s_{2} + s_{3}) = a · s_{2} + a · s_{3}.

(74) (a + b) · s_{1} = a · s_{1} + b · s_{1}.

(75) a · b · s_{1} = a · (b · s_{1}).

(76) 1 · s_{1} = s_{1}.

(77) (−1) · s_{1} = −s_{1}.

(78) s_{1} − x = s_{1} + −x.

(79) s_{2} − s_{3} = −(s_{3} − s_{2}).

(80) s_{1} = s_{1} − 0_{the vectors of X}.

(81) s_{1} = −−s_{1}.

(82) s_{2} − (s_{3} + s_{4}) = s_{2} − s_{3} − s_{4}.

(83) (s_{2} + s_{3}) − s_{4} = s_{2} + (s_{3} − s_{4}).

(84) s_{2} − (s_{3} − s_{4}) = (s_{2} − s_{3}) + s_{4}.

(85) a · (s_{2} − s_{3}) = a · s_{2} − a · s_{3}.

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

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

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

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

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

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

516 jan popiołek

[12] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, [13] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized

Mathematics, 1(3):445–449, 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.

Introduction to Banach and Hilbert Spaces - Part II

Jan Popiołek Warsaw University Białystok

Summary. A continuation of [8]. It contains the definitions of the convergent sequence and the limit of the sequence. The convergence with respect to the norm and the distance is also introduced. Last part is devoted to the following concepts: ball, closed ball and sphere.

MML Identifier: BHSP 2.

The articles [5], [14], [19], [3], [4], [1], [7], [6], [2], [20], [12], [18], [13], [11], [17], [16], [15], [10], [9], and [8] provide the notation and terminology for this paper. For simplicity we follow a convention: X is a real unitary space, x, y, z are points of X, g, g_{1}, g_{2} are points of X, a, q, r are real numbers, s_{1}, s_{2}, s_{3}, s′_{1} are sequences of X, and k, n, m are natural numbers. Let us consider X, s_{1}. We say that s_{1} is convergent if and only if: (Def.1) there exists g such that for every r such that r > 0 there exists m such

that for every n such that n ≥ m holds ρ(s_{1}(n), g) < r. The following propositions are true:

(1) If s_{1} is constant, then s_{1} is convergent.

(2) If s_{1} is convergent and there exists k such that for every n such that k ≤ n holds s′_{1}(n) = s_{1}(n), then s′_{1} is convergent.

(3) If s_{2} is convergent and s_{3} is convergent, then s_{2} + s_{3} is convergent.

(4) If s_{2} is convergent and s_{3} is convergent, then s_{2} − s_{3} is convergent.

(5) If s_{1} is convergent, then a · s_{1} is convergent.

(6) If s_{1} is convergent, then −s_{1} is convergent.

(7) If s_{1} is convergent, then s_{1} + x is convergent.

(8) If s_{1} is convergent, then s_{1} − x is convergent.

(9) s_{1} is convergent if and only if there exists g such that for every r such that r > 0 there exists m such that for every n such that n ≥ m holds ∥s_{1}(n) − g∥ < r.

Let us consider X, s_{1}. Let us assume that s_{1} is convergent. The functor lim s_{1} yields a point of X and is defined as follows: (Def.2) for every r such that r > 0 there exists m such that for every n such

that n ≥ m holds ρ(s_{1}(n), lim s_{1}) < r. Next we state a number of propositions:

(10) If s_{1} is constant and x ∈ rng s_{1}, then lim s_{1} = x.

(11) If s_{1} is constant and there exists n such that s_{1}(n) = x, then lim s_{1} = x.

(12) If s_{1} is convergent and there exists k such that for every n such that n ≥ k holds s′_{1}(n) = s_{1}(n), then lim s_{1} = lim s′_{1}.

(13) If s_{2} is convergent and s_{3} is convergent, then lim(s_{2} + s_{3}) = lim s_{2} + lim s_{3}.

(14) If s_{2} is convergent and s_{3} is convergent, then lim(s_{2} − s_{3}) = lim s_{2} − lim s_{3}.

(15) If s_{1} is convergent, then lim(a · s_{1}) = a · lim s_{1}.

(16) If s_{1} is convergent, then lim(−s_{1}) = −lim s_{1}.

(17) If s_{1} is convergent, then lim(s_{1} + x) = lim s_{1} + x.

(18) If s_{1} is convergent, then lim(s_{1} − x) = lim s_{1} − x.

(19) If s_{1} is convergent, then lim s_{1} = g if and only if for every r such that r > 0 there exists m such that for every n such that n ≥ m holds ∥s_{1}(n) − g∥ < r.

Let us consider X, s_{1}. The functor ∥s_{1}∥ yielding a sequence of real numbers is defined by: (Def.3) for every n holds ∥s_{1}∥(n) = ∥s_{1}(n)∥.

Next we state three propositions:

(20) If s_{1} is convergent, then ∥s_{1}∥ is convergent.

(21) If s_{1} is convergent and lim s_{1} = g, then ∥s_{1}∥ is convergent and lim∥s_{1}∥ = ∥g∥.

(22) If s_{1} is convergent and lim s_{1} = g, then ∥s_{1} − g∥ is convergent and

lim∥s_{1} − g∥ = 0. Let us consider X, s_{1}, x. The functor ρ(s_{1}, x) yielding a sequence of real

numbers is defined by: (Def.4) for every n holds (ρ(s_{1}, x))(n) = ρ(s_{1}(n), x).

We now state a number of propositions:

(23) If s_{1} is convergent and lim s_{1} = g, then ρ(s_{1}, g) is convergent.

(24) If s_{1} is convergent and lim s_{1} = g, then ρ(s_{1}, g) is convergent and lim ρ(s_{1}, g) = 0.

(25) If s_{2} is convergent and lim s_{2} = g_{1} and s_{3} is convergent and lim s_{3} = g_{2}, then ∥s_{2} + s_{3}∥ is convergent and lim∥s_{2} + s_{3}∥ = ∥g_{1} + g_{2}∥.

(26) If s_{2} is convergent and lim s_{2} = g_{1} and s_{3} is convergent and lim s_{3} = g_{2}, then ∥(s_{2} +s_{3})−(g_{1} +g_{2})∥ is convergent and lim∥(s_{2} +s_{3})−(g_{1} +g_{2})∥ = 0.

(27) If s_{2} is convergent and lim s_{2} = g_{1} and s_{3} is convergent and lim s_{3} = g_{2}, then ∥s_{2} − s_{3}∥ is convergent and lim∥s_{2} − s_{3}∥ = ∥g_{1} − g_{2}∥.

(28) If s_{2} is convergent and lim s_{2} = g_{1} and s_{3} is convergent and lim s_{3} = g_{2}, then ∥s_{2} − s_{3} − (g_{1} − g_{2})∥ is convergent and lim∥s_{2} − s_{3} − (g_{1} − g_{2})∥ = 0.

(29) If s_{1} is convergent and lim s_{1} = g, then ∥a · s_{1}∥ is convergent and lim∥a · s_{1}∥ = ∥a · g∥.

(30) If s_{1} is convergent and lim s_{1} = g, then ∥a · s_{1} − a · g∥ is convergent and lim∥a · s_{1} − a · g∥ = 0.

(31) If s_{1} is convergent and lim s_{1} = g, then ∥−s_{1}∥ is convergent and lim∥−s_{1}∥ = ∥−g∥.

(32) If s_{1} is convergent and lim s_{1} = g, then ∥−s_{1} − −g∥ is convergent and lim∥−s_{1} − −g∥ = 0.

(33) If s_{1} is convergent and lim s_{1} = g, then ∥(s_{1} +x)−(g +x)∥ is convergent and lim∥(s_{1} + x) − (g + x)∥ = 0.

(34) If s_{1} is convergent and lim s_{1} = g, then ∥s_{1} − x∥ is convergent and lim∥s_{1} − x∥ = ∥g − x∥.

(35) If s_{1} is convergent and lim s_{1} = g, then ∥s_{1} − x − (g − x)∥ is convergent and lim∥s_{1} − x − (g − x)∥ = 0.

(36) If s_{2} is convergent and lim s_{2} = g_{1} and s_{3} is convergent and lim s_{3} = g_{2}, then ρ(s_{2} + s_{3}, g_{1} + g_{2}) is convergent and lim ρ(s_{2} + s_{3}, g_{1} + g_{2}) = 0.

(37) If s_{2} is convergent and lim s_{2} = g_{1} and s_{3} is convergent and lim s_{3} = g_{2}, then ρ(s_{2} − s_{3}, g_{1} − g_{2}) is convergent and lim ρ(s_{2} − s_{3}, g_{1} − g_{2}) = 0.

(38) If s_{1} is convergent and lim s_{1} = g, then ρ(a · s_{1}, a · g) is convergent and lim ρ(a · s_{1}, a · g) = 0.

(39) If s_{1} is convergent and lim s_{1} = g, then ρ(s_{1} + x, g + x) is convergent

and lim ρ(s_{1} + x, g + x) = 0. Let us consider X, x, r. Let us assume that r ≥ 0. The functor Ball(x, r)

yielding a subset of X is defined by: (Def.5) Ball(x, r) = {y : ∥x − y∥ < r}, where y ranges over points of X.

Let us consider X, x, r. Let us assume that r ≥ 0. The functor \overline{Ball}(x, r) yielding a subset of X is defined by: (Def.6) \overline{Ball}(x, r) = \{y : ∥x − y∥ ≤ r\}, where y ranges over points of X.

Let us consider X, x, r. Let us assume that r ≥ 0. The functor Sphere(x, r) yields a subset of X and is defined as follows: (Def.7) Sphere(x, r) = {y : ∥x − y∥ = r}, where y ranges over points of X.

The following propositions are true:

(40) If r ≥ 0, then z ∈ Ball(x, r) if and only if ∥x − z∥ < r.

(41) If r ≥ 0, then z ∈ Ball(x, r) if and only if ρ(x, z) < r.

(42) If r > 0, then x ∈ Ball(x, r).

(43) If r ≥ 0, then if y ∈ Ball(x, r) and z ∈ Ball(x, r), then ρ(y, z) < 2 · r.

(44) If r ≥ 0, then if y ∈ Ball(x, r), then y − z ∈ Ball(x − z, r).

(45) If r ≥ 0, then if y ∈ Ball(x, r), then y − x ∈ Ball(0_{the vectors of X}, r).

(46) If r ≥ 0, then if y ∈ Ball(x, r) and r ≤ q, then y ∈ Ball(x, q).

(47) If r ≥ 0, then z ∈ \overline{Ball}(x, r) if and only if ∥x − z∥ ≤ r.

(48) If r ≥ 0, then z ∈ \overline{Ball}(x, r) if and only if ρ(x, z) ≤ r.

(49) If r ≥ 0, then x ∈ \overline{Ball}(x, r).

(50) If r ≥ 0, then if y ∈ Ball(x, r), then y ∈ \overline{Ball}(x, r).

(51) If r ≥ 0, then z ∈ Sphere(x, r) if and only if ∥x − z∥ = r.

(52) If r ≥ 0, then z ∈ Sphere(x, r) if and only if ρ(x, z) = r.

(53) If r ≥ 0, then if y ∈ Sphere(x, r), then y ∈ \overline{Ball}(x, r).

(54) If r ≥ 0, then Ball(x, r) ⊆ \overline{Ball}(x, r).

(55) If r ≥ 0, then Sphere(x, r) ⊆ \overline{Ball}(x, r).

(56) If r ≥ 0, then Ball(x, r) ∪ Sphere(x, r) = \overline{Ball}(x, r).

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

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

[8] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

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

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

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

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

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

[16] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

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

introduction to banach and hilbert … 521

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

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

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

522

Introduction to Banach and Hilbert Spaces - Part III

Jan Popiołek Warsaw University Białystok

Summary. A continuation of [11] and of [12]. First we define the following concepts: the Cauchy sequence, the bounded sequence and the subsequence. The last part consists definitions of the complete space and the Hilbert space.

MML Identifier: BHSP 3.

The articles [5], [18], [22], [3], [4], [1], [10], [8], [9], [7], [15], [2], [23], [16], [17], [14], [21], [20], [19], [13], [11], [12], and [6] provide the notation and terminology for this paper. For simplicity we follow the rules: X is a real unitary space, x is a point of X, g is a point of X, a, r are real numbers, M is a real number, s_{1}, s_{2}, s_{3}, s_{4} are sequences of X, N_{1} is an increasing sequence of naturals, and k, n, m are natural numbers. Let us consider X, s_{1}. We say that s_{1} is a Cauchy sequence if and only if: (Def.1) for every r such that r > 0 there exists k such that for all n, m such

that n ≥ k and m ≥ k holds ρ(s_{1}(n), s_{1}(m)) < r. One can prove the following propositions:

(1) If s_{1} is constant, then s_{1} is a Cauchy sequence.

(2) s_{1} is a Cauchy sequence if and only if for every r such that r > 0 there exists k such that for all n, m such that n ≥ k and m ≥ k holds ∥s_{1}(n) − s_{1}(m)∥ < r.

(3) If s_{2} is a Cauchy sequence and s_{3} is a Cauchy sequence, then s_{2} + s_{3} is a Cauchy sequence.

(4) If s_{2} is a Cauchy sequence and s_{3} is a Cauchy sequence, then s_{2} − s_{3} is a Cauchy sequence.

(5) If s_{1} is a Cauchy sequence, then a · s_{1} is a Cauchy sequence.

(6) If s_{1} is a Cauchy sequence, then −s_{1} is a Cauchy sequence.

(7) If s_{1} is a Cauchy sequence, then s_{1} + x is a Cauchy sequence.

(8) If s_{1} is a Cauchy sequence, then s_{1} − x is a Cauchy sequence.

(9) If s_{1} is convergent, then s_{1} is a Cauchy sequence.

Let us consider X, s_{2}, s_{3}. We say that s_{2} is compared to s_{3} if and only if: (Def.2) for every r such that r > 0 there exists m such that for every n such

that n ≥ m holds ρ(s_{2}(n), s_{3}(n)) < r. One can prove the following propositions:

(10) s_{1} is compared to s_{1}.

(11) If s_{2} is compared to s_{3}, then s_{3} is compared to s_{2}.

(12) If s_{2} is compared to s_{3} and s_{3} is compared to s_{4}, then s_{2} is compared to s_{4}.

(13) s_{2} is compared to s_{3} if and only if for every r such that r > 0 there exists m such that for every n such that n ≥ m holds ∥s_{2}(n)−s_{3}(n)∥ < r.

(14) If there exists k such that for every n such that n ≥ k holds s_{2}(n) = s_{3}(n), then s_{2} is compared to s_{3}.

(15) If s_{2} is a Cauchy sequence and s_{2} is compared to s_{3}, then s_{3} is a Cauchy sequence.

(16) If s_{2} is convergent and s_{2} is compared to s_{3}, then s_{3} is convergent.

(17) If s_{2} is convergent and lim s_{2} = g and s_{2} is compared to s_{3}, then s_{3} is

convergent and lim s_{3} = g. Let us consider X, s_{1}. We say that s_{1} is bounded if and only if:

(Def.3) there exists M such that M > 0 and for every n holds ∥s_{1}(n)∥ ≤ M.

One can prove the following propositions:

(18) If s_{2} is bounded and s_{3} is bounded, then s_{2} + s_{3} is bounded.

(19) If s_{1} is bounded, then −s_{1} is bounded.

(20) If s_{2} is bounded and s_{3} is bounded, then s_{2} − s_{3} is bounded.

(21) If s_{1} is bounded, then a · s_{1} is bounded.

(22) If s_{1} is constant, then s_{1} is bounded.

(23) For every m there exists M such that M > 0 and for every n such that n ≤ m holds ∥s_{1}(n)∥ < M.

(24) If s_{1} is convergent, then s_{1} is bounded.

(25) If s_{2} is bounded and s_{2} is compared to s_{3}, then s_{3} is bounded. Let us consider X, N_{1}, s_{1}. Then s_{1} · N_{1} is a sequence of X. Let us consider X, s_{2}, s_{1}. We say that s_{2} is a subsequence of s_{1} if and only

if: (Def.4) there exists N_{1} such that s_{2} = s_{1} · N_{1}.

One can prove the following propositions:

(26) For every n holds (s_{1} · N_{1})(n) = s_{1}(N_{1}(n)).

(27) s_{1} is a subsequence of s_{1}.

(28) If s_{2} is a subsequence of s_{3} and s_{3} is a subsequence of s_{4}, then s_{2} is a subsequence of s_{4}.

(29) If s_{1} is constant and s_{2} is a subsequence of s_{1}, then s_{2} is constant.

(30) If s_{1} is constant and s_{2} is a subsequence of s_{1}, then s_{1} = s_{2}.

(31) If s_{1} is bounded and s_{2} is a subsequence of s_{1}, then s_{2} is bounded.

(32) If s_{1} is convergent and s_{2} is a subsequence of s_{1}, then s_{2} is convergent.

(33) If s_{2} is a subsequence of s_{1} and s_{1} is convergent, then lim s_{2} = lim s_{1}.

(34) If s_{1} is a Cauchy sequence and s_{2} is a subsequence of s_{1}, then s_{2} is a

Cauchy sequence. Let us consider X, s_{1}, k. The functor s_{1} ↑ k yields a sequence of X and is

defined by: (Def.5) for every n holds (s_{1} ↑ k)(n) = s_{1}(n + k).

The following propositions are true:

(35) s_{1} ↑ 0 = s_{1}.

(36) s_{1} ↑ k ↑ m = s_{1} ↑ m ↑ k.

(37) s_{1} ↑ k ↑ m = s_{1} ↑ (k + m).

(38) (s_{2} + s_{3}) ↑ k = s_{2} ↑ k + s_{3} ↑ k.

(39) (−s_{1}) ↑ k = −s_{1} ↑ k.

(40) (s_{2} − s_{3}) ↑ k = s_{2} ↑ k − s_{3} ↑ k.

(41) (a · s_{1}) ↑ k = a · (s_{1} ↑ k).

(42) (s_{1} · N_{1}) ↑ k = s_{1} · (N_{1} ↑ k).

(43) s_{1} ↑ k is a subsequence of s_{1}.

(44) If s_{1} is convergent, then s_{1} ↑ k is convergent and lim(s_{1} ↑ k) = lim s_{1}.

(45) If s_{1} is convergent and there exists k such that s_{2} = s_{1} ↑ k, then s_{2} is convergent and lim s_{2} = lim s_{1}.

(46) If s_{1} is convergent and there exists k such that s_{1} = s_{2} ↑ k, then s_{2} is convergent.

(47) If s_{1} is a Cauchy sequence and there exists k such that s_{1} = s_{2} ↑ k, then s_{2} is a Cauchy sequence.

(48) If s_{1} is a Cauchy sequence, then s_{1} ↑ k is a Cauchy sequence.

(49) If s_{2} is compared to s_{3}, then s_{2} ↑ k is compared to s_{3} ↑ k.

(50) If s_{1} is bounded, then s_{1} ↑ k is bounded.

(51) If s_{1} is constant, then s_{1} ↑ k is constant.

Let us consider X. We say that X is a complete space if and only if: (Def.6) for every s_{1} such that s_{1} is a Cauchy sequence holds s_{1} is convergent.

The following propositions are true:

(52) If X is a complete space and s_{2} is a Cauchy sequence and s_{2} is compared to s_{3}, then s_{3} is a Cauchy sequence.

(53) If X is a complete space and s_{1} is a Cauchy sequence, then s_{1} is bounded.

Let us consider X. We say that X is a Hilbert space if and only if: (Def.7) X is a real unitary space and X is a complete space.

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] Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321–328, 1990.

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

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

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

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

[11] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511–516, 1991.

[12] Jan Popiołek. Introduction to Banach and Hilbert spaces - part II. Formalized Mathematics, 2(4):517–521, 1991.

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

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

[15] Konrad Raczkowski and Andrzej Ne‘dzusiak. Serieses. Formalized Mathematics, 2(4):449–452, 1991.

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

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

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

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

[20] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297–301, 1990.

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

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Category Ens

Czesław Byliński Warsaw University Białystok

Summary. If V is any non-empty set of sets, we define Ens_{V} to be the category with the objects of all sets X ∈ V, morphisms of all mappings from X into Y, with the ususal composition of mappings. By a mapping we mean a triple ⟨X, Y, f⟩ where f is a function from X into Y. The notations and concepts included correspond to those presented in [11,9]. We also introduce representable functors to illustrate properties of the category Ens.

MML Identifier: ENS 1.

The notation and terminology used here are introduced in the following papers: [15], [16], [13], [2], [3], [7], [5], [1], [14], [10], [12], [4], [8], and [6].

Mappings

In the sequel V denotes a non-empty set and A, B denote elements of V. Let us consider V. The functor Funcs V yielding a non-empty set of functions is defined by: (Def.1) Funcs V = ⋃{B^{A}}.

We now state three propositions:

(1) For an arbitrary f holds f ∈ Funcs V if and only if there exist A, B such that if B = ∅, then A = ∅ but f is a function from A into B.

(2) B^{A} ⊆ Funcs V.

(3) For every non-empty subset W of V holds Funcs W ⊆ Funcs V.

In the sequel f is an element of Funcs V. Let us consider V. The functor Maps V yielding a non-empty set is defined as follows: (Def.2) Maps V = {⟨⟨⟨⟨A, B⟩⟩, f⟩⟩ : (B = ∅ ⇒ A = ∅)∧f is a function from A into B}.

In the sequel m, m_{1}, m_{2}, m_{3} are elements of Maps V. One can prove the following four propositions:

(4) There exist f, A, B such that m = ⟨⟨⟨⟨A, B⟩⟩, f⟩⟩ but if B = ∅, then A = ∅ and f is a function from A into B.

(5) For every function f from A into B such that if B = ∅, then A = ∅ holds ⟨⟨⟨⟨A, B⟩⟩, f⟩⟩ ∈ Maps V.

(6) Maps V ⊆ [: [: V, V :], Funcs V :].

(7) For every non-empty subset W of V holds Maps W ⊆ Maps V. We now define three new functors. Let us consider V, m. The functor

graph(m) yields a function and is defined as follows: (Def.3) graph(m) = m_{2}. The functor dom m yields an element of V and is defined by: (Def.4) dom m = (m_{1})_{1}. The functor cod m yielding an element of V is defined by: (Def.5) cod m = (m_{1})_{2}.

The following three propositions are true:

(8) m = ⟨⟨⟨⟨ dom m, cod m⟩⟩, graph(m)⟩⟩.

(9) cod m ≠ ∅ or dom m = ∅ but graph(m) is a function from dom m into cod m.

(10) For every function f and for all sets A, B such that ⟨⟨⟨⟨A, B⟩⟩, f⟩⟩ ∈

Maps V holds if B = ∅, then A = ∅ but f is a function from A into B. Let us consider V, A. The functor id(A) yields an element of Maps V and is

defined by: (Def.6) id(A) = ⟨⟨⟨⟨A, A⟩⟩, id_{A} ⟩⟩.

The following proposition is true

(11) graph(id(A)) = id_{A} and dom id(A) = A and cod id(A) = A. Let us consider V, m_{1}, m_{2}. Let us assume that cod m_{1} = dom m_{2}. The

functor m_{2} · m_{1} yields an element of Maps V and is defined as follows: (Def.7) m_{2} · m_{1} = ⟨⟨⟨⟨ dom m_{1}, cod m_{2}⟩⟩, graph(m_{2}) · graph(m_{1})⟩⟩.

One can prove the following propositions:

(12) If dom m_{2} = cod m_{1}, then graph((m_{2} · m_{1})) = graph(m_{2}) · graph(m_{1}) and dom(m_{2} · m_{1}) = dom m_{1} and cod(m_{2} · m_{1}) = cod m_{2}.

(13) If dom m_{2} = cod m_{1} and dom m_{3} = cod m_{2}, then m_{3} · (m_{2} · m_{1}) = m_{3} · m_{2} · m_{1}.

