Preface

As