(14) m · id(dom m) = m and id(cod m) · m = m.

Let us consider V, A, B. The functor Maps(A, B) yields a set and is defined by: (Def.8) Maps(A, B) = {⟨⟨⟨⟨A, B⟩⟩, f⟩⟩ : ⟨⟨⟨⟨A, B⟩⟩, f⟩⟩ ∈ Maps V }, where f ranges

over elements of Funcs V. The following propositions are true:

(15) For every function f from A into B such that if B = ∅, then A = ∅ holds ⟨⟨⟨⟨A, B⟩⟩, f⟩⟩ ∈ Maps(A, B).

(16) If m ∈ Maps(A, B), then m = ⟨⟨⟨⟨A, B⟩⟩, graph(m)⟩⟩.

(17) Maps(A, B) ⊆ Maps V.

(18) Maps V = ⋃{Maps(A, B)}.

(19) m ∈ Maps(A, B) if and only if dom m = A and cod m = B.

(20) If m ∈ Maps(A, B), then graph(m) ∈ B^{A}.

Let us consider V, m. We say that m is a surjection if and only if: (Def.9) rng graph(m) = cod m.

Category Ens

We now define four new functors. Let us consider V. The functor Dom_{V} yields a function from Maps V into V and is defined by:

(Def.10) for every m holds Dom_{V} (m) = dom m.

The functor Cod_{V} yields a function from Maps V into V and is defined as follows: (Def.11) for every m holds Cod_{V} (m) = cod m.

The functor ·_{V} yields a partial function from [: Maps V, Maps V :] to Maps V and is defined as follows:

(Def.12) for all m_{2}, m_{1} holds ⟨⟨m_{2}, m_{1}⟩⟩ ∈ dom(·_{V}) if and only if dom m_{2} = cod m_{1} and for all m_{2}, m_{1} such that dom m_{2} = cod m_{1} holds ·_{V} (⟨⟨m_{2},

m_{1}⟩⟩) = m_{2} · m_{1}.

The functor Id_{V} yields a function from V into Maps V and is defined by: (Def.13) for every A holds Id_{V} (A) = id(A).

Let us consider V. The functor Ens_{V} yields a category structure and is defined by:

(Def.14) Ens_{V} = ⟨V, Maps V, Dom_{V} , Cod_{V} , ·_{V} , Id_{V} ⟩.

We now state the proposition

(21) ⟨V, Maps V, Dom_{V} , Cod_{V} , ·_{V} , Id_{V} ⟩ is a category.

Let us consider V. Then Ens_{V} is a category.

In the sequel a, b are objects of Ens_{V}. Next we state the proposition

(22) A is an object of Ens_{V}.

Let us consider V, A. The functor ^{@}A yielding an object of Ens_{V} is defined as follows:

(Def.15) ^{@}A = A.

One can prove the following proposition

(23) a is an element of V.

Let us consider V, a. The functor ^{@}a yields an element of V and is defined by:

(Def.16) ^{@}a = a.

In the sequel f, g denote morphisms of Ens_{V}. The following proposition is true (24) m is a morphism of Ens_{V}.

Let us consider V, m. The functor ^{@}m yields a morphism of Ens_{V} and is defined as follows:

(Def.17) ^{@}m = m.

One can prove the following proposition

(25) f is an element of Maps V.

Let us consider V, f. The functor ^{@}f yields an element of Maps V and is defined as follows:

(Def.18) ^{@}f = f.

One can prove the following propositions:

(26) dom f = dom(^{@}f) and cod f = cod(^{@}f).

(27) hom(a, b) = Maps(^{@}a,^{@}b).

(28) If dom g = cod f, then g · f = (^{@}g) · (^{@}f).

(29) id_{a} = id(^{@}a).

(30) If a = ∅, then a is an initial object.

(31) If ∅ ∈ V and a is an initial object, then a = ∅.

(32) For every universal class W and for every object a of Ens_{W} such that a is an initial object holds a = ∅.

(33) If there exists arbitrary x such that a = {x}, then a is a terminal object.

(34) If V ≠ {∅} and a is a terminal object, then there exists arbitrary x such that a = {x}.

(35) For every universal class W and for every object a of Ens_{W} such that a is a terminal object there exists arbitrary x such that a = {x}.

(36) f is monic if and only if graph((^{@}f)) is one-to-one.

(37) If f is epi and there exists A and there exist arbitrary x_{1}, x_{2} such that x_{1} ∈ A and x_{2} ∈ A and x_{1} ≠ x_{2}, then ^{@}f is a surjection.

(38) If ^{@}f is a surjection, then f is epi.

(39) For every universal class W and for every morphism f of Ens_{W} such that f is epi holds ^{@}f is a surjection.

(40) For every non-empty subset W of V holds Ens_{W} is full subcategory of Ens_{V}.

Representable Functors

We follow a convention: C will be a category, a, b, c will be objects of C, and f, g, h, f′, g′ will be morphisms of C. Let us consider C. The functor Hom(C) yields a non-empty set and is defined as follows:

(Def.19) Hom(C) = {hom(a, b)}.

We now state two propositions:

(41) hom(a, b) ∈ Hom(C).

(42) If hom(a, cod f) = ∅, then hom(a, dom f) = ∅ but if hom(dom f, a) = ∅,

then hom(cod f, a) = ∅. We now define two new functors. Let us consider C, a, f. The functor

hom(a, f) yielding a function from hom(a, dom f) into hom(a, cod f) is defined by:

(Def.20) for every g such that g ∈ hom(a, dom f) holds (hom(a, f))(g) = f · g. The functor hom(f, a) yields a function from hom(cod f, a) into hom(dom f, a) and is defined by:

(Def.21) for every g such that g ∈ hom(cod f, a) holds (hom(f, a))(g) = g · f.

We now state several propositions:

(43) hom(a, id_{c}) = id_{hom(a,c)}.

(44) hom(id_{c}, a) = id_{hom(c,a)}.

(45) If dom g = cod f, then hom(a, g · f) = hom(a, g) · hom(a, f).

(46) If dom g = cod f, then hom(g · f, a) = hom(f, a) · hom(g, a).

(47) ⟨⟨⟨⟨ hom(a, dom f), hom(a, cod f)⟩⟩, hom(a, f)⟩⟩ is an element of Maps Hom(C).

(48) ⟨⟨⟨⟨ hom(cod f, a), hom(dom f, a)⟩⟩, hom(f, a)⟩⟩ is an element of

Maps Hom(C). We now define two new functors. Let us consider C, a. The functor hom(a, −)

yields a function from the morphisms of C into Maps Hom(C) and is defined as follows:

(Def.22) for every f holds (hom(a, −))(f) = ⟨⟨⟨⟨ hom(a, dom f), hom(a, cod f)⟩⟩,

hom(a, f)⟩⟩. The functor hom(−, a) yields a function from the morphisms of C into

Maps Hom(C)

and is defined as follows: (Def.23) for every f holds (hom(−, a))(f) = ⟨⟨⟨⟨ hom(cod f, a), hom(dom f, a)⟩⟩,

hom(f, a)⟩⟩. The following propositions are true:

(49) If Hom(C) ⊆ V, then hom(a, −) is a functor from C to Ens_{V}.

(50) If Hom(C) ⊆ V, then hom(−, a) is a contravariant functor from C into Ens_{V}.

(51) If hom(dom f, cod f′) = ∅, then hom(cod f, dom f′) = ∅.

Let us consider C, f, g. The functor hom(f, g) yielding a function from hom(cod f, dom g) into hom(dom f, cod g) is defined by:

(Def.24) for every h such that h ∈ hom(cod f, dom g) holds (hom(f, g))(h) =

g · h · f. We now state several propositions:

(52) ⟨⟨⟨⟨ hom(cod f, dom g), hom(dom f, cod g)⟩⟩, hom(f, g)⟩⟩ is an element of Maps Hom(C).

(53) hom(id_{a}, f) = hom(a, f) and hom(f, id_{a}) = hom(f, a).

(54) hom(id_{a}, id_{b}) = id_{hom(a,b)}.

(55) hom(f, g) = hom(dom f, g) · hom(f, dom g).

(56) If cod g = dom f and dom g′ = cod f′, then hom(f·g, g′·f′) = hom(g, g′)·

hom(f, f′). Let us consider C. The functor hom_{C}(−, −) yielding a function from the

morphisms of [: C, C:] into Maps Hom(C) is defined as follows: (Def.25) for all f, g holds (hom_{C}(−, −))(⟨⟨f, g⟩⟩) =

⟨⟨⟨⟨ hom(cod f, dom g), hom(dom f, cod g)⟩⟩, hom(f, g)⟩⟩. The following two propositions are true:

(57) hom(a, −) = (curry(hom_{C}(−, −)))(id_{a}) and hom(−, a) = (curry′(hom_{C}(−, −)))(id_{a}).

(58) If Hom(C) ⊆ V, then hom_{C}(−, −) is a functor from [: C^{op}, C:] to Ens_{V}. We now define two new functors. Let us consider V, C, a. Let us assume

that Hom(C) ⊆ V. The functor hom_{V} (a, −) yields a functor from C to Ens_{V} and is defined by:

(Def.26) hom_{V} (a, −) = hom(a, −).

The functor hom_{V} (−, a) yields a contravariant functor from C into Ens_{V} and is defined as follows:

(Def.27) hom_{V} (−, a) = hom(−, a).

Let us consider V, C. Let us assume that Hom(C) ⊆ V. The functor hom_{V}^{C}(−, −) yielding a functor from [: C^{op}, C:] to Ens_{V} is defined as follows:

(Def.28) hom_{V}^{C}(−, −) = hom_{C}(−, −).

One can prove the following propositions:

(59) If Hom(C) ⊆ V, then (hom_{V} (a, −))(f) = ⟨⟨⟨⟨ hom(a, dom f), hom(a, cod f)⟩⟩, hom(a, f)⟩⟩.

(60) If Hom(C) ⊆ V, then (Obj(hom_{V} (a, −)))(b) = hom(a, b).

(61) If Hom(C) ⊆ V, then (hom_{V} (−, a))(f) = ⟨⟨⟨⟨ hom(cod f, a), hom(dom f, a)⟩⟩, hom(f, a)⟩⟩.

(62) If Hom(C) ⊆ V, then (Obj(hom_{V} (−, a)))(b) = hom(b, a).

(63) If Hom(C) ⊆ V, then (hom_{V}^{C}(−, −))(⟨⟨f^{op}, g⟩⟩) = ⟨⟨⟨⟨ hom(cod f, dom g), hom(dom f, cod g)⟩⟩, hom(f, g)⟩⟩.

(64) If Hom(C) ⊆ V, then (Obj(hom_{V}^{C}(−, −)))(⟨⟨a^{op}, b⟩⟩) = hom(a, b).

(65) If Hom(C) ⊆ V, then (hom_{V}^{C}(−, −))(a^{op}, −) = hom_{V} (a, −).

(66) If Hom(C) ⊆ V, then (hom_{V}^{C}(−, −))(−, a) = hom_{V} (−, a).

Acknowledgements

I would like to thank Andrzej Trybulec for his useful sugestions and valuable comments.

References

category ens 533

[1] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537–541, 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.

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[8] Czesław Byliński. Subcategories and products of categories. Formalized Mathematics, 1(4):725–732, 1990.

[9] Sunders MacLane. Categories for the working mathematician. Springer, Berlin/Heilderberg/New York, 1972.

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

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

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

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

Received August 1, 1991 534

A Borsuk Theorem on Homotopy Types

Andrzej Trybulec Warsaw University Białystok

Summary. We present a Borsuk’s theorem published first in [3] (compare also [4, pages 119–120]). It is slightly generalized, the assumption of metrizability is omitted. We introduce concepts needed for the formulation and the proof of theorems on upper semi-continuous decompositions, retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved.

MML Identifier: BORSUK 1.

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

Preliminaries

We follow a convention: X, Y, X_{1}, X_{2}, Y_{1}, Y_{2} will be sets, A will be a subset of X, and e, u will be arbitrary. The following propositions are true:

(1) If X meets Y_{1} and X ⊆ Y_{2}, then X meets Y_{1} ∩ Y_{2}.

(2) If e ∈ [: X_{1}, Y_{1} :] and e ∈ [: X_{2}, Y_{2} :], then e ∈ [: X_{1} ∩ X_{2}, Y_{1} ∩ Y_{2} :].

(3) id_{X} ^{∘}A = A.

(4) id_{X} ^{−1}A = A.

(5) For every function F such that X ⊆ F ^{−1} X_{1} holds F ^{∘} X ⊆ X_{1}.

(6) (X ⟼ u)^{∘} X_{1} ⊆ {u}.

(7) If [: X_{1}, X_{2} :] ⊆ [: Y_{1}, Y_{2} :] and [: X_{1}, X_{2} :] ≠ ∅, then X_{1} ⊆ Y_{1} and X_{2} ⊆ Y_{2}.

(8) If {e} meets X, then e ∈ X. The scheme NonUniqExD deals with a set A, a set B, and a binary predicate

P, and states that:

there exists a function f from A into B such that for every e such that e ∈ A holds P[e, f(e)] provided the following requirement is met:

• for every e such that e ∈ A there exists u such that u ∈ B and P[e, u].

We now state several propositions:

(9) If e ∈ 2^{[: X, Y :]}, then (^{∘} π_{1}(X × Y))(e) = π_{1}(X × Y)^{∘} e.

(10) If e ∈ 2^{[: X, Y :]}, then (^{∘} π_{2}(X × Y))(e) = π_{2}(X × Y)^{∘} e.

(11) If e ∈ [: X, Y :], then e = ⟨⟨e_{1}, e_{2}⟩⟩.

(12) For every subset X_{1} of X and for every subset Y_{1} of Y such that [: X_{1}, Y_{1} :] ≠ ∅ holds π_{1}(X ×Y)^{∘} [: X_{1}, Y_{1} :] = X_{1} and π_{2}(X ×Y)^{∘} [: X_{1}, Y_{1} :] = Y_{1}.

(13) For every subset X_{1} of X and for every subset Y_{1} of Y such that [: X_{1}, Y_{1} :] ≠ ∅ holds (^{∘} π_{1}(X × Y))([: X_{1}, Y_{1} :]) = X_{1} and (^{∘} π_{2}(X × Y))([: X_{1}, Y_{1} :]) = Y_{1}.

(14) Let A be a subset of [: X, Y :]. Then for every family H of subsets of [: X, Y :] such that for every e such that e ∈ H holds e ⊆ A and there exists a subset X_{1} of X and there exists a subset Y_{1} of Y such that e = [: X_{1}, Y_{1} :] holds [: ⋃((^{∘} π_{1}(X × Y))^{∘} H), ⋂((^{∘} π_{2}(X × Y))^{∘} H) :] ⊆ A.

(15) Let A be a subset of [: X, Y :]. Then for every family H of subsets of [: X, Y :] such that for every e such that e ∈ H holds e ⊆ A and there exists a subset X_{1} of X and there exists a subset Y_{1} of Y such that e = [: X_{1}, Y_{1} :] holds [: ⋂((^{∘} π_{1}(X × Y))^{∘} H), ⋃((^{∘} π_{2}(X × Y))^{∘} H) :] ⊆ A.

(16) For every set X and for every non-empty set Y and for every function f from X into Y and for every family H of subsets of X holds ⋃((^{∘} f)^{∘}H) = f ^{∘} ⋃ H.

In the sequel X, Y, Z denote non-empty sets. One can prove the following propositions:

(17) For every family a of subsets of X holds ⋃ ⋃ a = ⋃{⋃ A : A ∈ a}, where A ranges over subsets of X.

(18) For every family D of subsets of X such that ⋃ D = X for every subset A of D and for every subset B of X such that B = ⋃ A holds B^{c} ⊆ ⋃(A^{c}).

(19) For every function F from X into Y and for every function G from X into Z such that for all elements x, x′ of X such that F (x) = F (x′) holds G(x) = G(x′) there exists a function H from Y into Z such that H · F = G.

(20) For all X, Y, Z and for every element y of Y and for every function F from X into Y and for every function G from Y into Z holds F ^{−1} {y} ⊆ (G · F)^{−1} {G(y)}.

(21) For every function F from X into Y and for every element x of X and for every element z of Z holds [: F, id_{Z} :](⟨⟨x, z⟩⟩) = ⟨⟨F (x), z⟩⟩.

(22) For every function F from X into Y and for every subset A of X holds id_{X} ^{∘}A = A.

(23) For every function F from X into Y and for every subset A of X and for every subset B of Z holds [: F, id_{Z} :]^{∘} [: A, B:] = [: F ^{∘} A, B:].

(24) For every function F from X into Y and for every element y of Y and

for every element z of Z holds [: F, id_{Z} :]^{−1} {⟨⟨y, z⟩⟩} = [: F ^{−1} {y}, {z}:]. Let B, A be non-empty sets, and let x be an element of B. Then A ⟼ x is

a function from A into B.

Let Y be a non-empty set, and let y be an element of Y. Then {y} is a subset of Y.

Partitions

One can prove the following four propositions:

(25) For every partition D of X and for every subset A of D holds ⋃ A is a subset of X.

(26) For every partition D of X and for all subsets A, B of D holds ⋃(A ∩ B) = ⋃ A ∩ ⋃ B.

(27) For every partition D of X and for every subset A of D and for every subset B of X such that B = ⋃ A holds B^{c} = ⋃(A^{c}).

(28) For every equivalence relation E of X holds Classes E is non-empty.

Let us consider X, and let D be a non-empty partition of X. The projection onto D yielding a function from X into D is defined as follows: (Def.1) for every element p of X holds p ∈ (the projection onto D)(p).

Next we state several propositions:

(29) For every non-empty partition D of X and for every element p of X and for every element A of D such that p ∈ A holds A = (the projection onto D)(p).

(30) For every non-empty partition D of X and for every element p of D holds p = (the projection onto D)^{−1} {p}.

(31) For every non-empty partition D of X and for every subset A of D holds (the projection onto D)^{−1} A = ⋃ A.

(32) For every non-empty partition D of X and for every element W of D there exists an element W′ of X such that (the projection onto D)(W′) = W.

(33) For every non-empty partition D of X and for every subset W of X such that for every subset B of X such that B ∈ D and B meets W holds B ⊆ W holds W = (the projection onto D)^{−1}(the projection onto D)^{∘}W.

Topological Preliminaries

In the sequel X, Y denote topological spaces. We now state two propositions:

(34) Ω_{X} ≠ ∅_{X}.

(35) For every subspace Y of X holds the carrier of Y ⊆ the carrier of X. Let X, Y be topological spaces, and let F be a function from the carrier of

X into the carrier of Y. Let us note that one can characterize the predicate F is continuous by the following (equivalent) condition: (Def.2) for every point W of X and for every neighborhood G of F (W) there

exists a neighborhood H of W such that F ^{∘} H ⊆ G.

The following proposition is true

(36) For every point y of Y holds (the carrier of X) ⟼ y is continuous. Let us consider X, Y. A map from X into Y is called a continuous map from

X into Y if:

(Def.3) it is continuous.

Let X, Y, Z be topological spaces, and let F be a continuous map from X into Y, and let G be a continuous map from Y into Z. Then G·F is a continuous map from X into Z.

We now state two propositions:

(37) For every continuous map A from X into Y and for every subset G of Y holds A^{−1} Int G ⊆ Int(A^{−1} G).

(38) For every point W of Y and for every continuous map A from X into Y and for every neighborhood G of W holds A^{−1} G is a neighborhood of A^{−1} {W }.

Let X, Y be topological spaces, and let W be a point of Y, and let A be a continuous map from X into Y, and let G be a neighborhood of W. Then A^{−1} G is a neighborhood of A^{−1} {W }.

One can prove the following propositions:

(39) For every X and for all subsets A, B of the carrier of X and for every neighborhood U_{1} of B such that A ⊆ B holds U_{1} is a neighborhood of A.

(40) For every subset A of X and for every point x of X holds A is a neighborhood of x if and only if A is a neighborhood of {x}.

(41) For every point x of X holds {x} is compact.

(42) For every subspace Y of X and for every subset A of X and for every subset B of Y such that A = B holds A is compact if and only if B is compact.

Cartesian Products of Topological Spaces

Let us consider X, Y. The functor [: X, Y :] yielding a topological space is defined by:

(Def.4) the carrier of [: X, Y :] = [: the carrier of X, the carrier of Y :] and the topology of [: X, Y :] = {⋃ A : A ⊆ {[: X_{1}, Y_{1} :] : X_{1} ∈ the topology of X ∧ Y_{1} ∈ the topology of Y }}, where X_{1} ranges over subsets of X, and Y_{1} ranges over subsets of Y.

Next we state three propositions:

(43) The carrier of [: X, Y :] = [: the carrier of X, the carrier of Y :].

(44) The topology of [: X, Y :] = {⋃ A : A ⊆ {[: X_{1}, Y_{1} :] : X_{1} ∈ the topology of X ∧ Y_{1} ∈ the topology of Y }}, where X_{1} ranges over subsets of X, and Y_{1} ranges over subsets of Y.

(45) For every subset B of [: X, Y :] holds B is open if and only if there exists a family A of subsets of the carrier of [: X, Y :] such that B = ⋃ A and for every e such that e ∈ A there exists a subset X_{1} of X and there exists a subset Y_{1} of Y such that e = [: X_{1}, Y_{1} :] and X_{1} is open and Y_{1} is open.

Let X, Y be topological spaces, and let A be a subset of X, and let B be a subset of Y. Then [: A, B:] is a subset of [: X, Y :].

Let X, Y be topological spaces, and let x be a point of X, and let y be a point of Y. Then ⟨⟨x, y⟩⟩ is a point of [: X, Y :].

Next we state four propositions:

(46) For every subset V of X and for every subset W of Y such that V is open and W is open holds [: V, W :] is open.

(47) For every subset V of X and for every subset W of Y holds Int[: V, W :] = [: Int V, Int W :].

(48) For every point x of X and for every point y of Y and for every neighborhood V of x and for every neighborhood W of y holds [: V, W :] is a neighborhood of ⟨⟨x, y⟩⟩.

(49) For every subset A of X and for every subset B of Y and for every neighborhood V of A and for every neighborhood W of B holds [: V, W :] is a neighborhood of [: A, B:].

Let X, Y be topological spaces, and let x be a point of X, and let y be a point of Y, and let V be a neighborhood of x, and let W be a neighborhood of y. Then [: V, W :] is a neighborhood of ⟨⟨x, y⟩⟩.

Next we state the proposition

(50) For every point X_{3} of [: X, Y :] there exists a point W of X and there

exists a point T of Y such that X_{3} = ⟨⟨W, T ⟩⟩. Let X, Y be topological spaces, and let A be a subset of X, and let t be a

point of Y, and let V be a neighborhood of A, and let W be a neighborhood of t. Then [: V, W :] is a neighborhood of [: A, {t}:].

Let us consider X, Y, and let A be a subset of [: X, Y :]. The functor BaseAppr(A) yields a family of subsets of [: X, Y :] and is defined by: (Def.5) BaseAppr(A) = {[: X_{1}, Y_{1} :] : [: X_{1}, Y_{1} :] ⊆ A ∧ X_{1} is open∧Y_{1} is open},

where X_{1} ranges over subsets of X, and Y_{1} ranges over subsets of Y.

We now state several propositions:

(51) For every subset A of [: X, Y :] holds BaseAppr(A) is open.

(52) For all subsets A, B of [: X, Y :] such that A ⊆ B holds BaseAppr(A) ⊆ BaseAppr(B).

(53) For every subset A of [: X, Y :] holds ⋃ BaseAppr(A) ⊆ A.

(54) For every subset A of [: X, Y :] such that A is open holds A = ⋃ BaseAppr(A).

(55) For every subset A of [: X, Y :] holds Int A = ⋃ BaseAppr(A). We now define two new functors. Let us consider X, Y. The functor π_{1}(X, Y)

yielding a function from 2^{the carrier of [: X, Y :]} into 2^{the carrier of X} is defined by:

(Def.6) π_{1}(X, Y) = ^{∘} π_{1}( (the carrier of X)× the carrier of Y). The functor π_{2}(X, Y) yields a function from 2^{the carrier of [: X, Y :]} into 2^{the carrier of Y} and is defined as follows:

(Def.7) π_{2}(X, Y) = ^{∘} π_{2}( (the carrier of X)× the carrier of Y).

We now state a number of propositions:

(56) Let A be a subset of [: X, Y :]. Then for every family H of subsets of [: X, Y :] such that for every e such that e ∈ H holds e ⊆ A and there exists a subset X_{1} of X and there exists a subset Y_{1} of Y such that e = [: X_{1}, Y_{1} :] holds [: ⋃(π_{1}(X, Y)^{∘} H), ⋂(π_{2}(X, Y)^{∘} H) :] ⊆ A.

(57) For every family H of subsets of [: X, Y :] and for every set C such that C ∈ π_{1}(X, Y) ^{∘} H there exists a subset D of [: X, Y :] such that D ∈ H and C = π_{1}( (the carrier of X)× the carrier of Y)^{∘} D.

(58) For every family H of subsets of [: X, Y :] and for every set C such that C ∈ π_{2}(X, Y) ^{∘} H there exists a subset D of [: X, Y :] such that D ∈ H and C = π_{2}( (the carrier of X)× the carrier of Y)^{∘} D.

(59) For every subset D of [: X, Y :] such that D is open for every subset X_{1} of X and for every subset Y_{1} of Y holds if X_{1} = π_{1}( (the carrier of X)× the carrier of Y)^{∘} D, then X_{1} is open but if Y_{1} = π_{2}( (the carrier of X)× the carrier of Y)^{∘} D, then Y_{1} is open.

(60) For every family H of subsets of [: X, Y :] such that H is open holds π_{1}(X, Y)^{∘} H is open and π_{2}(X, Y)^{∘} H is open.

(61) For every family H of subsets of [: X, Y :] such that π_{1}(X, Y)^{∘} H = ∅ or π_{2}(X, Y)^{∘} H = ∅ holds H = ∅.

(62) For every family H of subsets of [: X, Y :] and for every subset X_{1} of X and for every subset Y_{1} of Y such that H is a cover of [: X_{1}, Y_{1} :] holds if Y_{1} ≠ ∅, then π_{1}(X, Y)^{∘}H is a cover of X_{1} but if X_{1} ≠ ∅, then π_{2}(X, Y)^{∘}H is a cover of Y_{1}.

(63) For every family H of subsets of X and for every subset Y of X such that H is a cover of Y there exists a family F of subsets of X such that F ⊆ H and F is a cover of Y and for every set C such that C ∈ F holds C ∩ Y ≠ ∅.

(64) For every family F of subsets of X and for every family H of subsets of [: X, Y :] such that F is finite and F ⊆ π_{1}(X, Y) ^{∘} H there exists a family G of subsets of [: X, Y :] such that G ⊆ H and G is finite and F = π_{1}(X, Y)^{∘} G.

(65) For every subset X_{1} of X and for every subset Y_{1} of Y such that [: X_{1}, Y_{1} :] ≠ ∅ holds π_{1}(X, Y)([: X_{1}, Y_{1} :]) = X_{1} and π_{2}(X, Y)([: X_{1}, Y_{1} :]) = Y_{1}.

(66) π_{1}(X, Y)(∅) = ∅ and π_{2}(X, Y)(∅) = ∅.

(67) For every point t of Y and for every subset A of the carrier of X such that A is compact for every neighborhood G of [: A, {t}:] there exists a neighborhood V of A and there exists a neighborhood W of t such that [: V, W :] ⊆ G.

Partitions of Topological Spaces

Let us consider X. The trivial decomposition of X yielding a non-empty partition of the carrier of X is defined by: (Def.8) the trivial decomposition of X = Classes(△_{the carrier of X}).

We now state the proposition

(68) For every subset A of X such that A ∈ the trivial decomposition of X

there exists a point x of X such that A = {x}. Let X be a topological space, and let D be a non-empty partition of the

carrier of X. The decomposition space of D yielding a topological space is defined as follows: (Def.9) the carrier of the decomposition space of D = D and the topology of

the decomposition space of D = {A : ⋃ A ∈ the topology of X}, where A ranges over subsets of D.

One can prove the following proposition

(69) For every non-empty partition D of the carrier of X and for every subset A of D holds ⋃ A ∈ the topology of X if and only if A ∈ the topology of the decomposition space of D.

Let X be a topological space, and let D be a non-empty partition of the carrier of X. The projection onto D yielding a continuous map from X into the decomposition space of D is defined as follows:

(Def.10) the projection onto D = the projection onto D.

We now state three propositions:

(70) For every non-empty partition D of the carrier of X and for every point W of X holds W ∈ (the projection onto D)(W).

(71) For every non-empty partition D of the carrier of X and for every point W of the decomposition space of D there exists a point W′ of X such that (the projection onto D)(W′) = W.

(72) For every non-empty partition D of the carrier of X holds rng(the pro-

jection onto D) = the carrier of the decomposition space of D. Let X_{4} be a topological space, and let X be a subspace of X_{4}, and let D be

a non-empty partition of the carrier of X. The trivial extension of D yields a non-empty partition of the carrier of X_{4} and is defined as follows:

(Def.11) the trivial extension of D = D ∪ {{p} : p ∉ the carrier of X}, where p ranges over points of X_{4}.

The following propositions are true:

(73) For every topological space X_{4} and for every subspace X of X_{4} and for every non-empty partition D of the carrier of X holds D ⊆ the trivial extension of D.

(74) For every topological space X_{4} and for every subspace X of X_{4} and for every non-empty partition D of the carrier of X and for every subset A of X_{4} such that A ∈ the trivial extension of D holds A ∈ D or there exists a point x of X_{4} such that x ∉ Ω_{X} and A = {x}.

(75) For every topological space X_{4} and for every subspace X of X_{4} and for every non-empty partition D of the carrier of X and for every point x of X_{4} such that x ∉ the carrier of X holds {x} ∈ the trivial extension of D.

(76) For every topological space X_{4} and for every subspace X of X_{4} and for every non-empty partition D of the carrier of X and for every point W of X_{4} such that W ∈ the carrier of X holds (the projection onto the trivial extension of D)(W) = (the projection onto D)(W).

(77) For every topological space X_{4} and for every subspace X of X_{4} and for every non-empty partition D of the carrier of X and for every point W of X_{4} such that W ∉ the carrier of X holds (the projection onto the trivial extension of D)(W) = {W }.

(78) For every topological space X_{4} and for every subspace X of X_{4} and for every non-empty partition D of the carrier of X and for all points W, W′ of X_{4} such that W ∉ the carrier of X and (the projection onto the trivial extension of D)(W) = (the projection onto the trivial extension of D)(W′) holds W = W′.

(79) For every topological space X_{4} and for every subspace X of X_{4} and for every non-empty partition D of the carrier of X and for every point e of X_{4} such that (the projection onto the trivial extension of D)(e) ∈ the carrier of the decomposition space of D holds e ∈ the carrier of X.

(80) For every topological space X_{4} and for every subspace X of X_{4} and for every non-empty partition D of the carrier of X and for every e such that e ∈ the carrier of X holds (the projection onto the trivial extension of D)(e) ∈ the carrier of the decomposition space of D.

Upper Semicontinuous Decompositions

Let X be a topological space. A non-empty partition of the carrier of X is said to be an upper semi-continuous decomposition of X if:

(Def.12) for every subset A of X such that A ∈ it for every neighborhood V of A there exists a subset W of X such that W is open and A ⊆ W and W ⊆ V and for every subset B of X such that B ∈ it and B meets W holds B ⊆ W.

We now state two propositions:

(81) For every upper semi-continuous decomposition D of X and for every point t of the decomposition space of D and for every neighborhood G of (the projection onto D)^{−1} {t} holds (the projection onto D)^{∘} G is a neighborhood of t.

(82) The trivial decomposition of X is an upper semi-continuous decompo-

sition of X. Let us consider X. A subspace of X is called a closed subspace of X if:

(Def.13) for every subset A of X such that A = the carrier of it holds A is closed.

Let X_{4} be a topological space, and let X be a closed subspace of X_{4}, and let D be an upper semi-continuous decomposition of X. Then the trivial extension of D is an upper semi-continuous decomposition of X_{4}.

Let X be a topological space. An upper semi-continuous decomposition of X is called an upper semi-continuous decomposition into compacta of X if:

(Def.14) for every subset A of X such that A ∈ it holds A is compact.

Let X_{4} be a topological space, and let X be a closed subspace of X_{4}, and let D be an upper semi-continuous decomposition into compacta of X. Then the trivial extension of D is an upper semi-continuous decomposition into compacta of X_{4}.

Let X be a topological space, and let Y be a closed subspace of X, and let D be an upper semi-continuous decomposition into compacta of Y. Then the decomposition space of D is a closed subspace of the decomposition space of the trivial extension of D.

Borsuk’s Theorems on the Decomposition of Retracts

The topological space is defined by: (Def.15) for every subset P of (the metric space of real numbers)_{top} such that

P = [0, 1] holds = (the metric space of real numbers)_{top} P.

Next we state the proposition

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

We now define two new functors. The point 0_{}of is defined by: (Def.16) 0_{}= 0.

The point 1_{}of is defined by: (Def.17) 1_{}= 1.

Let A be a topological space, and let B be a subspace of A, and let F be a continuous map from A into B. We say that F is a retraction if and only if:

(Def.18) for every point W of A such that W ∈ the carrier of B holds F (W) = W.

We now define two new predicates. Let X be a topological space, and let Y be a subspace of X. We say that Y is a retract of X if and only if:

(Def.19) there exists a continuous map F from X into Y such that F is a retrac-

tion. We say that Y is a strong deformation retract of X if and only if:

544 andrzej trybulec

(Def.20) there exists a continuous map H from [: X, :] into X such that for every point A of X holds H(⟨⟨A, 0_{}⟩⟩) = A and H(⟨⟨A, 1_{}⟩⟩) ∈ the carrier of Y but if A ∈ the carrier of Y, then for every point T of holds H(⟨⟨A, T ⟩⟩) = A.

We now state two propositions:

(84) For every topological space X_{4} and for every closed subspace X of X_{4} and for every upper semi-continuous decomposition D into compacta of X such that X is a retract of X_{4} holds the decomposition space of D is a retract of the decomposition space of the trivial extension of D.

(85) For every topological space X_{4} and for every closed subspace X of X_{4} and for every upper semi-continuous decomposition D into compacta of X such that X is a strong deformation retract of X_{4} holds the decomposition space of D is a strong deformation retract of the decomposition space of the trivial extension of D.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

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[3] Karol Borsuk. On the homotopy types of some decomposition spaces. Bull. Acad. Polon. Sci., (18):235–239, 1970.

[4] Karol Borsuk. Theory of Shape. Volume 59 of Monografie Matematyczne, PWN, Warsaw, 1975.

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

[9] Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1(2):409–420, 1990.

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a borsuk theorem on homotopy types 545

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Received August 1, 1991 546

Cartesian Product of Functions

Grzegorz Bancerek Warsaw University Białystok

Summary. A supplement of [3] and [2], i.e. some useful and explanatory properties of the product and also the curried and uncurried functions are shown. Besides, the functions yielding functions are considered: two different products and other operation of such functions are introduced. Finally, two facts are presented: quasi-distributivity of the

power of the set to other one w.r.t. the union (X^{⨄_{x} f(x)} ≈ ∏_{x} X^{f(x)}) and quasi-distributivity of the poroduct w.r.t. the raising to the power (∏_{x} f(x)^{X} ≈ (∏_{x} f(x))^{X}).

MML Identifier: FUNCT 6.

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

Properties of Cartesian product

For simplicity we follow the rules: x, y, y_{1}, y_{2}, z, a will be arbitrary, f, g, h, h′, f_{1}, f_{2} will denote functions, i will denote a natural number, X, Y, Z, V_{1}, V_{2} will denote sets, P will denote a permutation of X, D, D_{1}, D_{2}, D_{3} will denote non-empty sets, d_{1} will denote an element of D_{1}, d_{2} will denote an element of D_{2}, and d_{3} will denote an element of D_{3}. We now state a number of propositions:

(1) x ∈ ∏⟨X⟩ if and only if there exists y such that y ∈ X and x = ⟨y⟩.

(2) z ∈ ∏⟨X, Y ⟩ if and only if there exist x, y such that x ∈ X and y ∈ Y and z = ⟨x, y⟩.

(3) a ∈ ∏⟨X, Y, Z⟩ if and only if there exist x, y, z such that x ∈ X and y ∈ Y and z ∈ Z and a = ⟨x, y, z⟩.

(4) ∏⟨D⟩ = D^{1}.

(5) ∏⟨D_{1}, D_{2}⟩ = {⟨d_{1}, d_{2}⟩}.

(6) ∏⟨D, D⟩ = D^{2}.

(7) ∏⟨D_{1}, D_{2}, D_{3}⟩ = {⟨d_{1}, d_{2}, d_{3}⟩}.

(8) ∏⟨D, D, D⟩ = D^{3}.

(9) ∏(i ⟼ D) = D^{i}.

(10) ∏ f ⊆ (⋃ f)^{dom f}.

Curried and uncurried functions of some functions

The following propositions are true:

(11) If x ∈ dom f, then there exist y, z such that x = ⟨⟨y, z⟩⟩.

(12) ([: X, Y :] ⟼ z) = [: Y, X:] ⟼ z.

(13) curry f = curry′ f and uncurry f = uncurry′ f.

(14) If [: X, Y :] ≠ ∅, then curry([: X, Y :] ⟼ z) = X ⟼ (Y ⟼ z) and curry′([: X, Y :] ⟼ z) = Y ⟼ (X ⟼ z).

(15) uncurry(X ⟼ (Y ⟼ z)) = [: X, Y :] ⟼ z and uncurry′(X ⟼ (Y ⟼ z)) = [: Y, X:] ⟼ z.

(16) If x ∈ dom f and g = f(x), then rng g ⊆ rng uncurry f and rng g ⊆ rng uncurry′ f.

(17) dom uncurry(X ⟼ f) = [: X, dom f :] and rng uncurry(X ⟼ f) ⊆ rng f and dom uncurry′(X ⟼ f) = [: dom f, X:] and rng uncurry′(X ⟼ f) ⊆ rng f.

(18) If X ≠ ∅, then rng uncurry(X ⟼ f) = rng f and rng uncurry′(X ⟼ f) = rng f.

(19) If [: X, Y :] ≠ ∅ and f ∈ Z^{[: X, Y :]}, then curry f ∈ (Z^{Y})^{X} and curry′ f ∈ (Z^{X})^{Y}.

(20) If f ∈ (Z^{Y})^{X}, then uncurry f ∈ Z^{[: X, Y :]} and uncurry′ f ∈ Z^{[: Y, X :]}.

(21) If curry f ∈ (Z^{Y})^{X} or curry′ f ∈ (Z^{X})^{Y} but dom f ⊆ [: V_{1}, V_{2} :], then f ∈ Z^{[: X, Y :]}.

(22) If uncurry f ∈ Z^{[: X, Y :]} or uncurry′ f ∈ Z^{[: Y, X :]} but rng f ⊆ V_{1} ˙→V_{2} and dom f = X, then f ∈ (Z^{Y})^{X}.

(23) If f ∈ [: X, Y :] ˙→Z, then curry f ∈ X ˙→(Y ˙→Z) and curry′ f ∈ Y ˙→(X ˙→Z).

(24) If f ∈ X ˙→(Y ˙→Z), then uncurry f ∈ [: X, Y :] ˙→Z and uncurry′ f ∈ [: Y, X:] ˙→Z.

(25) If curry f ∈ X ˙→(Y ˙→Z) or curry′ f ∈ Y ˙→(X ˙→Z) but dom f ⊆ [: V_{1}, V_{2} :], then f ∈ [: X, Y :] ˙→Z.

(26) If uncurry f ∈ [: X, Y :] ˙→Z or uncurry′ f ∈ [: Y, X:] ˙→Z but rng f ⊆ V_{1} ˙→V_{2} and dom f ⊆ X, then f ∈ X ˙→(Y ˙→Z).

Functions yielding functions

Let X be a set. The functor Sub_{f} X is defined as follows: (Def.1) x ∈ Sub_{f} X if and only if x ∈ X and x is a function.

Next we state four propositions:

(27) Sub_{f} X ⊆ X.

(28) x ∈ f ^{−1} Sub_{f} rng f if and only if x ∈ dom f and f(x) is a function.

(29) Sub_{f} ∅ = ∅ and Sub_{f}{f} = {f} and Sub_{f}{f, g} = {f, g} and Sub_{f}{f, g, h} = {f, g, h}.

(30) If Y ⊆ Sub_{f} X, then Sub_{f} Y = Y.

We now define three new functors. Let f be a function. The functor dom_{κ} f(κ) yielding a function is defined by: (Def.2) dom(dom_{κ} f(κ)) = f ^{−1} Sub_{f} rng f and for every x such that x ∈ f ^{−1}

Sub_{f} rng f holds (dom_{κ} f(κ))(x) = π_{1}(f(x)). The functor rng_{κ} f(κ) yields a function and is defined as follows: (Def.3) dom(rng_{κ} f(κ)) = f ^{−1} Sub_{f} rng f and for every x such that x ∈ f ^{−1}

Sub_{f} rng f holds (rng_{κ} f(κ))(x) = π_{2}(f(x)). The functor ⋂ f is defined as follows: (Def.4) ⋂ f = ⋂ rng f.

Next we state a number of propositions:

(31) If x ∈ dom f and g = f(x), then x ∈ dom(dom_{κ} f(κ)) and (dom_{κ} f(κ))(x) = dom g and x ∈ dom(rng_{κ} f(κ)) and (rng_{κ} f(κ))(x) = rng g.

(32) dom_{κ} (κ) = and rng_{κ} (κ) = .

(33) dom_{κ}⟨f⟩(κ) = ⟨dom f⟩ and rng_{κ}⟨f⟩(κ) = ⟨rng f⟩.

(34) dom_{κ}⟨f, g⟩(κ) = ⟨dom f, dom g⟩ and rng_{κ}⟨f, g⟩(κ) = ⟨rng f, rng g⟩.

(35) dom_{κ}⟨f, g, h⟩(κ) = ⟨dom f, dom g, dom h⟩ and rng_{κ}⟨f, g, h⟩(κ) = ⟨rng f, rng g, rng h⟩.

(36) dom_{κ}(X ⟼ f)(κ) = X ⟼ dom f and rng_{κ}(X ⟼ f)(κ) = X ⟼ rng f.

(37) If f ≠ , then x ∈ ⋂ f if and only if for every y such that y ∈ dom f holds x ∈ f(y).

(38) ⋃ = ∅ and ⋂ = ∅.

(39) ⋃⟨X⟩ = X and ⋂⟨X⟩ = X.

(40) ⋃⟨X, Y ⟩ = X ∪ Y and ⋂⟨X, Y ⟩ = X ∩ Y.

(41) ⋃⟨X, Y, Z⟩ = X ∪ Y ∪ Z and ⋂⟨X, Y, Z⟩ = X ∩ Y ∩ Z.

(42) ⋃(∅ ⟼ Y) = ∅ and ⋂(∅ ⟼ Y) = ∅.

(43) If X ≠ ∅, then ⋃(X ⟼ Y) = Y and ⋂(X ⟼ Y) = Y.

Let f be a function, and let x, y be arbitrary. The functor f(x)(y) is defined by: (Def.5) f(x)(y) = (uncurry f)(⟨⟨x, y⟩⟩).

We now state several propositions:

(44) If x ∈ dom f and g = f(x) and y ∈ dom g, then f(x)(y) = g(y).

(45) If x ∈ dom f, then ⟨f⟩(1)(x) = f(x) and ⟨f, g⟩(1)(x) = f(x) and ⟨f, g, h⟩(1)(x) = f(x).

(46) If x ∈ dom g, then ⟨f, g⟩(2)(x) = g(x) and ⟨f, g, h⟩(2)(x) = g(x).

(47) If x ∈ dom h, then ⟨f, g, h⟩(3)(x) = h(x).

(48) If x ∈ X and y ∈ dom f, then (X ⟼ f)(x)(y) = f(y).

Cartesian product of functions with the same domain

Let f be a function. The functor ∏^{∗} f yielding a function is defined as follows: (Def.6) ∏^{∗} f = curry(uncurry′ f [: ⋂(dom_{κ} f(κ)), dom f :]).

We now state several propositions:

(49) dom ∏^{∗} f = ⋂(dom_{κ} f(κ)) and rng ∏^{∗} f ⊆ ∏(rng_{κ} f(κ)).

(50) If x ∈ dom ∏^{∗} f, then (∏^{∗} f)(x) is a function.

(51) If x ∈ dom ∏^{∗} f and g = (∏^{∗} f)(x), then dom g = f ^{−1} Sub_{f} rng f and for every y such that y ∈ dom g holds ⟨⟨y, x⟩⟩ ∈ dom uncurry f and g(y) = (uncurry f)(⟨⟨y, x⟩⟩).

(52) If x ∈ dom ∏^{∗} f, then for every g such that g ∈ rng f holds x ∈ dom g.

(53) If g ∈ rng f and for every g such that g ∈ rng f holds x ∈ dom g, then x ∈ dom ∏^{∗} f.

(54) If x ∈ dom f and g = f(x) and y ∈ dom ∏^{∗} f and h = (∏^{∗} f)(y), then g(y) = h(x).

(55) If x ∈ dom f and f(x) is a function and y ∈ dom ∏^{∗} f, then f(x)(y) = (∏^{∗} f)(y)(x).

Cartesian product of functions

Let f be a function. The functor ∏^{∘} f yielding a function is defined by the conditions (Def.7). (Def.7) (i) dom ∏^{∘} f = ∏(dom_{κ} f(κ)),

(ii) for every g such that g ∈ ∏(dom_{κ} f(κ)) there exists h such that (∏^{∘} f)(g) = h and dom h = f ^{−1} Sub_{f} rng f and for every x such that x ∈ dom h holds h(x) = (uncurry f)(⟨⟨x, g(x)⟩⟩).

The following propositions are true:

(56) If g ∈ ∏(dom_{κ} f(κ)) and x ∈ dom g, then (∏^{∘} f)(g)(x) = f(x)(g(x)).

(57) If x ∈ dom f and g = f(x) and h ∈ ∏(dom_{κ} f(κ)) and h′ = (∏^{∘} f)(h), then h(x) ∈ dom g and h′(x) = g(h(x)) and h′ ∈ ∏(rng_{κ} f(κ)).

(58) rng ∏^{∘} f = ∏(rng_{κ} f(κ)).

(59) If ∉ rng f, then ∏^{∘} f is one-to-one if and only if for every g such that g ∈ rng f holds g is one-to-one.

Properties of Cartesian products of functions

The following propositions are true:

(60) ∏^{∗} = and ∏^{∘} = { } ⟼ .

(61) dom ∏^{∗}⟨h⟩ = dom h and for every x such that x ∈ dom h holds (∏^{∗}⟨h⟩)(x) = ⟨h(x)⟩.

(62) dom ∏^{∗}⟨f_{1}, f_{2}⟩ = dom f_{1}∩dom f_{2} and for every x such that x ∈ dom f_{1}∩ dom f_{2} holds (∏^{∗}⟨f_{1}, f_{2}⟩)(x) = ⟨f_{1}(x), f_{2}(x)⟩.

(63) If X ≠ ∅, then dom ∏^{∗}(X ⟼ f) = dom f and for every x such that x ∈ dom f holds (∏^{∗}(X ⟼ f))(x) = X ⟼ f(x).

(64) dom ∏^{∘}⟨h⟩ = ∏⟨dom h⟩ and rng ∏^{∘}⟨h⟩ = ∏⟨rng h⟩ and for every x such that x ∈ dom h holds (∏^{∘}⟨h⟩)(⟨x⟩) = ⟨h(x)⟩.

(65) (i) dom ∏^{∘}⟨f_{1}, f_{2}⟩ = ∏⟨dom f_{1}, dom f_{2}⟩, (ii) rng ∏^{∘}⟨f_{1}, f_{2}⟩ = ∏⟨rng f_{1}, rng f_{2}⟩, (iii) for all x, y such that x ∈ dom f_{1} and y ∈ dom f_{2} holds (∏^{∘}⟨f_{1}, f_{2}⟩)(⟨x,

y⟩) = ⟨f_{1}(x), f_{2}(y)⟩.

(66) dom ∏^{∘}(X ⟼ f) = (dom f)^{X} and rng ∏^{∘}(X ⟼ f) = (rng f)^{X} and for every g such that g ∈ (dom f)^{X} holds (∏^{∘}(X ⟼ f))(g) = f · g.

(67) If x ∈ dom f_{1} and x ∈ dom f_{2}, then for all y_{1}, y_{2} holds ⟨f_{1}, f_{2}⟩(x) = ⟨⟨y_{1}, y_{2}⟩⟩ if and only if (∏^{∗}⟨f_{1}, f_{2}⟩)(x) = ⟨y_{1}, y_{2}⟩.

(68) If x ∈ dom f_{1} and y ∈ dom f_{2}, then for all y_{1}, y_{2} holds [: f_{1}, f_{2} :](⟨⟨x, y⟩⟩) = ⟨⟨y_{1}, y_{2}⟩⟩ if and only if (∏^{∘}⟨f_{1}, f_{2}⟩)(⟨x, y⟩) = ⟨y_{1}, y_{2}⟩.

(69) If dom f = X and dom g = X and for every x such that x ∈ X holds f(x) ≈ g(x), then ∏ f ≈ ∏ g.

(70) If dom f = dom h and dom g = rng h and h is one-to-one and for every x such that x ∈ dom h holds f(x) ≈ g(h(x)), then ∏ f ≈ ∏ g.

(71) If dom f = X, then ∏ f ≈ ∏(f · P).

Function yielding powers

Let us consider f, X. The functor X^{f} yielding a function is defined by: (Def.8) dom(X^{f}) = dom f and for every x such that x ∈ dom f holds X^{f}(x) =

X^{f(x)}. We now state several propositions:

(72) If ∅ ∉ rng f, then ∅^{f} = dom f ⟼ ∅.

(73) X^{}= .

(74) Y^{⟨X⟩} = ⟨Y^{X}⟩.

(75) Z^{⟨X,Y ⟩} = ⟨Z^{X}, Z^{Y} ⟩.

(76) Z^{X⟼Y} = X ⟼ Z^{Y}.

(77) X^{⋃ disjoin f} ≈ ∏(X^{f}).

Let us consider X, f. The functor f^{X} yielding a function is defined by:

552 grzegorz bancerek

(Def.9) dom(f^{X}) = dom f and for every x such that x ∈ dom f holds f^{X}(x) = f(x)^{X}.

Next we state several propositions:

(78) f^{∅} = dom f ⟼ { }.

(79) ^{X} = .

(80) ⟨Y ⟩^{X} = ⟨Y^{X}⟩.

(81) ⟨Y, Z⟩^{X} = ⟨Y^{X}, Z^{X}⟩.

(82) (Y ⟼ Z)^{X} = Y ⟼ Z^{X}.

(83) ∏(f^{X}) ≈ (∏ f)^{X}.

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

[2] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3):537–541, 1990.

[3] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

[4] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

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

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

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

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

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

[10] 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.

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

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

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

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

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

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

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

Introduction to Modal Propositional Logic

Alicia de la Cruz Universidad Politecnica de Madrid

MML Identifier: MODAL 1.

The terminology and notation used here are introduced in the following papers: [15], [11], [2], [14], [16], [13], [7], [5], [6], [8], [10], [12], [1], [9], [3], [4], and [17]. For simplicity we follow a convention: x, y will be arbitrary, n, m, k will denote natural numbers, t_{1} will denote a tree decorated by [: , qua a non-empty set :], w, s, t will denote finite sequences of elements of , X will denote a set, and D will denote a non-empty set. Next we state the proposition

(1) If X is finite, then card X = 2 if and only if there exist x, y such that X = {x, y} and x ≠ y.

Let Z be a tree. The root of Z yields an element of Z and is defined as follows: (Def.1) the root of Z = ε.

Let us consider D, and let T be a tree decorated by D. The root of T yields an element of D and is defined by: (Def.2) the root of T = T (the root of dom T).

Next we state a number of propositions:

(2) ⟨n⟩ = ⟨m⟩ if and only if n = m.

(3) If n ≠ m, then ⟨n⟩ and ⟨m⟩^{}s are not comparable.

(4) For every s such that s ≠ ε there exist w, n such that s = ⟨n⟩^{}w.

(5) If n ≠ m, then ⟨n⟩ ⟨m⟩^{}s.

(6) If n ≠ m, then ⟨n⟩ ⟨m⟩^{}s.

(7) ⟨n⟩ ⟨m⟩.

(8) If w ≠ ε, then s ≺ s^{}w.

(9) The elementary tree of 1 = {ε, ⟨0⟩}.

(10) The elementary tree of 2 = {ε, ⟨0⟩, ⟨1⟩}.

(11) For every tree Z and for all n, m such that n ≤ m and ⟨m⟩ ∈ Z holds ⟨n⟩ ∈ Z.

(12) If w^{}t ≺ w^{}s, then t ≺ s.

(13) t_{1} ∈ ^{∗} ˙→[: , qua a non-empty set :].

(14) For all trees Z, Z_{1} and for every element z of Z holds z ∈ Z(z/Z_{1}).

(15) For all trees Z, Z_{1}, Z_{2} and for every element z of Z such that Z(z/Z_{1}) = Z(z/Z_{2}) holds Z_{1} = Z_{2}.

(16) For all trees Z, Z_{1}, Z_{2} decorated by D and for every element z of dom Z such that Z(z/Z_{1}) = Z(z/Z_{2}) holds Z_{1} = Z_{2}.

(17) For all trees Z_{1}, Z_{2} and for every finite sequence p of elements of such that p ∈ Z_{1} for every element v of Z_{1}(p/Z_{2}) and for every element w of Z_{1} such that v = w and w ≺ p holds succ v = succ w.

(18) For all trees Z_{1}, Z_{2} and for every finite sequence p of elements of such that p ∈ Z_{1} for every element v of Z_{1}(p/Z_{2}) and for every element w of Z_{1} such that v = w and p and w are not comparable holds succ v = succ w.

(19) For all trees Z_{1}, Z_{2} and for every finite sequence p of elements of such that p ∈ Z_{1} for every element v of Z_{1}(p/Z_{2}) and for every element w of Z_{2} such that v = p^{}w holds succ v ≈ succ w.

(20) For every tree Z_{1} and for every finite sequence p of elements of such that p ∈ Z_{1} for every element v of Z_{1} and for every element w of Z_{1} p such that v = p^{}w holds succ v ≈ succ w.

(21) For every tree Z and for every element p of Z such that Z is finite holds succ p is finite.

(22) For every tree Z such that Z is finite and the branch degree of the root of Z = 0 holds card Z = 1 and Z = {ε}.

(23) For every tree Z such that Z is finite and the branch degree of the root of Z = 1 holds succ(the root of Z) = {⟨0⟩}.

(24) For every tree Z such that Z is finite and the branch degree of the root of Z = 2 holds succ(the root of Z) = {⟨0⟩, ⟨1⟩}.

In the sequel s′, w′ will be elements of ^{∗} . One can prove the following propositions:

(25) For every tree Z and for every element o of Z such that o ≠ the root of Z holds Z o ≈ {o^{}s′ : o^{}s′ ∈ Z} and the root of Z ∉ {o^{}w′ : o^{}w′ ∈ Z}.

(26) For every tree Z and for every element o of Z such that o ≠ the root of Z and Z is finite holds card(Z o) < card Z.

(27) For every tree Z and for every element z of Z such that succ(the root of Z) = {z} and Z is finite holds Z = (the elementary tree of 1)(⟨0⟩/(Z z)).

(28) For every tree Z decorated by D and for every element z of dom Z such that succ(the root of dom Z) = {z} and dom Z is finite holds Z = ( the elementary tree of 1 ⟼ the root of Z)(⟨0⟩/(Z z)).

(29) For every tree Z and for all elements x_{1}, x_{2} of Z such that Z is finite and x_{1} = ⟨0⟩ and x_{2} = ⟨1⟩ and succ(the root of Z) = {x_{1}, x_{2}} holds Z = (the elementary tree of 2)(⟨0⟩/(Z x_{1}))(⟨1⟩/(Z x_{2})).

(30) Let Z be a tree decorated by D. Then for all elements x_{1}, x_{2} of dom Z such that dom Z is finite and x_{1} = ⟨0⟩ and x_{2} = ⟨1⟩ and succ(the root of dom Z) = {x_{1}, x_{2}} holds Z = ( the elementary tree of 2 ⟼ the root of Z)(⟨0⟩/(Z x_{1}))(⟨1⟩/(Z x_{2})).

The non-empty set V is defined by: (Def.3) V = [: {3}, :].

A variable is an element of V. The non-empty set C is defined as follows:

(Def.4) C = [: {0, 1, 2}, :].

A conective is an element of C. One can prove the following proposition

(31) C ∩ V = ∅.

In the sequel p, q denote variables. Let T be a tree, and let v be an element of T. Then the branch degree of v is a natural number.

Let D be a non-empty set. A non-empty set is called a non-empty set of trees decorated by D if: (Def.5) for every x such that x ∈ it holds x is a tree decorated by D.

Let D_{0} be a non-empty set, and let D be a non-empty set of trees decorated by D_{0}. We see that the element of D is a tree decorated by D_{0}.

The non-empty set WFF of trees decorated by [: , qua a non-empty set :] is defined by the condition (Def.6). (Def.6) Let x be a tree decorated by [: , qua a non-empty set :]. Then x ∈

WFF if and only if the following conditions are satisfied:

(i) dom x is finite, (ii) for every element v of dom x holds the branch degree of v ≤ 2 but if

the branch degree of v = 0, then x(v) = ⟨⟨0, 0⟩⟩ or there exists k such that x(v) = ⟨⟨3, k⟩⟩ but if the branch degree of v = 1, then x(v) = ⟨⟨1, 0⟩⟩ or x(v) = ⟨⟨1, 1⟩⟩ but if the branch degree of v = 2, then x(v) = ⟨⟨2, 0⟩⟩.

A MP-formula is an element of WFF. In the sequel A, A_{1}, B, B_{1}, C denote MP-formulae. Let us consider A, and

let a be an element of dom A. Then A a is a MP-formula.

Let a be an element of C. The functor Arity(a) yielding a natural number is defined by: (Def.7) Arity(a) = a_{1}.

Let D be a non-empty set, and let T, T_{1} be trees decorated by D, and let p be a finite sequence of elements of . Let us assume that p ∈ dom T. The functor T (p ← T_{1}) yields a tree decorated by D and is defined by: (Def.8) T (p ← T_{1}) = T (p/T_{1}).

The following propositions are true:

(32) (The elementary tree of 1 ⟼ ⟨⟨1, 0⟩⟩)(⟨0⟩/A) is a MP-formula.

(33) (The elementary tree of 1 ⟼ ⟨⟨1, 1⟩⟩)(⟨0⟩/A) is a MP-formula.

(34) (The elementary tree of 2 ⟼ ⟨⟨2, 0⟩⟩)(⟨0⟩/A)(⟨1⟩/B) is a MP-formula. We now define three new functors. Let us consider A. The functor ¬A yields

a MP-formula and is defined as follows: (Def.9) ¬A = ( the elementary tree of 1 ⟼ ⟨⟨1, 0⟩⟩)(⟨0⟩/A). The functor A yields a MP-formula and is defined as follows:

(Def.10) A = ( the elementary tree of 1 ⟼ ⟨⟨1, 1⟩⟩)(⟨0⟩/A).

Let us consider B. The functor A ∧ B yielding a MP-formula is defined as follows:

(Def.11) A ∧ B = ( the elementary tree of 2 ⟼ ⟨⟨2, 0⟩⟩)(⟨0⟩/A)(⟨1⟩/B).

We now define three new functors. Let us consider A. The functor A yields a MP-formula and is defined as follows:

(Def.12) A = ¬ ¬A.

Let us consider B. The functor A ∨ B yields a MP-formula and is defined as follows:

(Def.13) A ∨ B = ¬(¬A ∧ ¬B).

The functor A ⇒ B yields a MP-formula and is defined by: (Def.14) A ⇒ B = ¬(A ∧ ¬B).

The following propositions are true:

(35) The elementary tree of 0 ⟼ ⟨⟨3, n⟩⟩ is a MP-formula.

(36) The elementary tree of 0 ⟼ ⟨⟨0, 0⟩⟩ is a MP-formula.

Let us consider p. The functor ^{@}p yields a MP-formula and is defined by: (Def.15) ^{@}p = the elementary tree of 0 ⟼ p.

We now state four propositions:

(37) If ^{@}p = ^{@}q, then p = q.

(38) If ¬A = ¬B, then A = B.

(39) If A = B, then A = B.

(40) If A ∧ B = A_{1} ∧ B_{1}, then A = A_{1} and B = B_{1}.

The MP-formula VERUM is defined by: (Def.16) VERUM = the elementary tree of 0 ⟼ ⟨⟨0, 0⟩⟩.

Next we state several propositions:

(41) card dom A ≠ 0.

(42) If card dom A = 1, then A = VERUM or there exists p such that A = ^{@}p.

(43) If card dom A ≥ 2, then there exists B such that A = ¬B or A = B or there exist B, C such that A = B ∧ C.

(44) card dom A < card dom ¬A.

(45) card dom A < card dom A.

(46) card dom A < card dom(A ∧ B) and card dom B < card dom(A ∧ B).

We now define four new attributes. A MP-formula is atomic if: (Def.17) there exists p such that it = ^{@}p.

A MP-formula is negative if: (Def.18) there exists A such that it = ¬A.

A MP-formula is necessitive if: (Def.19) there exists A such that it = A.

A MP-formula is conjunctive if: (Def.20) there exist A, B such that it = A ∧ B.

The scheme MP Ind deals with a unary predicate P, and states that:

for every element A of WFF holds P[A] provided the parameter satisfies the following conditions:

• P[VERUM],

• for every variable p holds P[^{@}p],

• for every element A of WFF such that P[A] holds P[¬A],

• for every element A of WFF such that P[A] holds P[ A],

• for all elements A, B of WFF such that P[A] and P[B] holds P[A ∧ B].

The following propositions are true:

(47) For every element A of WFF holds A = VERUM or A is a MP-formula or A is a MP-formula or A is a MP-formula or A is a MP-formula.

(48) A = VERUM or there exists p such that A = ^{@}p or there exists B such that A = ¬B or there exists B such that A = B or there exist B, C such that A = B ∧ C.

(49) ^{@}p ≠ ¬A and ^{@}p ≠ A and ^{@}p ≠ A ∧ B.

(50) ¬A ≠ B and ¬A ≠ B ∧ C.

(51) A ≠ B ∧ C.

(52) VERUM ≠ ^{@}p and VERUM ≠ ¬A and VERUM ≠ A and VERUM ≠

A ∧ B. The scheme MP Func Ex deals with a non-empty set A, an element B of

A, a unary functor F yielding an element of A, a unary functor G yielding an element of A, a unary functor H yielding an element of A, and a binary functor I yielding an element of A and states that:

there exists a function f from WFF into A such that f(VERUM) = B and for every variable p holds f(^{@}p) = F(p) and for every element A of WFF and for every element d of A such that f(A) = d holds f(¬A) = G(d) and for every element A of WFF and for every element d of A such that f(A) = d holds f( A) = H(d) and for all elements A, B of WFF and for all elements d_{1}, d_{2} of A such that d_{1} = f(A) and d_{2} = f(B) holds f(A ∧ B) = I(d_{1}, d_{2}) for all values of the parameters.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

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

[3] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.

[4] Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397–402, 1991.

[5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 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. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

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

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

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

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

[17] Wojciech Zielonka. Preliminaries to the Lambek calculus. Formalized Mathematics, 2(3):413–418, 1991.

Totally Bounded Metric Spaces

Alicia de la Cruz Universidad Politecnica de Madrid

MML Identifier: TBSP 1.

The papers [19], [9], [1], [4], [20], [2], [18], [13], [5], [8], [14], [21], [7], [15], [12], [11], [17], [6], [10], [16], and [3] provide the terminology and notation for this paper. For simplicity we follow the rules: M is a metric space, c, g are elements of the carrier of M, F is a family of subsets of the carrier of M, A, B are subsets of the carrier of M, f is a function, n, m, p, k are natural numbers, and r, s, L are real numbers. Next we state four propositions:

(1) For every L such that 0 < L and L < 1 for all n, m such that n ≤ m holds L^{m} ≤ L^{n}.

(2) For every L such that 0 < L and L < 1 for every k holds L^{k} ≤ 1 and 0 < L^{k}.

(3) For every L such that 0 < L and L < 1 for every s such that 0 < s there exists n such that L^{n} < s.

(4) For every set X such that X is finite and X ≠ ∅ and for all sets Y, Z such that Y ∈ X and Z ∈ X holds Y ⊆ Z or Z ⊆ Y there exists a set V such that V ∈ X and for every set Z such that Z ∈ X holds V ⊆ Z.

Let us consider M, F. Then ⋃ F is a subset of the carrier of M.

Let D be a non-empty set. Then Ω_{D} is a subset of D. Then ∅_{D} is a subset of D.

Let us consider M. We say that M is totally bounded if and only if: (Def.1) for every r such that r > 0 there exists F such that F is finite and the

carrier of M = ⋃ F and for every A such that A ∈ F there exists g such that A = Ball(g, r).

Let us consider M. A function is called a sequence of M if: (Def.2) dom it = and rng it ⊆ the carrier of M.

In the sequel S_{1} will denote a sequence of M. The following proposition is true (5) f is a sequence of M if and only if dom f = and for every n holds

f(n) is an element of the carrier of M. Let us consider M, S_{1}, n. Then S_{1}(n) is an element of the carrier of M. Let us consider M, S_{1}. We say that S_{1} is convergent if and only if:

(Def.3) there exists an element x of the carrier of M such that for every r such that r > 0 there exists n such that for every m such that n ≤ m holds ρ(S_{1}(m), x) < r.

Let us consider M, S_{1}. Let us assume that S_{1} is convergent. The functor lim S_{1} yields an element of the carrier of M and is defined by: (Def.4) for every r such that r > 0 there exists n such that for every m such

that m ≥ n holds ρ(S_{1}(m), lim S_{1}) < r.

The following proposition is true

(6) For every S_{1} such that S_{1} is convergent holds lim S_{1} = g if and only if for every r such that 0 < r there exists n such that for every m such that n ≤ m holds ρ(S_{1}(m), g) < r.

Let us consider M, S_{1}. We say that S_{1} is a Cauchy sequence if and only if: (Def.5) for every r such that r > 0 there exists p such that for all n, m such

that p ≤ n and p ≤ m holds ρ(S_{1}(n), S_{1}(m)) < r.

Let us consider M. We say that M is complete if and only if: (Def.6) for every S_{1} such that S_{1} is a Cauchy sequence holds S_{1} is convergent.

We now state two propositions:

(7) For every S_{1} such that S_{1} is convergent holds S_{1} is a Cauchy sequence.

(8) For every S_{1} holds S_{1} is a Cauchy sequence if and only if for every r such that r > 0 there exists p such that for all n, k such that p ≤ n holds ρ(S_{1}(n + k), S_{1}(n)) < r.

Let us consider M. A function from the carrier of M into the carrier of M is called a contraction of M if: (Def.7) there exists L such that 0 < L and L < 1 and for all points x, y of M

holds ρ(it(x), it(y)) ≤ L · ρ(x, y).

We now state four propositions:

(9) For every contraction f of M such that M is complete there exists c such that f(c) = c and for every element y of the carrier of M such that f(y) = y holds y = c.

(10) If M_{top} is compact, then M is complete.

(11) For every contraction f of M such that M_{top} is compact there exists an element c of the carrier of M such that f(c) = c and for every element y of the carrier of M such that f(y) = y holds y = c.

(12) If M_{top} is compact, then M is totally bounded.

We now define two new predicates. Let us consider M. We say that M is bounded if and only if:

(Def.8) there exists r such that 0 < r and for all points x, y of M holds ρ(x, y) ≤

r. Let us consider A. We say that A is bounded if and only if: (Def.9) (i) there exists r such that 0 < r and for all points x, y of M such that

x ∈ A and y ∈ A holds ρ(x, y) ≤ r if A ≠ ∅. One can prove the following propositions:

(13) If A ≠ ∅, then A is bounded if and only if there exists r such that 0 < r and for all points x, y of M such that x ∈ A and y ∈ A holds ρ(x, y) ≤ r.

(14) ∅_{the carrier of M} is bounded.

(15) If A ≠ ∅, then A is bounded if and only if there exist r, c such that 0 < r and c ∈ A and for every point z of M such that z ∈ A holds ρ(c, z) ≤ r.

(16) If 0 < r, then g ∈ Ball(g, r) and Ball(g, r) ≠ ∅.

(17) If r ≤ 0, then Ball(g, r) = ∅.

(18) If 0 < r, then Ball(g, r) is bounded.

(19) Ball(g, r) is bounded.

(20) If A is bounded and B is bounded, then A ∪ B is bounded.

(21) If A is bounded and B ⊆ A, then B is bounded.

(22) If A = {g}, then A is bounded.

(23) If A is finite, then A is bounded.

(24) If F is finite and for every A such that A ∈ F holds A is bounded, then ⋃ F is bounded.

(25) M is bounded if and only if Ω_{the carrier of M} is bounded.

(26) If M is totally bounded, then M is bounded.

Let us consider M, A. Let us assume that A ≠ ∅ and A is bounded. The functor ∨A yields a real number and is defined as follows:

(Def.10) for all points x, y of M such that x ∈ A and y ∈ A holds ρ(x, y) ≤ ∨A and for every s such that for all points x, y of M such that x ∈ A and y ∈ A holds ρ(x, y) ≤ s holds ∨A ≤ s.

We now state several propositions:

(27) Suppose A ≠ ∅ and A is bounded. Then ∨A = r if and only if for all points x, y of M such that x ∈ A and y ∈ A holds ρ(x, y) ≤ r and for every s such that for all points x, y of M such that x ∈ A and y ∈ A holds ρ(x, y) ≤ s holds r ≤ s.

(28) If A = {g}, then ∨A = 0.

(29) If A ≠ ∅ and A is bounded, then 0 ≤ ∨A.

(30) If A ≠ ∅ and A is bounded, then ∨A = 0 if and only if there exists a point g of M such that A = {g}.

(31) If 0 < r, then ∨ Ball(g, r) ≤ 2 · r.

(32) If A ≠ ∅ and A is bounded and B ≠ ∅ and B ⊆ A, then B is bounded and ∨B ≤ ∨A.

(33) If A ≠ ∅ and A is bounded and B ≠ ∅ and B is bounded and A∩B ≠ ∅, then A ∪ B is bounded and ∨(A ∪ B) ≤ ∨A + ∨B.

Let us consider M, S_{1}. Then rng S_{1} is a subset of the carrier of M. One can prove the following proposition

(34) If S_{1} is a Cauchy sequence, then rng S_{1} is bounded.

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] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

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

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

[8] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

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

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

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

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

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

[16] Konrad Raczkowski and Andrzej Ne‘dzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

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

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

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

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

[21] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231–237, 1990.

Categories of Groups

Michał Muzalewski

Summary. We define the category of groups and its subcategories: category of Abelian groups and category of groups with the operator of \frac{1}{2} . The carriers of the groups are included in a universum. The universum is a parameter of the categories.

MML Identifier: GRCAT 1.

The articles [13], [2], [14], [3], [1], [11], [7], [5], [4], [12], [10], [6], [9], and [8] provide the notation and terminology for this paper. For simplicity we follow the rules: x, y will be arbitrary, D will be a non-empty set, U_{1} will be a universal class, and G, H will be group structures. Let us consider x. Then {x} is a non-empty set.

The following propositions are true:

(1) For all sets X, Y, A and for all x, y such that ⟨⟨x, y⟩⟩ ∈ A and A ⊆ [: X, Y :] holds x is an element of X and y is an element of Y.

(2) For all sets X, Y, A and for an arbitrary z such that z ∈ A and A ⊆ [: X, Y :] there exists an element x of X and there exists an element y of Y such that z = ⟨⟨x, y⟩⟩.

(3) For all elements u_{1}, u_{2}, u_{3}, u_{4} of U_{1} holds ⟨⟨u_{1}, u_{2}, u_{3}⟩⟩ is an element of U_{1} and ⟨⟨u_{1}, u_{2}, u_{3}, u_{4}⟩⟩ is an element of U_{1}.

(4) For all x, y such that x ∈ y and y ∈ U_{1} holds x ∈ U_{1}.

In this article we present several logical schemes. The scheme PartLambda2 deals with a set A, a set B, a set C, a binary functor F, and a binary predicate P, 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 and P[x, y] and for all x, y such that ⟨⟨x, y⟩⟩ ∈ dom f holds f(⟨⟨x, y⟩⟩) = F(x, y) provided the following requirement is met:

• for all x, y such that x ∈ A and y ∈ B and P[x, y] holds F(x, y) ∈ C.

The scheme PartLambda2D deals with a non-empty set A, a non-empty set B, a set C, a binary functor F, and a binary predicate P, and states that:

there exists a partial function f from [: A, B:] to C such that for every element x of A and for every element y of B holds ⟨⟨x, y⟩⟩ ∈ dom f if and only if P[x, y] and for every element x of A and for every element y of B such that ⟨⟨x, y⟩⟩ ∈ dom f holds f(⟨⟨x, y⟩⟩) = F(x, y) provided the parameters satisfy the following condition:

• for every element x of A and for every element y of B such that P[x, y] holds F(x, y) ∈ C.

We now define three new functors. op_{2} is a binary operation on {∅}.

op_{1} is a unary operation on {∅}.

op_{0} is an element of {∅}.

We now state three propositions:

(5) op_{2}(∅, ∅) = ∅ and op_{1}(∅) = ∅ and op_{0} = ∅.

(6) {∅} ∈ U_{1} and ⟨⟨{∅}, {∅}⟩⟩ ∈ U_{1} and [: {∅}, {∅}:] ∈ U_{1} and op_{2} ∈ U_{1} and op_{1} ∈ U_{1}.

(7) ⟨\{∅\}, op_{2}, op_{1}, op_{0}⟩ is a group with the operator \frac{1}{2}.

The trivial group being a group with the operator \frac{1}{2} is defined as follows: (Def.1) the trivial group= ⟨{∅}, op_{2}, op_{1}, op_{0}⟩.

We now state the proposition

(8) If G = the trivial group , then for every element x of G holds x = ∅ and for all elements x, y of G holds x + y = ∅ and for every element x of G holds −x = ∅ and 0_{G} = ∅.

In the sequel C denotes a category and O denotes a non-empty subset of the objects of C. Let us consider C, O. The functor Morphs O yields a non-empty subset of the morphisms of C and is defined by: (Def.2) Morphs O = ⋃{hom(a, b) : a ∈ O ∧ b ∈ O}, where a ranges over objects

of C, and b ranges over objects of C.

We now define four new functors. Let us consider C, O. The functor dom O yielding a function from Morphs O into O is defined by: (Def.3) dom O = (the dom-map of C) Morphs O. The functor cod O yields a function from Morphs O into O and is defined by: (Def.4) cod O = (the cod-map of C) Morphs O. The functor comp O yielding a partial function from [: Morphs O, Morphs O qua a non-empty set :] to Morphs O is defined as follows: (Def.5) comp O = (the composition of C) [: Morphs O, Morphs O:]. The functor I_{O} yielding a function from O into Morphs O is defined by: (Def.6) I_{O} = (the id-map of C) O.

Next we state the proposition

(9) ⟨O, Morphs O, dom O, cod O, comp O, I_{O}⟩ is full subcategory of C.

Let us consider C, O. The functor cat O yielding a subcategory of C is defined as follows: (Def.7) cat O = ⟨O, Morphs O, dom O, cod O, comp O, I_{O}⟩.

Next we state the proposition

(10) The objects of cat O = O.

Let us consider G, H. A map from G into H is a function from the carrier of G into the carrier of H.

Let G_{1}, G_{2}, G_{3} be group structures, and let f be a map from G_{1} into G_{2}, and let g be a map from G_{2} into G_{3}. Then g · f is a map from G_{1} into G_{3}.

Let us consider G. The functor id_{G} yields a map from G into G and is defined by: (Def.8) id_{G} = id_{(the carrier of G)}.

One can prove the following two propositions:

(11) For every element x of G holds id_{G}(x) = x.

(12) For every map f from G into H holds f · id_{G} = f and id_{H} ·f = f. Let us consider G, H. The functor zero(G, H) yielding a map from G into

H is defined by: (Def.9) zero(G, H) = (the carrier of G) ⟼ 0_{H}.

Let us consider G, H, and let f be a map from G into H. We say that f is additive if and only if:

(Def.10) for all elements x, y of G holds f(x + y) = f(x) + f(y).

One can prove the following propositions:

(13) For all G_{1}, G_{2}, G_{3} being group structures and for every map f from G_{1} into G_{2} and for every map g from G_{2} into G_{3} and for every element x of G_{1} holds (g · f)(x) = g(f(x)).

(14) For all G_{1}, G_{2}, G_{3} being group structures and for every map f from G_{1} into G_{2} and for every map g from G_{2} into G_{3} such that f is additive and g is additive holds g · f is additive.

(15) For every element x of G holds (zero(G, H))(x) = 0_{H}.

(16) For every group H holds zero(G, H) is additive.

In the sequel G, H are groups. We consider group morphism structures which are systems

⟨a dom-map, a cod-map, a Fun⟩, where the dom-map, the cod-map are a group and the Fun is a map from the dom-map into the cod-map.

We now define two new functors. Let f be a group morphism structure. The functor dom f yielding a group is defined as follows:

(Def.11) dom f = the dom-map of f.

The functor cod f yields a group and is defined by: (Def.12) cod f = the cod-map of f.

Let f be a group morphism structure. The functor fun f yields a map from dom f into cod f and is defined by:

(Def.13) fun f = the Fun of f.

Next we state the proposition

(17) For every f being a group morphism structure and for all groups G_{1}, G_{2} and for every map f_{0} from G_{1} into G_{2} such that f = ⟨G_{1}, G_{2}, f_{0}⟩ holds dom f = G_{1} and cod f = G_{2} and fun f = f_{0}.

Let us consider G, H. The functor ZERO G yielding a group morphism structure is defined as follows:

(Def.14) ZERO G = ⟨G, H, zero(G, H)⟩.

A group morphism structure is said to be a morphism of groups if: (Def.15) fun it is additive.

One can prove the following proposition

(18) For every morphism F of groups holds the Fun of F is additive. Let us consider G, H. Then ZERO G is a morphism of groups. Let us consider G, H. A morphism of groups is said to be a morphism from

G to H if: (Def.16) dom it = G and cod it = H.

We now state three propositions:

(19) For every f being a group morphism structure such that dom f = G and cod f = H and fun f is additive holds f is a morphism from G to H.

(20) For every map f from G into H such that f is additive holds ⟨G, H, f⟩ is a morphism from G to H.

(21) id_{G} is additive.

Let us consider G. The functor I_{G} yields a morphism from G to G and is defined by:

(Def.17) I_{G} = ⟨G, G, id_{G}⟩.

Let us consider G, H. Then ZERO G is a morphism from G to H. We now state several propositions:

(22) For every morphism F from G to H there exists a map f from G into H such that F = ⟨G, H, f⟩ and f is additive.

(23) For every morphism F from G to H there exists a map f from G into H such that F = ⟨G, H, f⟩.

(24) For every morphism F of groups there exist G, H such that F is a morphism from G to H.

(25) For every morphism F of groups there exist groups G, H and there exists a map f from G into H such that F is a morphism from G to H and F = ⟨G, H, f⟩ and f is additive.

(26) For all morphisms g, f of groups such that dom g = cod f there exist groups G_{1}, G_{2}, G_{3} such that g is a morphism from G_{2} to G_{3} and f is a morphism from G_{1} to G_{2}.

(27) For every morphism F of groups holds F is a morphism from dom F to cod F.

Let G, F be morphisms of groups. Let us assume that dom G = cod F. The functor G · F yielding a morphism of groups is defined by:

(Def.18) for all groups G_{1}, G_{2}, G_{3} and for every map g from G_{2} into G_{3} and for every map f from G_{1} into G_{2} such that G = ⟨G_{2}, G_{3}, g⟩ and F = ⟨G_{1}, G_{2}, f⟩ holds G · F = ⟨G_{1}, G_{3}, g · f⟩.

Next we state the proposition

(28) For all groups G_{1}, G_{2}, G_{3} and for every morphism G from G_{2} to G_{3} and for every morphism F from G_{1} to G_{2} holds G · F is a morphism from G_{1} to G_{3}.

Let G_{1}, G_{2}, G_{3} be groups, and let G be a morphism from G_{2} to G_{3}, and let F be a morphism from G_{1} to G_{2}. Then G · F is a morphism from G_{1} to G_{3}.

The following propositions are true:

(29) For all groups G_{1}, G_{2}, G_{3} and for every morphism G from G_{2} to G_{3} and for every morphism F from G_{1} to G_{2} and for every map g from G_{2} into G_{3} and for every map f from G_{1} into G_{2} such that G = ⟨G_{2}, G_{3}, g⟩ and F = ⟨G_{1}, G_{2}, f⟩ holds G · F = ⟨G_{1}, G_{3}, g · f⟩.

(30) For all morphisms f, g of groups such that dom g = cod f there exist groups G_{1}, G_{2}, G_{3} and there exists a map f_{0} from G_{1} into G_{2} and there exists a map g_{0} from G_{2} into G_{3} such that f = ⟨G_{1}, G_{2}, f_{0}⟩ and g = ⟨G_{2}, G_{3}, g_{0}⟩ and g · f = ⟨G_{1}, G_{3}, g_{0} · f_{0}⟩.

(31) For all morphisms f, g of groups such that dom g = cod f holds dom(g · f) = dom f and cod(g · f) = cod g.

(32) For all groups G_{1}, G_{2}, G_{3}, G_{4} and for every morphism f from G_{1} to G_{2} and for every morphism g from G_{2} to G_{3} and for every morphism h from G_{3} to G_{4} holds h · (g · f) = h · g · f.

(33) For all morphisms f, g, h of groups such that dom h = cod g and dom g = cod f holds h · (g · f) = h · g · f.

(34) dom(I_{G}) = G and cod(I_{G}) = G and for every morphism f of groups such that cod f = G holds I_{G} · f = f and for every morphism g of groups such that dom g = G holds g · I_{G} = g.

A non-empty set is called a non-empty set of groups if: (Def.19) for every element x of it holds x is a group.

In the sequel V will be a non-empty set of groups. Let us consider V. We see that the element of V is a group.

We now state two propositions:

(35) For every morphism f of groups and for every element x of {f} holds x is a morphism of groups.

(36) For every morphism f from G to H and for every element x of {f}

holds x is a morphism from G to H. A non-empty set is called a non-empty set of morphisms of groups if:

(Def.20) for every element x of it holds x is a morphism of groups.

Let M be a non-empty set of morphisms of groups. We see that the element of M is a morphism of groups.

We now state the proposition

(37) For every morphism f of groups holds {f} is a non-empty set of mor-

phisms of groups. Let us consider G, H. A non-empty set of morphisms of groups is called a

non-empty set of morphisms from G into H if: (Def.21) for every element x of it holds x is a morphism from G to H.

The following two propositions are true:

(38) D is a non-empty set of morphisms from G into H if and only if for every element x of D holds x is a morphism from G to H.

(39) For every morphism f from G to H holds {f} is a non-empty set of

morphisms from G into H. Let us consider G, H. The functor Morphs(G, H) yields a non-empty set of

morphisms from G into H and is defined by: (Def.22) x ∈ Morphs(G, H) if and only if x is a morphism from G to H.

Let us consider G, H, and let M be a non-empty set of morphisms from G into H. We see that the element of M is a morphism from G to H.

Let us consider x, y. The predicate P_{ob} x, y is defined by: (Def.23) there exist arbitrary x_{1}, x_{2}, x_{3}, x_{4} such that x = ⟨⟨x_{1}, x_{2}, x_{3}, x_{4}⟩⟩ and

there exists G such that y = G and x_{1} = the carrier of G and x_{2} = the addition of G and x_{3} = the reverse-map of G and x_{4} = the zero of G.

One can prove the following two propositions:

(40) For arbitrary x, y_{1}, y_{2} such that P_{ob} x, y_{1} and P_{ob} x, y_{2} holds y_{1} = y_{2}.

(41) There exists x such that x ∈ U_{1} and P_{ob} x,the trivial group . Let us consider U_{1}. The functor GroupObj(U_{1}) yields a non-empty set and

is defined as follows: (Def.24) for every y holds y ∈ GroupObj(U_{1}) if and only if there exists x such

that x ∈ U_{1} and P_{ob} x, y.

The following propositions are true:

(42) The trivial group∈ GroupObj(U_{1}).

(43) For every element x of GroupObj(U_{1}) holds x is a group.

Let us consider U_{1}. Then GroupObj(U_{1}) is a non-empty set of groups. Let us consider V. The functor Morphs V yielding a non-empty set of mor-

phisms of groups is defined by: (Def.25) for every x holds x ∈ Morphs V if and only if there exist elements G,

H of V such that x is a morphism from G to H.

Let us consider V, and let F be an element of Morphs V. Then dom F is an element of V. Then cod F is an element of V.

Let us consider V, and let G be an element of V. The functor I_{G} yields an element of Morphs V and is defined by:

(Def.26) I_{G} = I_{G}.

We now define three new functors. Let us consider V. The functor dom V yields a function from Morphs V into V and is defined as follows:

(Def.27) for every element f of Morphs V holds (dom V)(f) = dom f. The functor cod V yields a function from Morphs V into V and is defined as follows:

(Def.28) for every element f of Morphs V holds (cod V)(f) = cod f.

The functor I_{V} yielding a function from V into Morphs V is defined as follows: (Def.29) for every element G of V holds I_{V} (G) = I_{G}.

One can prove the following two propositions:

(44) For all elements g, f of Morphs V such that dom g = cod f there exist elements G_{1}, G_{2}, G_{3} of V such that g is a morphism from G_{2} to G_{3} and f is a morphism from G_{1} to G_{2}.

(45) For all elements g, f of Morphs V such that dom g = cod f holds g · f ∈

Morphs V. Let us consider V. The functor comp V yields a partial function from

[: Morphs V, Morphs V :] to Morphs V and is defined by: (Def.30) for all elements g, f of Morphs V holds ⟨⟨g, f⟩⟩ ∈ dom comp V if and

only if dom g = cod f and for all elements g, f of Morphs V such that ⟨⟨g, f⟩⟩ ∈ dom comp V holds (comp V)(⟨⟨g, f⟩⟩) = g · f.

Let us consider U_{1}. The functor GroupCat(U_{1}) yielding a category structure is defined by:

(Def.31) GroupCat(U_{1}) = ⟨GroupObj(U_{1}), Morphs GroupObj(U_{1}), dom GroupObj(U_{1}), cod GroupObj(U_{1}), comp GroupObj(U_{1}), I_{GroupObj(U_{1})}⟩.

Next we state several propositions:

(46) For all morphisms f, g of GroupCat(U_{1}) holds ⟨⟨g, f⟩⟩ ∈ dom (the composition of GroupCat(U_{1})) if and only if dom g = cod f.

(47) For every morphism f of GroupCat(U_{1}) and for every element f′ of Morphs GroupObj(U_{1}) and for every object b of GroupCat(U_{1}) and for every element b′ of GroupObj(U_{1}) holds f is an element of Morphs GroupObj(U_{1}) and f′ is a morphism of GroupCat(U_{1}) and b is an element of GroupObj(U_{1}) and b′ is an object of GroupCat(U_{1}).

(48) For every object b of GroupCat(U_{1}) and for every element b′ of GroupObj(U_{1}) such that b = b′ holds id_{b} = I_{b′}.

(49) For every morphism f of GroupCat(U_{1}) and for every element f′ of Morphs GroupObj(U_{1}) such that f = f′ holds dom f = dom f′ and cod f = cod f′.

(50) Let f, g be morphisms of GroupCat(U_{1}). Let f′, g′ be elements of

Morphs GroupObj(U_{1}). Suppose f = f′ and g = g′. Then

(i) dom g = cod f if and only if dom g′ = cod f′, (ii) dom g = cod f if and only if ⟨⟨g′, f′⟩⟩ ∈ dom comp GroupObj(U_{1}), (iii) if dom g = cod f, then g · f = g′ · f′, (iv) dom f = dom g if and only if dom f′ = dom g′,

(v) cod f = cod g if and only if cod f′ = cod g′. Let us consider U_{1}. Then GroupCat(U_{1}) is a category. Let us consider U_{1}. The functor AbGroupObj(U_{1}) yielding a non-empty

subset of the objects of GroupCat(U_{1}) is defined as follows: (Def.32) AbGroupObj(U_{1}) = {G : ⋁_{H} G = H}, where G ranges over elements of

the objects of GroupCat(U_{1}), and H ranges over Abelian groups.

One can prove the following proposition

(51) The trivial group∈ AbGroupObj(U_{1}).

Let us consider U_{1}. The functor AbGroupCat(U_{1}) yielding a subcategory of GroupCat(U_{1}) is defined as follows:

(Def.33) AbGroupCat(U_{1}) = cat AbGroupObj(U_{1}).

We now state the proposition

(52) The objects of AbGroupCat(U_{1}) = AbGroupObj(U_{1}).

Let us consider U_{1}. The functor \frac{1}{2} GroupObj(U_{1}) yields a non-empty subset of the objects of AbGroupCat(U_{1}) and is defined as follows:

(Def.34) \frac{1}{2} GroupObj(U_{1}) = \{G : ⋁_{H} G = H\}, where G ranges over elements of the objects of AbGroupCat(U_{1}), and H ranges over groups with the operator \frac{1}{2}.

Let us consider U_{1}. The functor \frac{1}{2} GroupCat(U_{1}) yields a subcategory of AbGroupCat(U_{1}) and is defined by:

(Def.35) \frac{1}{2} GroupCat(U_{1}) = cat \frac{1}{2} GroupObj(U_{1}).

Next we state two propositions:

(53) The objects of \frac{1}{2} GroupCat(U_{1}) = \frac{1}{2} GroupObj(U_{1}).

(54) The trivial group∈ \frac{1}{2} GroupObj(U_{1}).

References

[1] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 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. Introduction to categories and functors. Formalized Mathematics, 1(2):409–420, 1990.

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

[6] Czesław Byliński. Subcategories and products of categories. Formalized Mathematics, 1(4):725–732, 1990.

categories of groups 571

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

[8] Michał Muzalewski. Atlas of Midpoint Algebra. Formalized Mathematics, 2(4):487–491, 1991.

[9] Michał Muzalewski and Wojciech Skaba. Groups, rings, left- and right-modules. Formalized Mathematics, 2(2):275–278, 1991.

[10] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595–600, 1990.

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

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

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

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

572

Homomorphisms and Isomorphisms of Groups. Quotient Group

Wojciech A. Trybulec Warsaw University Michał J. Trybulec Warsaw University

Summary. Quotient group, homomorphisms and isomorphisms of groups are introduced. The so called isomorphism theorems are proved following [7].

MML Identifier: GROUP 6.

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

(1) For all non-empty sets A, B and for every function f from A into B holds f is one-to-one if and only if for all elements a, b of A such that f(a) = f(b) holds a = b.

Let G be a group, and let A be a subgroup of G. We see that the subgroup of A is a subgroup of G.

Let G be a group, and let A be a subgroup of G. We see that the normal subgroup of A is a subgroup of A.

Let G be a group. Then {1}_{G} is a normal subgroup of G. Then Ω_{G} is a normal subgroup of G.

For simplicity we adopt the following rules: n is a natural number, i is an integer, G, H, I are groups, A, B are subgroups of G, N, M are normal subgroups of G, a, a_{1}, a_{2}, a_{3}, b are elements of G, c is an element of H, f is a function from the carrier of G into the carrier of H, x is arbitrary, and A_{1}, A_{2} are subsets of G. One can prove the following propositions:

(2) For every subgroup X of A and for every element x of A such that x = a holds x · X = a · X qua a subgroup of G and X · x = (X qua a subgroup of G) · a.

(3) For all subgroups X, Y of A holds (X qua a subgroup of G) ∩ Y qua a subgroup of G = X ∩ Y.

(4) a · b · a^{−1} = b^{a^{−1}} and a · (b · a^{−1}) = b^{a^{−1}}.

(5) If b ∈ N, then b^{a} ∈ N.

(6) a · A · A = a · A and a · (A · A) = a · A and A · A · a = A · a and A · (A · a) = A · a.

(7) If A_{1} = {[a, b]}, then G^{c} = gr(A_{1}).

(8) G^{c} is a subgroup of B if and only if for all a, b holds [a, b] ∈ B.

(9) If N is a subgroup of B, then N is a normal subgroup of B. Let us consider G, B, M. Let us assume that M is a subgroup of B. The

functor (M)_{B} yielding a normal subgroup of B is defined as follows: (Def.1) (M)_{B} = M.

One can prove the following proposition

(10) B ∩ N is a normal subgroup of B and N ∩ B is a normal subgroup of

B. Let us consider G, B, N. Then B ∩ N is a normal subgroup of B. Let us consider G, N, B. Then N ∩ B is a normal subgroup of B. A group is trivial if:

(Def.2) there exists x such that the carrier of it = {x}.

One can prove the following propositions:

(11) {1}_{G} is trivial.

(12) G is trivial if and only if ord(G) = 1 and G is finite.

(13) If G is trivial, then {1}_{G} = G.

Let us consider G, N. The functor Cosets N yielding a non-empty set is defined by: (Def.3) Cosets N = the left cosets of N.

In the sequel W_{1}, W_{2} denote elements of Cosets N. One can prove the following propositions:

(14) Cosets N = the left cosets of N and Cosets N = the right cosets of N.

(15) If x ∈ Cosets N, then there exists a such that x = a · N and x = N · a.

(16) a · N ∈ Cosets N and N · a ∈ Cosets N.

(17) If x ∈ Cosets N, then x is a subset of G.

(18) If A_{1} ∈ Cosets N and A_{2} ∈ Cosets N, then A_{1} · A_{2} ∈ Cosets N. Let us consider G, N. The functor CosOp N yields a binary operation on

Cosets N and is defined by: (Def.4) for all W_{1}, W_{2}, A_{1}, A_{2} such that W_{1} = A_{1} and W_{2} = A_{2} holds

(CosOp N)(W_{1}, W_{2}) = A_{1} · A_{2}. In the sequel O is a binary operation on Cosets N. One can prove the fol-

lowing two propositions:

(19) If for all W_{1}, W_{2}, A_{1}, A_{2} such that W_{1} = A_{1} and W_{2} = A_{2} holds O(W_{1}, W_{2}) = A_{1} · A_{2}, then O = CosOp N.

(20) For all W_{1}, W_{2}, A_{1}, A_{2} such that W_{1} = A_{1} and W_{2} = A_{2} holds

(CosOp N)(W_{1}, W_{2}) = A_{1} · A_{2}. Let us consider G, N. The functor ^{G}/_{N} yields a half group structure and is

defined as follows: (Def.5) ^{G}/_{N} = ⟨Cosets N, CosOp N⟩.

One can prove the following propositions:

(21) ^{G}/_{N} = ⟨Cosets N, CosOp N⟩.

(22) The carrier of ^{G}/_{N} = Cosets N.

(23) The operation of ^{G}/_{N} = CosOp N.

In the sequel S, T_{1}, T_{2} denote elements of ^{G}/_{N}. Let us consider G, N, S. The functor ^{@}S yields a subset of G and is defined by: (Def.6) ^{@}S = S.

One can prove the following two propositions:

(24) (^{@}T_{1}) · (^{@}T_{2}) = T_{1} · T_{2}.

(25) ^{@}T_{1} · T_{2} = (^{@}T_{1}) · (^{@}T_{2}).

Let us consider G, N. Then ^{G}/_{N} is a group.

In the sequel S will denote an element of ^{G}/_{N}. The following propositions are true:

(26) There exists a such that S = a · N and S = N · a.

(27) N · a is an element of ^{G}/_{N} and a · N is an element of ^{G}/_{N} and \overline{N} is an element of ^{G}/_{N}.

(28) x ∈ ^{G}/_{N} if and only if there exists a such that x = a · N and x = N · a.

(29) 1_{^{G}/_{N}} = \overline{N}.

(30) If S = a · N, then S^{−1} = a^{−1} · N.

(31) If the left cosets of N is finite, then ^{G}/_{N} is finite.

(32) Ord(^{G}/_{N}) = |• : N|.

(33) If the left cosets of N is finite, then ord(^{G}/_{N}) = |• : N|_{}.

(34) If M is a subgroup of B, then ^{B}/_{(M)_{B}} is a subgroup of ^{G}/_{M}.

(35) If M is a subgroup of N, then ^{N}/_{(M)_{N}} is a normal subgroup of ^{G}/_{M}.

(36) ^{G}/_{N} is an Abelian group if and only if G^{c} is a subgroup of N. Let us consider G, H. A function from the carrier of G into the carrier of H

is called a homomorphism from G to H if: (Def.7) it(a · b) = it(a) · it(b).

One can prove the following proposition

(37) If for all a, b holds f(a · b) = f(a) · f(b), then f is a homomorphism

from G to H. In the sequel g, h will be homomorphisms from G to H, g_{1} will be a homo-

morphism from H to G, and h_{1} will be a homomorphism from H to I. One can prove the following propositions:

(38) dom g = the carrier of G and rng g ⊆ the carrier of H.

(39) g(a · b) = g(a) · g(b).

(40) g(1_{G}) = 1_{H}.

(41) g(a^{−1}) = g(a)^{−1}.

(42) g(a^{b}) = g(a)^{g(b)}.

(43) g([a, b]) = [g(a), g(b)].

(44) g([a_{1}, a_{2}, a_{3}]) = [g(a_{1}), g(a_{2}), g(a_{3})].

(45) g(a^{n}) = g(a)^{n}.

(46) g(a^{i}) = g(a)^{i}.

(47) id_{(the carrier of G)} is a homomorphism from G to G.

(48) h_{1} · h is a homomorphism from G to I.

Let us consider G, H, I, h, h_{1}. Then h_{1} · h is a homomorphism from G to I. Let us consider G, H, g. Then rng g is a subset of H. Let us consider G, H. The functor G → {1}_{H} yields a homomorphism from

G to H and is defined by: (Def.8) for every a holds (G → {1}_{H})(a) = 1_{H}.

The following proposition is true

(49) h_{1} · (G → {1}_{H}) = G → {1}_{I} and (H → {1}_{I}) · h = G → {1}_{I}. Let us consider G, N. The canonical homomorphism onto cosets of N yield-

ing a homomorphism from G to ^{G}/_{N} is defined as follows: (Def.9) for every a holds (the canonical homomorphism onto cosets of N)(a) =

a · N. Let us consider G, H, g. The functor Ker g yields a normal subgroup of G

and is defined by: (Def.10) the carrier of Ker g = {a : g(a) = 1_{H}}.

The following three propositions are true:

(50) a ∈ Ker h if and only if h(a) = 1_{H}.

(51) Ker(G → {1}_{H}) = G.

(52) Ker(the canonical homomorphism onto cosets of N) = N. Let us consider G, H, g. The functor Im g yields a subgroup of H and is

defined as follows: (Def.11) the carrier of Im g = g^{∘} (the carrier of G).

Next we state a number of propositions:

(53) rng g = the carrier of Im g.

(54) x ∈ Im g if and only if there exists a such that x = g(a).

(55) Im g = gr(rng g).

(56) Im(G → {1}_{H}) = {1}_{H}.

(57) Im(the canonical homomorphism onto cosets of N) = ^{G}/_{N}.

(58) h is a homomorphism from G to Im h.

(59) If G is finite, then Im g is finite.

(60) If G is an Abelian group, then Im g is an Abelian group.

(61) Ord(Im g) ≤ Ord(G).

(62) If G is finite, then ord(Im g) ≤ ord(G).

We now define two new predicates. Let us consider G, H, h. We say that h is a monomorphism if and only if:

(Def.12) h is one-to-one.

We say that h is an epimorphism if and only if: (Def.13) rng h = the carrier of H.

We now state several propositions:

(63) If h is a monomorphism and c ∈ Im h, then h(h^{−1}(c)) = c.

(64) If h is a monomorphism, then h^{−1}(h(a)) = a.

(65) If h is a monomorphism, then h^{−1} is a homomorphism from Im h to G.

(66) h is a monomorphism if and only if Ker h = {1}_{G}.

(67) h is an epimorphism if and only if Im h = H.

(68) If h is an epimorphism, then for every c there exists a such that h(a) = c.

(69) The canonical homomorphism onto cosets of N is an epimorphism.

Let us consider G, H, h. We say that h is an isomorphism if and only if: (Def.14) h is an epimorphism and h is a monomorphism.

One can prove the following propositions:

(70) h is an isomorphism if and only if rng h = the carrier of H and h is one-to-one.

(71) If h is an isomorphism, then dom h = the carrier of G and rng h = the carrier of H.

(72) If h is an isomorphism, then h^{−1} is a homomorphism from H to G.

(73) If h is an isomorphism and g_{1} = h^{−1}, then g_{1} is an isomorphism.

(74) If h is an isomorphism and h_{1} is an isomorphism, then h_{1} · h is an isomorphism.

(75) The canonical homomorphism onto cosets of {1}_{G} is an isomorphism.

Let us consider G, H. We say that G and H are isomorphic if and only if: (Def.15) there exists h such that h is an isomorphism.

We now state a number of propositions:

(76) G and G are isomorphic.

(77) If G and H are isomorphic, then H and G are isomorphic.

(78) If G and H are isomorphic and H and I are isomorphic, then G and I are isomorphic.

(79) If h is a monomorphism, then G and Im h are isomorphic.

(80) If G is trivial and H is trivial, then G and H are isomorphic.

(81) {1}_{G} and {1}_{H} are isomorphic.

578 wojciech a. trybulec et al.

(82) G and ^{G}/_{{1}_{G}} are isomorphic and ^{G}/_{{1}_{G}} and G are isomorphic.

(83) ^{G}/_{Ω_{G}} is trivial.

(84) If G and H are isomorphic, then Ord(G) = Ord(H).

(85) If G and H are isomorphic but G is finite or H is finite, then G is finite and H is finite.

(86) If G and H are isomorphic but G is finite or H is finite, then ord(G) = ord(H).

(87) If G and H are isomorphic but G is trivial or H is trivial, then G is trivial and H is trivial.

(88) If G and H are isomorphic but G is an Abelian group or H is an Abelian group, then G is an Abelian group and H is an Abelian group.

(89) ^{G}/_{Ker g} and Im g are isomorphic and Im g and ^{G}/_{Ker g} are isomorphic.

(90) There exists a homomorphism h from ^{G}/_{Ker g} to Im g such that h is an isomorphism and g = h· the canonical homomorphism onto cosets of Ker g.

(91) For every normal subgroup J of ^{G}/_{M} such that J = ^{N}/_{(M)_{N}} and M is a subgroup of N holds ^{(^{G}/_{M})}/_{J} and ^{G}/_{N} are isomorphic.

(92) ^{(B⊔N)}/_{(N)_{B⊔N}} and ^{B}/_{(B∩N)} are isomorphic.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

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

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

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

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

[6] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

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

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

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

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

[13] Wojciech A. Trybulec. Commutator and center of a group. Formalized Mathematics, 2(4):461–466, 1991.

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

[15] Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41–47, 1991.

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

homomorphisms and isomorphisms of groups. … 579 580

Rings and Modules - Part II

Michał Muzalewski Warsaw University Białystok

Summary. We define the trivial left module, morphism of left modules and the field Z_{3}. We proof some elementary facts.

MML Identifier: MOD 2.

The terminology and notation used in this paper are introduced in the following articles: [14], [13], [4], [5], [6], [2], [3], [1], [7], [9], [11], [12], [10], and [8]. For simplicity we adopt the following convention: x, y, z are arbitrary, D is a nonempty set, R, R_{1}, R_{2}, R_{3} are associative rings, G is a left module structure over R, H is a left module structure over R, S is a left module structure over R, G_{1} is a left module structure over R_{1}, G_{2} is a left module structure over R_{2}, G_{3} is a left module structure over R_{3}, and U_{1} is a universal class. Let us consider x. Then {x} is a non-empty set.

Let us consider R. lop(R) is a function from [: the carrier of R, the carrier of the trivial group :] into the carrier of the trivial group.

Let us consider R. The functor _{R}Θ yields a left module over R and is defined by: (Def.1) _{R}Θ = ⟨the trivial group, lop(R)⟩.

Next we state the proposition

(1) For every vector x of _{R}Θ holds x = Θ_{_{R}Θ}.

Let us consider R_{1}, R_{2}, G_{1}, G_{2}. A map from G_{1} into G_{2} is a map from the carrier of G_{1} into the carrier of G_{2}.

Let us consider R_{1}, R_{2}, R_{3}, G_{1}, G_{2}, G_{3}, and let f be a map from G_{1} into G_{2}, and let g be a map from G_{2} into G_{3}. Then g · f is a map from G_{1} into G_{3}.

Let us consider R, G. The functor id_{G} yielding a map from G into G is defined as follows: (Def.2) id_{G} = id_{(the carrier of G)}.

The following propositions are true:

(2) For every vector x of G holds id_{G}(x) = x.

(3) For every map f from G_{1} into G_{2} holds f · id_{G_{1}} = f and id_{G_{2}} ·f = f. Let us consider R_{1}, R_{2}, G_{1}, G_{2}. The functor zero(G_{1}, G_{2}) yields a map from

G_{1} into G_{2} and is defined as follows: (Def.3) zero(G_{1}, G_{2}) = zero( the carrier of G_{1}, the carrier of G_{2}).

Let us consider R, and let G, H be left module structures over R, and let f be a map from G into H. We say that f is linear if and only if: (Def.4) for all vectors x, y of G holds f(x + y) = f(x) + f(y) and for every

scalar a of R and for every vector x of G holds f(a · x) = a · f(x).

The following propositions are true:

(4) For every map f from G into H such that f is linear holds f is additive.

(5) For every map f from G_{1} into G_{2} and for every map g from G_{2} into G_{3} and for every vector x of G_{1} holds (g · f)(x) = g(f(x)).

(6) For every map f from G into H and for every map g from H into S such that f is linear and g is linear holds g · f is linear.

For simplicity we adopt the following rules: R, R_{1}, R_{2} denote associative rings, G denotes a left module over R, H denotes a left module over R, G_{1} denotes a left module over R_{1}, and G_{2} denotes a left module over R_{2}. The following propositions are true:

(7) For every vector x of G_{1} holds (zero(G_{1}, G_{2}))(x) = Θ_{G_{2}}.

(8) zero(G, H) is linear.

In the sequel G_{1} will denote a left module over R, G_{2} will denote a left module over R, and G_{3} will denote a left module over R. Let us consider R. We consider left module morphism structures over R which are systems

⟨a dom-map, a cod-map, a Fun⟩, where the dom-map, the cod-map are a left module over R and the Fun is a map from the dom-map into the cod-map.

In the sequel f will be a left module morphism structure over R. We now define two new functors. Let us consider R, f. The functor dom f yields a left module over R and is defined as follows: (Def.5) dom f = the dom-map of f. The functor cod f yields a left module over R and is defined as follows: (Def.6) cod f = the cod-map of f.

Let us consider R, f. The functor fun f yields a map from dom f into cod f and is defined by: (Def.7) fun f = the Fun of f.

One can prove the following proposition

(9) For every map f_{0} from G_{1} into G_{2} such that f = ⟨G_{1}, G_{2}, f_{0}⟩ holds dom f = G_{1} and cod f = G_{2} and fun f = f_{0}.

Let us consider R, G, H. The functor ZERO G yielding a left module morphism structure over R is defined as follows: (Def.8) ZERO G = ⟨G, H, zero(G, H)⟩.

Let us consider R. A left module morphism structure over R is said to be a left module morphism of R if: (Def.9) fun it is linear.

One can prove the following proposition

(10) For every left module morphism F of R holds the Fun of F is linear. Let us consider R, G, H. Then ZERO G is a left module morphism of R. Let us consider R, G, H. A left module morphism of R is said to be a

morphism from G to H if: (Def.10) dom it = G and cod it = H.

One can prove the following three propositions:

(11) If dom f = G and cod f = H and fun f is linear, then f is a morphism from G to H.

(12) For every map f from G into H such that f is linear holds ⟨G, H, f⟩ is a morphism from G to H.

(13) id_{G} is linear.

Let us consider R, G. The functor I_{G} yields a morphism from G to G and is defined by:

(Def.11) I_{G} = ⟨G, G, id_{G}⟩.

Let us consider R, G, H. Then ZERO G is a morphism from G to H. The following propositions are true:

(14) For every morphism F from G to H there exists a map f from G into H such that F = ⟨G, H, f⟩ and f is linear.

(15) For every morphism F from G to H there exists a map f from G into H such that F = ⟨G, H, f⟩.

(16) For every left module morphism F of R there exist G, H such that F is a morphism from G to H.

(17) For every left module morphism F of R there exist left modules G, H over R and there exists a map f from G into H such that F is a morphism from G to H and F = ⟨G, H, f⟩ and f is linear.

(18) For all left module morphisms g, f of R such that dom g = cod f there exist G_{1}, G_{2}, G_{3} such that g is a morphism from G_{2} to G_{3} and f is a morphism from G_{1} to G_{2}.

(19) For every left module morphism F of R holds F is a morphism from

dom F to cod F. Let us consider R, and let G, F be left module morphisms of R. Let us

assume that dom G = cod F. The functor G · F yields a left module morphism of R and is defined as follows:

(Def.12) for all left modules G_{1}, G_{2}, G_{3} over R and for every map g from G_{2} into G_{3} and for every map f from G_{1} into G_{2} such that G = ⟨G_{2}, G_{3}, g⟩ and F = ⟨G_{1}, G_{2}, f⟩ holds G · F = ⟨G_{1}, G_{3}, g · f⟩.

Next we state the proposition

(20) For every morphism G from G_{2} to G_{3} and for every morphism F from

G_{1} to G_{2} holds G · F is a morphism from G_{1} to G_{3}. Let us consider R, G_{1}, G_{2}, G_{3}, and let G be a morphism from G_{2} to G_{3}, and

let F be a morphism from G_{1} to G_{2}. The functor F [G] yielding a morphism from G_{1} to G_{3} is defined by:

(Def.13) F [G] = G · F.

We now state several propositions:

(21) Let G be a morphism from G_{2} to G_{3}. Then for every morphism F from G_{1} to G_{2} and for every map g from G_{2} into G_{3} and for every map f from G_{1} into G_{2} such that G = ⟨G_{2}, G_{3}, g⟩ and F = ⟨G_{1}, G_{2}, f⟩ holds F [G] = ⟨G_{1}, G_{3}, g · f⟩ and G · F = ⟨G_{1}, G_{3}, g · f⟩.

(22) Let f, g be left module morphisms of R. Then if dom g = cod f, then there exist left modules G_{1}, G_{2}, G_{3} over R and there exists a map f_{0} from G_{1} into G_{2} and there exists a map g_{0} from G_{2} into G_{3} such that f = ⟨G_{1}, G_{2}, f_{0}⟩ and g = ⟨G_{2}, G_{3}, g_{0}⟩ and g · f = ⟨G_{1}, G_{3}, g_{0} · f_{0}⟩.

(23) For all left module morphisms f, g of R such that dom g = cod f holds dom(g · f) = dom f and cod(g · f) = cod g.

(24) For all left modules G_{1}, G_{2}, G_{3}, G_{4} over R and for every morphism f from G_{1} to G_{2} and for every morphism g from G_{2} to G_{3} and for every morphism h from G_{3} to G_{4} holds h · (g · f) = h · g · f.

(25) For all left module morphisms f, g, h of R such that dom h = cod g and dom g = cod f holds h · (g · f) = h · g · f.

(26) dom(I_{G}) = G and cod(I_{G}) = G and for every left module morphism f of R such that cod f = G holds I_{G} · f = f and for every left module morphism g of R such that dom g = G holds g · I_{G} = g.

(27) {x, y, z} is a non-empty set.

Let us consider x, y, z. Then {x, y, z} is a non-empty set. We now state four propositions:

(28) For all elements u, v, w of U_{1} holds {u, v, w} is an element of U_{1}.

(29) For every element u of U_{1} holds succ u is an element of U_{1}.

(30) \overline{0} is an element of U_{1} and \overline{1} is an element of U_{1} and \overline{2} is an element of U_{1}.

(31) \overline{0} ≠ \overline{1} and \overline{0} ≠ \overline{2} and \overline{1} ≠ \overline{2}.

In the sequel a, b will be elements of \{\overline{0}, \overline{1}, \overline{2}\}. We now define three new functors. Let us consider a. The functor −a yields an element of \{\overline{0}, \overline{1}, \overline{2}\} and is defined as follows:

(Def.14) (i) −a = \overline{0} if a = \overline{0}, (ii) −a = \overline{2} if a = \overline{1}, (iii) −a = \overline{1} if a = \overline{2}. Let us consider b. The functor a + b yields an element of \{\overline{0}, \overline{1}, \overline{2}\} and is defined by:

(Def.15) (i) a + b = b if a = \overline{0}, (ii) a + b = a if b = \overline{0}, (iii) a + b = \overline{2} if a = \overline{1} and b = \overline{1}, (iv) a + b = \overline{0} if a = \overline{1} and b = \overline{2},

(v) a + b = \overline{0} if a = \overline{2} and b = \overline{1}, (vi) a + b = \overline{1} if a = \overline{2} and b = \overline{2}. The functor a · b yielding an element of \{\overline{0}, \overline{1}, \overline{2}\} is defined by:

(Def.16) (i) a · b = \overline{0} if b = \overline{0}, (ii) a · b = \overline{0} if a = \overline{0}, (iii) a · b = a if b = \overline{1}, (iv) a · b = b if a = \overline{1},

(v) a · b = \overline{1} if a = \overline{2} and b = \overline{2}.

We now define five new functors. The binary operation add_{3} on \{\overline{0}, \overline{1}, \overline{2}\} is defined by:

(Def.17) add_{3}(a, b) = a + b.

The binary operation mult_{3} on \{\overline{0}, \overline{1}, \overline{2}\} is defined by:

(Def.18) mult_{3}(a, b) = a · b.

The unary operation compl_{3} on \{\overline{0}, \overline{1}, \overline{2}\} is defined as follows:

(Def.19) compl_{3}(a) = −a.

The element unit_{3} of \{\overline{0}, \overline{1}, \overline{2}\} is defined as follows:

(Def.20) unit_{3} = \overline{1}.

The element zero_{3} of \{\overline{0}, \overline{1}, \overline{2}\} is defined as follows:

(Def.21) zero_{3} = \overline{0}.

The field structure Z_{3} is defined by:

(Def.22) Z_{3} = ⟨\{\overline{0}, \overline{1}, \overline{2}\}, mult_{3}, add_{3}, compl_{3}, unit_{3}, zero_{3}⟩.

Next we state several propositions:

(32) 0_{Z_{3}} = \overline{0} and 1_{Z_{3}} = \overline{1} and 0_{Z_{3}} is an element of \{\overline{0}, \overline{1}, \overline{2}\} and 1_{Z_{3}} is an element of \{\overline{0}, \overline{1}, \overline{2}\} and the addition of Z_{3} = add_{3} and the multiplication of Z_{3} = mult_{3} and the reverse-map of Z_{3} = compl_{3}.

(33) For all scalars x, y of Z_{3} and for all elements X, Y of \{\overline{0}, \overline{1}, \overline{2}\} such that X = x and Y = y holds x + y = X + Y and x · y = X · Y and −x = −X.

(34) Let x, y, z be scalars of Z_{3}. Let X, Y, Z be elements of \{\overline{0}, \overline{1}, \overline{2}\}. Suppose X = x and Y = y and Z = z. Then x + y + z = X + Y + Z and x+(y +z) = X +(Y +Z) and x·y ·z = X ·Y ·Z and x·(y ·z) = X ·(Y ·Z).

(35) Let x, y, z, a, b be elements of \{\overline{0}, \overline{1}, \overline{2}\}. Suppose a = \overline{0} and b = \overline{1}.

Then

(i) x + y = y + x, (ii) x + y + z = x + (y + z), (iii) x + a = x, (iv) x + −x = a,

(v) x · y = y · x, (vi) x · y · z = x · (y · z), (vii) x · b = x, (viii) if x ≠ a, then there exists an element y of \{\overline{0}, \overline{1}, \overline{2}\} such that x · y = b, (ix) a ≠ b,

(x) x · (y + z) = x · y + x · z.

(36) Let F be a field structure. Suppose that

(i) for all scalars x, y, z of F holds x+y = y +x and x+y +z = x+(y +z) and x + 0_{F} = x and x + −x = 0_{F} and x · y = y · x and x · y · z = x · (y · z) and x · 1_{F} = x but if x ≠ 0_{F}, then there exists a scalar y of F such that x · y = 1_{F} and 0_{F} ≠ 1_{F} and x · (y + z) = x · y + x · z. Then F is a field.

(37) Z_{3} is a Fano field.

Let us note that it makes sense to consider the following constant. Then Z_{3} is a Fano field.

In the sequel D′ is a non-empty set. One can prove the following propositions:

(38) For every function f from D into D′ such that D ∈ U_{1} and D′ ∈ U_{1} holds f ∈ U_{1}.

(39) For every G being a field structure such that the carrier of G ∈ U_{1} holds the addition of G is an element of U_{1} and the reverse-map of G is an element of U_{1} and the zero of G is an element of U_{1} and the multiplication of G is an element of U_{1} and the unity of G is an element of U_{1}.

(40) The carrier of Z_{3} ∈ U_{1} and the addition of Z_{3} is an element of U_{1} and the reverse-map of Z_{3} is an element of U_{1} and the zero of Z_{3} is an element of U_{1} and the multiplication of Z_{3} is an element of U_{1} and the unity of Z_{3} is an element of U_{1}.

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

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

[3] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

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

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

rings and modules - part ii 587

[9] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[10] Michał Muzalewski and Wojciech Skaba. Groups, rings, left- and right-modules. Formalized Mathematics, 2(2):275–278, 1991.

[11] Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97–104, 1991.

[12] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595–600, 1990.

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

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

588

Free Modules

Michał Muzalewski Warsaw University Białystok

Summary. We define free modules and prove that every left module over Skew-Field is free.

MML Identifier: MOD 3.

The papers [20], [5], [3], [2], [4], [19], [16], [14], [15], [1], [18], [6], [7], [8], [12], [11], [9], [10], [13], and [17] provide the terminology and notation for this paper. One can prove the following propositions:

(1) For every ring R and for every scalar a of R such that −a = 0_{R} holds a = 0_{R}.

(2) For every integral domain R holds 0_{R} ≠ −1_{R}. For simplicity we follow the rules: x is arbitrary, R is an associative ring, V

is a left module over R, L, L_{1}, L_{2} are linear combinations of V, a is a scalar of R, v, w are vectors of V, F is a finite sequence of elements of the carrier of the carrier of V, and C is a finite subset of V. We now state several propositions:

(3) If −v = w, then v = −w.

(4) ∑(0_{LC_{V}}) = Θ_{V}.

(5) L_{1} + L_{2} = L_{2} + L_{1}.

(6) If support L ⊆ C, then there exists F such that F is one-to-one and rng F = C and ∑ L = ∑(LF).

(7) ∑(a · L) = a · ∑ L.

(8) ∑(−L) = −∑ L.

(9) ∑(L_{1} − L_{2}) = ∑ L_{1} − ∑ L_{2}.

(10) L + 0_{LC_{V}} = L and 0_{LC_{V}} + L = L. In the sequel W denotes a submodule of V, A, B denote subsets of V, and l

denotes a linear combination of A. Let us consider R, V, A. The functor Lin(A) yielding a submodule of V is defined as follows:

(Def.1) the carrier of the carrier of Lin(A) = {∑ l}.

The following propositions are true:

(11) x ∈ Lin(A) if and only if there exists l such that x = ∑ l.

(12) If x ∈ A, then x ∈ Lin(A).

(13) Lin(∅_{the carrier of the carrier of V}) = 0_{V}.

(14) If Lin(A) = 0_{V}, then A = ∅ or A = {Θ_{V} }.

(15) If 0_{R} ≠ 1_{R} and A = the carrier of the carrier of W, then Lin(A) = W.

(16) If 0_{R} ≠ 1_{R} and A = the carrier of the carrier of V, then Lin(A) = V.

(17) If A ⊆ B, then Lin(A) is a submodule of Lin(B).

(18) If Lin(A) = V and A ⊆ B, then Lin(B) = V.

(19) Lin(A ∪ B) = Lin(A) + Lin(B).

(20) Lin(A ∩ B) is a submodule of Lin(A) ∩ Lin(B).

Let us consider R, V. A subset of V is base if: (Def.2) it is linearly independent and Lin(it) = V.

Let us consider R. A left module over R is free if: (Def.3) there exists a subset B of it such that B is base.

We now state the proposition

(21) 0_{V} is free.

Let us consider R. A left module over R is called a free left R-module if: (Def.4) it is free.

For simplicity we adopt the following convention: R will denote a skew field, a, b will denote scalars of R, V will denote a left module over R, v, v_{1}, v_{2} will denote vectors of V, and A, B will denote subsets of V. The following propositions are true:

(22) 0_{R} ≠ −1_{R}.

(23) {v} is linearly independent if and only if v ≠ Θ_{V}.

(24) v_{1} ≠ v_{2} and {v_{1}, v_{2}} is linearly independent if and only if v_{2} ≠ Θ_{V} and for every a holds v_{1} ≠ a · v_{2}.

(25) v_{1} ≠ v_{2} and {v_{1}, v_{2}} is linearly independent if and only if for all a, b such that a · v_{1} + b · v_{2} = Θ_{V} holds a = 0_{R} and b = 0_{R}.

(26) If A is linearly independent, then there exists B such that A ⊆ B and B is base.

(27) If Lin(A) = V, then there exists B such that B ⊆ A and B is base.

(28) V is free.

Let us consider R, V. A subset of V is called a basis of V if: (Def.5) it is base.

In the sequel I is a basis of V. The following two propositions are true:

(29) If A is linearly independent, then there exists I such that A ⊆ I.

(30) If Lin(A) = V, then there exists I such that I ⊆ A. free modules 591

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] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 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] Michał Muzalewski and Wojciech Skaba. Finite sums of vectors in left module over associative ring. Formalized Mathematics, 2(2):279–282, 1991.

[9] Michał Muzalewski and Wojciech Skaba. Linear combinations in left module over associative ring. Formalized Mathematics, 2(2):295–300, 1991.

[10] Michał Muzalewski and Wojciech Skaba. Linear independence in left module over domain. Formalized Mathematics, 2(2):301–303, 1991.

[11] Michał Muzalewski and Wojciech Skaba. Operations on submodules in left module over associative ring. Formalized Mathematics, 2(2):289–293, 1991.

[12] Michał Muzalewski and Wojciech Skaba. Submodules and cosets of submodules in left module over associative ring. Formalized Mathematics, 2(2):283–287, 1991.

[13] Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97–104, 1991.

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

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

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

[17] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313–319, 1990.

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

592

Oriented Metric-Affine Plane - Part I

Jarosław Zajkowski Warsaw University Białystok

Summary. We present (in Euclidean and Minkowskian geometry) definitions and some properties of the oriented orthogonality relation. Next we consider consistence of Euclidean space and consistence of Minkowskian space.

MML Identifier: ANALORT.

The terminology and notation used in this paper have been introduced in the following articles: [1], [6], [7], [5], [3], [2], and [4]. We adopt the following rules: V will denote a real linear space, u, u_{1}, u_{2}, v, v_{1}, v_{2}, w, w_{1}, x, y will denote vectors of V, and n will denote a real number. Let us consider V, x, y. Let us assume that x, y span the space. Let us consider u. The functor ρ_{x,y}^{M}(u) yielding a vector of V is defined as follows: (Def.1) ρ_{x,y}^{M}(u) = π_{x,y}^{1}(u) · x + (−π_{x,y}^{2}(u)) · y.

The following propositions are true:

(1) If x, y span the space, then ρ_{x,y}^{M}(u + v) = ρ_{x,y}^{M}(u) + ρ_{x,y}^{M}(v).

(2) If x, y span the space, then ρ_{x,y}^{M}(n · u) = n · ρ_{x,y}^{M}(u).

(3) If x, y span the space, then ρ_{x,y}^{M}(0_{V}) = 0_{V}.

(4) If x, y span the space, then ρ_{x,y}^{M}(−u) = −ρ_{x,y}^{M}(u).

(5) If x, y span the space, then ρ_{x,y}^{M}(u − v) = ρ_{x,y}^{M}(u) − ρ_{x,y}^{M}(v).

(6) If x, y span the space and ρ_{x,y}^{M}(u) = ρ_{x,y}^{M}(v), then u = v.

(7) If x, y span the space, then ρ_{x,y}^{M}(ρ_{x,y}^{M}(u)) = u.

(8) If x, y span the space, then there exists v such that u = ρ_{x,y}^{M}(v). Let us consider V, x, y. Let us assume that x, y span the space. Let us

consider u. The functor ρ_{x,y}^{E}(u) yielding a vector of V is defined by: (Def.2) ρ_{x,y}^{E}(u) = π_{x,y}^{2}(u) · x + (−π_{x,y}^{1}(u)) · y.

Next we state several propositions:

(9) If x, y span the space, then ρ_{x,y}^{E}(−v) = −ρ_{x,y}^{E}(v).

(10) If x, y span the space, then ρ_{x,y}^{E}(u + v) = ρ_{x,y}^{E}(u) + ρ_{x,y}^{E}(v).

(11) If x, y span the space, then ρ_{x,y}^{E}(u − v) = ρ_{x,y}^{E}(u) − ρ_{x,y}^{E}(v).

(12) If x, y span the space, then ρ_{x,y}^{E}(n · u) = n · ρ_{x,y}^{E}(u).

(13) If x, y span the space and ρ_{x,y}^{E}(u) = ρ_{x,y}^{E}(v), then u = v.

(14) If x, y span the space, then ρ_{x,y}^{E}(ρ_{x,y}^{E}(u)) = −u.

(15) If x, y span the space, then there exists v such that ρ_{x,y}^{E}(v) = u. We now define two new predicates. Let us consider V, x, y, u, v, u_{1}, v_{1}. Let

us assume that x, y span the space. We say that the segments u, v and u_{1}, v_{1} are E-coherently orthogonal in the basis x, y if and only if:

(Def.3) ρ_{x,y}^{E}(u), ρ_{x,y}^{E}(v) \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1}.

We say that the segments u, v and u_{1}, v_{1} are M-coherently orthogonal in the basis x, y if and only if:

(Def.4) ρ_{x,y}^{M}(u), ρ_{x,y}^{M}(v) \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1}.

One can prove the following propositions:

(16) If x, y span the space, then if u, v \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1}, then ρ_{x,y}^{E}(u), ρ_{x,y}^{E}(v) \begin{matrix} \\ \\ ∥ \end{matrix} ρ_{x,y}^{E}(u_{1}), ρ_{x,y}^{E}(v_{1}).

(17) If x, y span the space, then if u, v \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1}, then ρ_{x,y}^{M}(u), ρ_{x,y}^{M}(v) \begin{matrix} \\ \\ ∥ \end{matrix} ρ_{x,y}^{M}(u_{1}), ρ_{x,y}^{M}(v_{1}).

(18) If x, y span the space, then if the segments u, u_{1} and v, v_{1} are Ecoherently orthogonal in the basis x, y, then the segments v, v_{1} and u_{1}, u are E-coherently orthogonal in the basis x, y.

(19) If x, y span the space, then if the segments u, u_{1} and v, v_{1} are Mcoherently orthogonal in the basis x, y, then the segments v, v_{1} and u, u_{1} are M-coherently orthogonal in the basis x, y.

(20) If x, y span the space, then the segments u, u and v, w are E-coherently orthogonal in the basis x, y.

(21) If x, y span the space, then the segments u, u and v, w are M-coherently orthogonal in the basis x, y.

(22) If x, y span the space, then the segments u, v and w, w are E-coherently orthogonal in the basis x, y.

(23) If x, y span the space, then the segments u, v and w, w are M-coherently orthogonal in the basis x, y.

(24) If x, y span the space, then u, v, ρ_{x,y}^{E}(u) and ρ_{x,y}^{E}(v) are orthogonal w.r.t. x, y.

(25) If x, y span the space, then the segments u, v and ρ_{x,y}^{E}(u), ρ_{x,y}^{E}(v) are E-coherently orthogonal in the basis x, y.

(26) If x, y span the space, then the segments u, v and ρ_{x,y}^{M}(u), ρ_{x,y}^{M}(v) are M-coherently orthogonal in the basis x, y.

(27) If x, y span the space, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1} if and only if there exist u_{2}, v_{2} such that u_{2} ≠ v_{2} and the segments u_{2}, v_{2} and u, v are E-coherently orthogonal in the basis x, y and the segments u_{2}, v_{2} and u_{1}, v_{1} are Ecoherently orthogonal in the basis x, y.

(28) If x, y span the space, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1} if and only if there exist u_{2}, v_{2} such that u_{2} ≠ v_{2} and the segments u_{2}, v_{2} and u, v are M-coherently orthogonal in the basis x, y and the segments u_{2}, v_{2} and u_{1}, v_{1} are Mcoherently orthogonal in the basis x, y.

(29) If x, y span the space, then u, v, u_{1} and v_{1} are orthogonal w.r.t. x, y if and only if the segments u, v and u_{1}, v_{1} are E-coherently orthogonal in the basis x, y or the segments u, v and v_{1}, u_{1} are E-coherently orthogonal in the basis x, y.

(30) If x, y span the space and the segments u, v and u_{1}, v_{1} are E-coherently orthogonal in the basis x, y and the segments u, v and v_{1}, u_{1} are Ecoherently orthogonal in the basis x, y, then u = v or u_{1} = v_{1}.

(31) If x, y span the space and the segments u, v and u_{1}, v_{1} are M-coherently orthogonal in the basis x, y and the segments u, v and v_{1}, u_{1} are Mcoherently orthogonal in the basis x, y, then u = v or u_{1} = v_{1}.

(32) If x, y span the space and the segments u, v and u_{1}, v_{1} are E-coherently orthogonal in the basis x, y and the segments u, v and u_{1}, w are Ecoherently orthogonal in the basis x, y, then the segments u, v and v_{1}, w are E-coherently orthogonal in the basis x, y or the segments u, v and w, v_{1} are E-coherently orthogonal in the basis x, y.

(33) If x, y span the space and the segments u, v and u_{1}, v_{1} are M-coherently orthogonal in the basis x, y and the segments u, v and u_{1}, w are Mcoherently orthogonal in the basis x, y, then the segments u, v and v_{1}, w are M-coherently orthogonal in the basis x, y or the segments u, v and w, v_{1} are M-coherently orthogonal in the basis x, y.

(34) If x, y span the space and the segments u, v and u_{1}, v_{1} are E-coherently orthogonal in the basis x, y, then the segments v, u and v_{1}, u_{1} are Ecoherently orthogonal in the basis x, y.

(35) If x, y span the space and the segments u, v and u_{1}, v_{1} are M-coherently orthogonal in the basis x, y, then the segments v, u and v_{1}, u_{1} are Mcoherently orthogonal in the basis x, y.

(36) If x, y span the space and the segments u, v and u_{1}, v_{1} are E-coherently orthogonal in the basis x, y and the segments u, v and v_{1}, w are Ecoherently orthogonal in the basis x, y, then the segments u, v and u_{1}, w are E-coherently orthogonal in the basis x, y.

(37) If x, y span the space and the segments u, v and u_{1}, v_{1} are M-coherently orthogonal in the basis x, y and the segments u, v and v_{1}, w are Mcoherently orthogonal in the basis x, y, then the segments u, v and u_{1}, w are M-coherently orthogonal in the basis x, y.

(38) If x, y span the space, then for every u, v, w there exists u_{1} such that w ≠ u_{1} and the segments w, u_{1} and u, v are E-coherently orthogonal in the basis x, y.

(39) If x, y span the space, then for every u, v, w there exists u_{1} such that w ≠ u_{1} and the segments w, u_{1} and u, v are M-coherently orthogonal in the basis x, y.

(40) If x, y span the space, then for every u, v, w there exists u_{1} such that w ≠ u_{1} and the segments u, v and w, u_{1} are E-coherently orthogonal in the basis x, y.

(41) If x, y span the space, then for every u, v, w there exists u_{1} such that w ≠ u_{1} and the segments u, v and w, u_{1} are M-coherently orthogonal in the basis x, y.

(42) If x, y span the space and the segments u, u_{1} and v, v_{1} are E-coherently orthogonal in the basis x, y and the segments w, w_{1} and v, v_{1} are Ecoherently orthogonal in the basis x, y and the segments w, w_{1} and u_{2}, v_{2} are E-coherently orthogonal in the basis x, y, then w = w_{1} or v = v_{1} or the segments u, u_{1} and u_{2}, v_{2} are E-coherently orthogonal in the basis x, y.

(43) If x, y span the space and the segments u, u_{1} and v, v_{1} are M-coherently orthogonal in the basis x, y and the segments w, w_{1} and v, v_{1} are Mcoherently orthogonal in the basis x, y and the segments w, w_{1} and u_{2}, v_{2} are M-coherently orthogonal in the basis x, y, then w = w_{1} or v = v_{1} or the segments u, u_{1} and u_{2}, v_{2} are M-coherently orthogonal in the basis x, y.

(44) If x, y span the space and the segments u, u_{1} and v, v_{1} are E-coherently orthogonal in the basis x, y, then the segments v, v_{1} and u, u_{1} are Ecoherently orthogonal in the basis x, y or the segments v, v_{1} and u_{1}, u are E-coherently orthogonal in the basis x, y.

(45) If x, y span the space and the segments u, u_{1} and v, v_{1} are M-coherently orthogonal in the basis x, y, then the segments v, v_{1} and u, u_{1} are Mcoherently orthogonal in the basis x, y or the segments v, v_{1} and u_{1}, u are M-coherently orthogonal in the basis x, y.

(46) If x, y span the space and the segments u, u_{1} and v, v_{1} are E-coherently orthogonal in the basis x, y and the segments v, v_{1} and w, w_{1} are Ecoherently orthogonal in the basis x, y and the segments u_{2}, v_{2} and w, w_{1} are E-coherently orthogonal in the basis x, y, then the segments u, u_{1} and u_{2}, v_{2} are E-coherently orthogonal in the basis x, y or v = v_{1} or w = w_{1}.

Next we state several propositions:

(47) If x, y span the space and the segments u, u_{1} and v, v_{1} are M-coherently orthogonal in the basis x, y and the segments v, v_{1} and w, w_{1} are Mcoherently orthogonal in the basis x, y and the segments u_{2}, v_{2} and w, w_{1} are M-coherently orthogonal in the basis x, y, then the segments u, u_{1} and u_{2}, v_{2} are M-coherently orthogonal in the basis x, y or v = v_{1} or

w = w_{1}.

(48) If x, y span the space and the segments u, u_{1} and v, v_{1} are E-coherently orthogonal in the basis x, y and the segments v, v_{1} and w, w_{1} are Ecoherently orthogonal in the basis x, y and the segments u, u_{1} and u_{2}, v_{2} are E-coherently orthogonal in the basis x, y, then the segments u_{2}, v_{2} and w, w_{1} are E-coherently orthogonal in the basis x, y or v = v_{1} or u = u_{1}.

(49) If x, y span the space and the segments u, u_{1} and v, v_{1} are M-coherently orthogonal in the basis x, y and the segments v, v_{1} and w, w_{1} are Mcoherently orthogonal in the basis x, y and the segments u, u_{1} and u_{2}, v_{2} are M-coherently orthogonal in the basis x, y, then the segments u_{2}, v_{2} and w, w_{1} are M-coherently orthogonal in the basis x, y or v = v_{1} or u = u_{1}.

(50) Suppose x, y span the space. Given v, w, u_{1}, v_{1}, w_{1}. Suppose the segments v, v_{1} and w, u_{1} are not E-coherently orthogonal in the basis x, y and the segments v, v_{1} and u_{1}, w are not E-coherently orthogonal in the basis x, y and the segments u_{1}, w_{1} and u_{1}, w are E-coherently orthogonal in the basis x, y. Then there exists u_{2} such that the segments v, v_{1} and v, u_{2} are E-coherently orthogonal in the basis x, y or the segments v, v_{1} and u_{2}, v are E-coherently orthogonal in the basis x, y but the segments u_{1}, w_{1} and u_{1}, u_{2} are E-coherently orthogonal in the basis x, y or the segments u_{1}, w_{1} and u_{2}, u_{1} are E-coherently orthogonal in the basis x, y.

(51) If x, y span the space, then there exist u, v, w such that the segments u, v and u, w are E-coherently orthogonal in the basis x, y and for all v_{1}, w_{1} such that the segments v_{1}, w_{1} and u, v are E-coherently orthogonal in the basis x, y holds the segments v_{1}, w_{1} and u, w are not E-coherently orthogonal in the basis x, y and the segments v_{1}, w_{1} and w, u are not E-coherently orthogonal in the basis x, y or v_{1} = w_{1}.

(52) Suppose x, y span the space. Given v, w, u_{1}, v_{1}, w_{1}. Suppose h the segments v, v_{1} and w, u_{1} are not M-coherently orthogonal in the basis x, y and h the segments v, v_{1} and u_{1}, w are not M-coherently orthogonal in the basis x, y and the segments u_{1}, w_{1} and u_{1}, w are M-coherently orthogonal in the basis x, y. Then there exists u_{2} such that the segments v, v_{1} and v, u_{2} are M-coherently orthogonal in the basis x, y or the segments v, v_{1} and u_{2}, v are M-coherently orthogonal in the basis x, y but the segments u_{1}, w_{1} and u_{1}, u_{2} are M-coherently orthogonal in the basis x, y or the segments u_{1}, w_{1} and u_{2}, u_{1} are M-coherently orthogonal in the basis x, y.

(53) If x, y span the space, then there exist u, v, w such that the segments u, v and u, w are M-coherently orthogonal in the basis x, y and for all v_{1}, w_{1} such that the segments v_{1}, w_{1} and u, v are M-coherently orthogonal in the basis x, y holds h the segments v_{1}, w_{1} and u, w are not M-coherently orthogonal in the basis x, y and h the segments v_{1}, w_{1} and w, u are not M-coherently orthogonal in the basis x, y or v_{1} = w_{1}.

In the sequel u_{3}, v_{3} will be arbitrary. Let us consider V, x, y. Let us assume that x, y span the space. The Euclidean oriented orthogonality defined over V,x,y yielding a binary relation on [: the vectors of V, the vectors of V :] is defined as follows: (Def.5) ⟨⟨u_{3}, v_{3}⟩⟩ ∈the Euclidean oriented orthogonality defined over V,x,y if and

only if there exist u_{1}, u_{2}, v_{1}, v_{2} such that u_{3} = ⟨⟨u_{1}, u_{2}⟩⟩ and v_{3} = ⟨⟨v_{1}, v_{2}⟩⟩ and the segments u_{1}, u_{2} and v_{1}, v_{2} are E-coherently orthogonal in the basis x, y.

Let us consider V, x, y. Let us assume that x, y span the space. The Minkowskian oriented orthogonality defined over V,x,y yields a binary relation on [: the vectors of V, the vectors of V :] and is defined by: (Def.6) ⟨⟨u_{3}, v_{3}⟩⟩ ∈the Minkowskian oriented orthogonality defined over V,x,y if

and only if there exist u_{1}, u_{2}, v_{1}, v_{2} such that u_{3} = ⟨⟨u_{1}, u_{2}⟩⟩ and v_{3} = ⟨⟨v_{1}, v_{2}⟩⟩ and the segments u_{1}, u_{2} and v_{1}, v_{2} are M-coherently orthogonal in the basis x, y.

Let us consider V, x, y. Let us assume that x, y span the space. The functor CESpace(V, x, y) yields an affine structure and is defined by: (Def.7) CESpace(V, x, y) = ⟨ the vectors of V,the Euclidean oriented orthogo-

nality defined over V,x,y⟩.

Let us consider V, x, y. Let us assume that x, y span the space. The functor CMSpace(V, x, y) yielding an affine structure is defined by: (Def.8) CMSpace(V, x, y) = ⟨ the vectors of V,the Minkowskian oriented or-

thogonality defined over V,x,y⟩.

Let A_{1} be an affine structure, and let p, q, r, s be elements of the points of A_{1}. The predicate p, q⊤^{>}r, s is defined as follows: (Def.9) ⟨⟨⟨⟨p, q⟩⟩, ⟨⟨r, s⟩⟩⟩⟩ ∈ the congruence of A_{1}.

One can prove the following propositions:

(54) If x, y span the space, then for every u_{3} holds u_{3} is an element of the points of CESpace(V, x, y) if and only if u_{3} is a vector of V.

(55) If x, y span the space, then for every u_{3} holds u_{3} is an element of the

points of CMSpace(V, x, y) if and only if u_{3} is a vector of V. In the sequel p, q, r, s are elements of the points of CESpace(V, x, y). Next

we state the proposition

(56) If x, y span the space and u = p and v = q and u_{1} = r and v_{1} = s, then p, q⊤^{>}r, s if and only if the segments u, v and u_{1}, v_{1} are E-coherently orthogonal in the basis x, y.

In the sequel p, q, r, s will be elements of the points of CMSpace(V, x, y). We now state the proposition

(57) If x, y span the space and u = p and v = q and u_{1} = r and v_{1} = s, then p, q⊤^{>}r, s if and only if the segments u, v and u_{1}, v_{1} are M-coherently orthogonal in the basis x, y. oriented metric-affine plane - part i 599

References

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

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical metric affine spaces and planes. Formalized Mathematics, 1(5):891–899, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. A construction of analytical ordered trapezium spaces. Formalized Mathematics, 2(3):315–322, 1991.

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

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

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

600

The Euclidean Space

Agata Darmochwał Warsaw University Białystok

Summary. The general definition of Euclidean Space.

MML Identifier: EUCLID.

The papers [14], [6], [9], [8], [12], [1], [5], [10], [3], [13], [4], [15], [16], [7], [11], and [2] provide the notation and terminology for this paper. In the sequel k, n denote natural numbers and r denotes a real number. Let us consider n. The functor R^{n} yields a non-empty set of finite sequences of and is defined as follows: (Def.1) R^{n} = ^{n}.

In the sequel x will denote a finite sequence of elements of . The function | |_{}from into is defined as follows: (Def.2) for every r holds | |_{}(r) = |r|.

Let us consider x. The functor |x| yields a finite sequence of elements of and is defined as follows: (Def.3) |x| = | |_{}· x.

Let us consider n. The functor ⟨\underbrace{0,…, 0}_{n}⟩ yields a finite sequence of elements

of and is defined by: (Def.4) ⟨\underbrace{0,…, 0}_{n}⟩ = n ⟼ 0 qua a real number .

Let us consider n. Then ⟨\underbrace{0,…, 0}_{n}⟩ is an element of R^{n}.

In the sequel x, x_{1}, x_{2}, y denote elements of R^{n}. One can prove the following proposition

(1) x is an element of ^{n}.

Let us consider n, x. Then −x is an element of R^{n}.

Let us consider n, x, y. Then x + y is an element of R^{n}. Then x − y is an element of R^{n}.

Let us consider n, r, x. Then r · x is an element of R^{n}.

Let us consider n, x. Then |x| is an element of ^{n}.

Let us consider n, x. Then ^{2}x is an element of ^{n}.

Let x be a finite sequence of elements of . The functor |x| yielding a real number is defined by: (Def.5) |x| = \sqrt{∑^{2}|x|}.

Next we state a number of propositions:

(2) len x = n.

(3) dom x = Seg n.

(4) If k ∈ Seg n, then x(k) ∈ .

(5) If for every k such that k ∈ Seg n holds x_{1}(k) = x_{2}(k), then x_{1} = x_{2}.

(6) If k ∈ Seg n and r = x(k), then |x|(k) = |r|.

(7) |⟨\underbrace{0,…, 0}_{n}⟩| = n ⟼ 0 qua a real number .

(8) |−x| = |x|.

(9) |r · x| = |r| · |x|.

(10) |⟨\underbrace{0,…, 0}_{n}⟩| = 0.

(11) If |x| = 0, then x = ⟨\underbrace{0,…, 0}_{n}⟩.

(12) |x| ≥ 0.

(13) |−x| = |x|.

(14) |r · x| = |r| · |x|.

(15) |x_{1} + x_{2}| ≤ |x_{1}| + |x_{2}|.

(16) |x_{1} − x_{2}| ≤ |x_{1}| + |x_{2}|.

(17) |x_{1}| − |x_{2}| ≤ |x_{1} + x_{2}|.

(18) |x_{1}| − |x_{2}| ≤ |x_{1} − x_{2}|.

(19) |x_{1} − x_{2}| = 0 if and only if x_{1} = x_{2}.

(20) If x_{1} ≠ x_{2}, then |x_{1} − x_{2}| > 0.

(21) |x_{1} − x_{2}| = |x_{2} − x_{1}|.

(22) |x_{1} − x_{2}| ≤ |x_{1} − x| + |x − x_{2}|. Let us consider n. The functor ρ^{n} yields a function from [: R^{n}, R^{n}:] into

and is defined by: (Def.6) for all elements x, y of R^{n} holds ρ^{n}(x, y) = |x − y|.

Next we state two propositions:

(23) ^{2}(x − y) = ^{2}(y − x).

(24) ρ^{n} is a metric of R^{n}.

Let us consider n. The functor E^{n} yields a metric space and is defined by: (Def.7) E^{n} = ⟨R^{n}, ρ^{n}⟩.

Let us consider n. The functor E_{T}^{n} yielding a topological space is defined by: (Def.8) E_{T}^{n} = E_{top}^{n}.

We adopt the following rules: p, p_{1}, p_{2}, p_{3} will denote points of E_{T}^{n} and x, x_{1}, x_{2}, y, y_{1}, y_{2} will denote real numbers. One can prove the following four propositions:

(25) The carrier of E_{T}^{n} = R^{n}.

(26) p is a function from Seg n into .

(27) p is a finite sequence of elements of .

(28) For every finite sequence f such that f = p holds len f = n.

Let us consider n. The functor 0_{E_{T}^{n}} yielding a point of E_{T}^{n} is defined by:

(Def.9) 0_{E_{T}^{n}} = ⟨\underbrace{0,…, 0}_{n}⟩.

Let us consider n, p_{1}, p_{2}. The functor p_{1} + p_{2} yields a point of E_{T}^{n} and is defined as follows:

(Def.10) for all elements p′_{1}, p′_{2} of R^{n} such that p′_{1} = p_{1} and p′_{2} = p_{2} holds

p_{1} + p_{2} = p′_{1} + p′_{2}. One can prove the following propositions:

(29) p_{1} + p_{2} = p_{2} + p_{1}.

(30) p_{1} + p_{2} + p_{3} = p_{1} + (p_{2} + p_{3}).

(31) 0_{E_{T}^{n}} + p = p and p + 0_{E_{T}^{n}} = p.

Let us consider x, n, p. The functor x · p yields a point of E_{T}^{n} and is defined as follows:

(Def.11) for every element p′ of R^{n} such that p′ = p holds x · p = x · p′.

Next we state several propositions:

(32) x · 0_{E_{T}^{n}} = 0_{E_{T}^{n}}.

(33) 1 · p = p and 0 · p = 0_{E_{T}^{n}}.

(34) x · y · p = x · (y · p).

(35) If x · p = 0_{E_{T}^{n}}, then x = 0 or p = 0_{E_{T}^{n}}.

(36) x · (p_{1} + p_{2}) = x · p_{1} + x · p_{2}.

(37) (x + y) · p = x · p + y · p.

(38) If x · p_{1} = x · p_{2}, then x = 0 or p_{1} = p_{2}.

Let us consider n, p. The functor −p yields a point of E_{T}^{n} and is defined as follows:

(Def.12) for every element p′ of R^{n} such that p′ = p holds −p = −p′.

We now state several propositions:

(39) −−p = p.

(40) p + −p = 0_{E_{T}^{n}} and −p + p = 0_{E_{T}^{n}}.

(41) If p_{1} + p_{2} = 0_{E_{T}^{n}}, then p_{1} = −p_{2} and p_{2} = −p_{1}.

(42) −(p_{1} + p_{2}) = −p_{1} + −p_{2}.

(43) −p = (−1) · p.

(44) −x · p = (−x) · p and −x · p = x · −p.

Let us consider n, p_{1}, p_{2}. The functor p_{1} − p_{2} yields a point of E_{T}^{n} and is defined by:

(Def.13) for all elements p′_{1}, p′_{2} of R^{n} such that p′_{1} = p_{1} and p′_{2} = p_{2} holds

p_{1} − p_{2} = p′_{1} − p′_{2}. One can prove the following propositions:

(45) p_{1} − p_{2} = p_{1} + −p_{2}.

(46) p − p = 0_{E_{T}^{n}}.

(47) If p_{1} − p_{2} = 0_{E_{T}^{n}}, then p_{1} = p_{2}.

(48) −(p_{1} − p_{2}) = p_{2} − p_{1} and −(p_{1} − p_{2}) = −p_{1} + p_{2}.

(49) p_{1} + (p_{2} − p_{3}) = (p_{1} + p_{2}) − p_{3}.

(50) p_{1} − (p_{2} + p_{3}) = p_{1} − p_{2} − p_{3}.

(51) p_{1} − (p_{2} − p_{3}) = (p_{1} − p_{2}) + p_{3}.

(52) p = (p + p_{1}) − p_{1} and p = (p − p_{1}) + p_{1}.

(53) x · (p_{1} − p_{2}) = x · p_{1} − x · p_{2}.

(54) (x − y) · p = x · p − y · p.

In the sequel p, p_{1}, p_{2} will be points of E_{T}^{2}. Next we state the proposition (55) There exist x, y such that p = ⟨x, y⟩.

We now define two new functors. Let us consider p. The functor p_{1} yields a real number and is defined by:

(Def.14) for every finite sequence f such that p = f holds p_{1} = f(1).

The functor p_{2} yielding a real number is defined by: (Def.15) for every finite sequence f such that p = f holds p_{2} = f(2).

Let us consider x, y. The functor [x, y] yields a point of E_{T}^{2} and is defined as follows:

(Def.16) [x, y] = ⟨x, y⟩.

The following propositions are true:

(56) [x, y]_{1} = x and [x, y]_{2} = y.

(57) p = [p_{1}, p_{2}].

(58) 0_{E_{T}^{2}} = [0, 0].

(59) p_{1} + p_{2} = [p_{1}_{1} + p_{2}_{1}, p_{1}_{2} + p_{2}_{2}].

(60) [x_{1}, y_{1}] + [x_{2}, y_{2}] = [x_{1} + x_{2}, y_{1} + y_{2}].

(61) x · p = [x · p_{1}, x · p_{2}].

(62) x · [x_{1}, y_{1}] = [x · x_{1}, x · y_{1}].

(63) −p = [−p_{1}, −p_{2}].

the euclidean space 605

(64) −[x_{1}, y_{1}] = [−x_{1}, −y_{1}].

(65) p_{1} − p_{2} = [p_{1}_{1} − p_{2}_{1}, p_{1}_{2} − p_{2}_{2}].

(66) [x_{1}, y_{1}] − [x_{2}, y_{2}] = [x_{1} − x_{2}, y_{1} − y_{2}].

References

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

[2] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

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

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

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

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

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

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

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

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

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

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

606

Metric Spaces as Topological Spaces - Fundamental Concepts

Agata Darmochwał^{1} Warsaw University Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. Some notions connected with metric spaces and the relationship between metric spaces and topological spaces. Compactness of topological spaces is transferred for the case of metric spaces [13]. Some basic theorems about translations of topological notions for metric spaces are proved. One-dimensional topological space 1 is introduced, too.

MML Identifier: TOPMETR.

The papers [21], [11], [1], [22], [20], [4], [5], [6], [12], [10], [3], [14], [16], [23], [9], [7], [2], [15], [18], [17], [19], and [8] provide the notation and terminology for this paper. For simplicity we follow a convention: a, b, r will denote real numbers, n will denote a natural number, T will denote a topological space, and F will denote a family of subsets of T. One can prove the following proposition

(1) F is a cover of T if and only if the carrier of T ⊆ ⋃ F.

In the sequel A will be a subspace of T. Next we state three propositions:

(2) For every point p of A holds p is a point of T.

(3) If T is a T_{2} space, then A is a T_{2} space.

(4) For all subspaces A, B of T such that the carrier of A ⊆ the carrier of B holds A is a subspace of B.

In the sequel P, Q denote subsets of T and p denotes a point of T. We now state several propositions:

(5) If P ≠ ∅_{T}, then T P is a subspace of T P ∪ Q qua a subset of T but if Q ≠ ∅_{T}, then T Q is a subspace of T P ∪ Q qua a subset of T.

(6) If P ≠ ∅ and p ∈ P, then for every neighborhood Q of p and for every point p′ of T P and for every subset Q′ of T P such that Q′ = Q ∩ P and p′ = p holds Q′ is a neighborhood of p′.

1The article was written during my work at Shinshu University, 1991.

(7) For all topological spaces A, B, C and for every map f from A into C such that f is continuous and C is a subspace of B for every map h from A into B such that h = f holds h is continuous.

(8) For all topological spaces A, B and for every map f from A into B and for every subspace C of B such that f is continuous and rng f ⊆ the carrier of C for every map h from A into C such that h = f holds h is continuous.

(9) For all topological spaces A, B and for every map f from A into B and for every subset C of B such that f is continuous and rng f ⊆ C and C ≠ ∅ for every map h from A into B C such that h = f holds h is continuous.

(10) For all topological spaces T, S and for every map f from T into S such that f is continuous for every subset P of T and for every map h from T P into S such that P ≠ ∅_{T} and h = f P holds h is continuous.

In the sequel M will denote a metric space and p will denote a point of M. One can prove the following proposition

(11) If r > 0, then p ∈ Ball(p, r).

We now define two new modes. Let us consider M. A subset of M is sets of points of M.

A family of subsets of M is a family of subsets of the carrier of M. Let us consider M. A metric space is said to be a subspace of M if:

(Def.1) the carrier of it ⊆ the carrier of M and for all points x, y of it holds (the distance of it)(x, y) = (the distance of M)(x, y).

In the sequel A will be a subspace of M. One can prove the following propositions:

(12) For every point p of A holds p is a point of M.

(13) For every point x of M and for every point x′ of A such that x = x′

holds Ball(x′, r) = Ball(x, r)∩ the carrier of A. Let M be a metric space, and let A be a non-empty subset of M. The functor

M A yielding a subspace of M is defined as follows:

(Def.2) the carrier of M A = A.

Let us consider a, b. Let us assume that a ≤ b. The functor [a, b]_{M} yields a subspace of the metric space of real numbers and is defined by:

(Def.3) for every non-empty subset P of the metric space of real numbers such that P = [a, b] holds [a, b]_{M} = (the metric space of real numbers) P.

We now state the proposition

(14) If a ≤ b, then the carrier of [a, b]_{M} = [a, b].

In the sequel F, G will be families of subsets of M. We now define two new predicates. Let us consider M, F. We say that F is a family of balls if and only if:

(Def.4) for an arbitrary P such that P ∈ F there exist p, r such that P =

Ball(p, r). We say that F is a cover of M if and only if: (Def.5) the carrier of M ⊆ ⋃ F.

The following propositions are true:

(15) For all points p, q of the metric space of real numbers and for all real numbers x, y such that x = p and y = q holds ρ(p, q) = |x − y|.

(16) The carrier of M = the carrier of M_{top} and the topology of M_{top} = the open set family of M.

(17) For every family F of subsets of M holds F is a family of subsets of M_{top}.

(18) For every family F of subsets of M_{top} holds F is a family of subsets of M.

(19) A_{top} is a subspace of M_{top}.

(20) For every subset P of E_{T}^{n} and for every non-empty subset Q of E^{n} such that P = Q holds (E_{T}^{n}) P = (E^{n} Q)_{top}.

(21) For every subset P of M_{top} such that P = Ball(p, r) holds P is open.

(22) For every subset P of M_{top} holds P is open if and only if for every point

p of M such that p ∈ P there exists r such that r > 0 and Ball(p, r) ⊆ P. Let us consider M. We say that M is compact if and only if:

(Def.6) M_{top} is compact.

We now state the proposition

(23) M is compact if and only if for every F such that F is a family of balls and F is a cover of M there exists G such that G ⊆ F and G is a cover of M and G is finite.

The topological space ^{1} is defined as follows: (Def.7) ^{1} = (the metric space of real numbers)_{top}.

One can prove the following proposition

(24) The carrier of ^{1} = .

Let us consider a, b. Let us assume that a ≤ b. The functor [a, b]_{T} yields a subspace of ^{1} and is defined by: (Def.8) [a, b]_{T} = ([a, b]_{M})_{top}.

We now state three propositions:

(25) If a ≤ b, then the carrier of [a, b]_{T} = [a, b].

(26) If a ≤ b, then for every subset P of ^{1} such that P = [a, b] holds [a, b]_{T} = ^{1} P.

(27) [0, 1]_{T} = .

Let us note that it makes sense to consider the following constant. Then is a subspace of ^{1}.

The following proposition is true 610 agata darmochwał and yatsuka nakamura

(28) For every map f from ^{1} into ^{1} such that there exist real numbers a, b such that for every real number x holds f(x) = a · x + b holds f is continuous.

References

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

[2] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

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

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

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

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

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

[10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

[13] John L. Kelly. Topologie. Volume I,II, von Nostrand, 1955.

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

[15] Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93–96, 1991.

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

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

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

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

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

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

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

[23] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231–237, 1990.

Heine–Borel’s Covering Theorem

Agata Darmochwał^{1} Warsaw University Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. Heine–Borel’s covering theorem, also known as Borel– Lebesgue theorem [3], is proved. Some useful theorems on real inequalities, intervals, sequences and notion of power sequence which are necessary for the theorem are also proved.

MML Identifier: HEINE.

The terminology and notation used in this paper have been introduced in the following articles: [23], [11], [1], [5], [6], [12], [9], [4], [24], [18], [19], [8], [7], [2], [20], [16], [13], [15], [14], [21], [22], [17], and [10]. We follow a convention: a, b, x, y, z denote real numbers and k, n denote natural numbers. We now state several propositions:

(1) For every subspace A of the metric space of real numbers and for all points p, q of A and for all x, y such that x = p and y = q holds ρ(p, q) = |x − y|.

(2) If x ≤ y and y ≤ z, then [x, y] ∪ [y, z] = [x, z].

(3) If x ≥ 0 and a + x ≤ b, then a ≤ b.

(4) If x ≥ 0 and a − x ≥ b, then a ≥ b.

(5) If x > 0, then x^{k} > 0.

In the sequel s_{1} will be a sequence of real numbers. Next we state the proposition

(6) If s_{1} is increasing and rng s_{1} ⊆ , then n ≤ s_{1}(n).

Let us consider s_{1}, k. The functor k^{s_{1}} yielding a sequence of real numbers is defined by: (Def.1) for every n holds k^{s_{1}}(n) = k^{s_{1}(n)}.

We now state several propositions:

1The article was written during my work at Shinshu University, 1991.

612 agata darmochwał and yatsuka nakamura

(7) 2^{n} ≥ n + 1.

(8) 2^{n} > n.

(9) If s_{1} is divergent to +∞, then 2^{s_{1}} is divergent to +∞.

(10) For every topological space T such that the carrier of T is finite holds T is compact.

(11) If a ≤ b, then [a, b]_{T} is compact.

References

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

[2] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

[3] Nicolas Bourbaki. Elements de Mathematique. Volume Topologie Generale, HERMANN, troisieme edition edition, 1960.

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

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

[9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

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

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

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

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

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

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

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

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

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

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

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

[21] Konrad Raczkowski and Andrzej Ne‘dzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.

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

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

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

heine–borel’s covering theorem 613 Received November 21, 1991 614

Some Facts about Union of Two Functions and Continuity of Union of Functions

Yatsuka Nakamura Shinshu University Nagano Agata Darmochwał^{1} Warsaw University Białystok

Summary. Proofs of two theorems connected with the union of any two functions and the proofs of two theorems on the continuity of the union of two continuous functions between topological spaces. The theorem stating that the union of two subsets of R^{2}, which are homeomorphic to unit interval and have only one terminal joined point, is also homeomorphic to unit interval is proved, too.

MML Identifier: TOPMETR2.

The notation and terminology used in this paper have been introduced in the following papers: [14], [9], [15], [13], [2], [3], [4], [11], [7], [5], [12], [10], [1], [6], and [8]. In the sequel x, y, z are real numbers. Next we state the proposition

(1) If x ≤ y and y ≤ z, then [x, y] ∩ [y, z] = {y}. In the sequel f, g will be functions and x_{1}, x_{2} will be arbitrary. Next we

state two propositions:

(2) If f is one-to-one and g is one-to-one and for all x_{1}, x_{2} such that x_{1} ∈ dom g and x_{2} ∈ dom f \ dom g holds g(x_{1}) ≠ f(x_{2}), then f +· g is one-to-one.

(3) If f ^{∘} (dom f ∩ dom g) ⊆ rng g, then rng f ∪ rng g = rng(f +· g). We follow the rules: T, T_{1}, T_{2}, S will be topological spaces and p, p_{1}, p_{2} will

be points of T. Next we state two propositions:

(4) Let T_{1}, T_{2} be subspaces of T. Let f be a map from T_{1} into S. Let g be a map from T_{2} into S. Suppose Ω_{T_{1}} ∪ Ω_{T_{2}} = Ω_{T} and Ω_{T_{1}} ∩ Ω_{T_{2}} = {p} and T_{1} is compact and T_{2} is compact and T is a T_{2} space and f is continuous and g is continuous and f(p) = g(p). Then there exists a map h from T into S such that h = f +· g and h is continuous.

1The article was written during my work at Shinshu University, 1991.

616 yatsuka nakamura and agata darmochwał

(5) Let f be a map from T_{1} into S. Let g be a map from T_{2} into S. Suppose that

(i) T_{1} is a subspace of T, (ii) T_{2} is a subspace of T, (iii) Ω_{T_{1}} ∪ Ω_{T_{2}} = Ω_{T}, (iv) Ω_{T_{1}} ∩ Ω_{T_{2}} = {p_{1}, p_{2}},

(v) T_{1} is compact, (vi) T_{2} is compact, (vii) T is a T_{2} space, (viii) f is continuous, (ix) g is continuous,

(x) f(p_{1}) = g(p_{1}), (xi) f(p_{2}) = g(p_{2}).

Then there exists a map h from T into S such that h = f +· g and h is continuous.

In the sequel P, Q denote subsets of E_{T}^{2}. One can prove the following proposition

(6) Let f be a map from into (E_{T}^{2}) P. Let g be a map from into (E_{T}^{2}) Q. Suppose P ∩ Q = {p} and f is a homeomorphism and f(1) = p and g is a homeomorphism and g(0) = p. Then there exists a map h from into (E_{T}^{2}) P ∪ Q qua a subset of E_{T}^{2} such that h is a homeomorphism and h(0) = f(0) and h(1) = g(1).

References

[1] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

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

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

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

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

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

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

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

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

some facts about union of two functions and … 617

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

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

Index of MML Identifiers

ALI2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .505 ANALORT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 BHSP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .511 BHSP 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517 BHSP 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .523 BORSUK 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 ENS 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .527 EUCLID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 FUNCT 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 GRCAT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 GROUP 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 GROUP 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 HEINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 MATRIX 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 MEASURE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 METRIC 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 MIDSP 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 MODAL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 MOD 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .579 MOD 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .587 NATTRA 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 NAT LAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 PCOMPS 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 QUIN 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .507 SERIES 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 TBSP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .559 TOPMETR2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 TOPMETR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Contents Form. Math. 2 (4)

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 Serieses

By Konrad Raczkowski and Andrzej Ne‘dzusiak . . . . . . . . 449

The Lattice of Natural Numbers and The Sublattice of it. The Set of Prime Numbers

By Marek Chmur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

Commutator and Center of a Group

By Wojciech A. Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Natural Transformations. Discrete Categories

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Matrices. Abelian Group of Matrices

By Katarzyna Jankowska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

Paracompact and Metrizable Spaces

By Leszek Borys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Atlas of Midpoint Algebra

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

Several Properties of the σ-additive Measure

By Józef Białas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Metrics in the Cartesian Product - Part II

By Stanisława Kanas and Adam Lecko . . . . . . . . . . . . . . . . . . . 499

Fix Point Theorem for Compact Spaces

By Alicia de la Cruz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

Quadratic Inequalities

By Jan Popiołek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Continued on inside back cover

Introduction to Banach and Hilbert Spaces - Part I

By Jan Popiołek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Introduction to Banach and Hilbert Spaces - Part II

By Jan Popiołek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Introduction to Banach and Hilbert Spaces - Part III

By Jan Popiołek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

Category Ens

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

A Borsuk Theorem on Homotopy Types

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

Cartesian Product of Functions

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

Introduction to Modal Propositional Logic

By Alicia de la Cruz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

Totally Bounded Metric Spaces

By Alicia de la Cruz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

Categories of Groups

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

Homomorphisms and Isomorphisms of Groups. Quotient Group

By Wojciech A. Trybulec et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Rings and Modules - Part II

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

Free Modules

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

Oriented Metric-Affine Plane - Part I

By Jarosław Zajkowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

The Euclidean Space

By Agata Darmochwał . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

Metric Spaces as Topological Spaces - Fundamental Concepts

By Agata Darmochwał and Yatsuka Nakamura . . . . . . . . 607

Heine–Borel’s Covering Theorem

By Agata Darmochwał and Yatsuka Nakamura . . . . . . . . 611

Some Facts about Union of Two Functions and Continuity of Union of Functions

By Yatsuka Nakamura and Agata Darmochwał . . . . . . . . 615

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

Continued on inside back cover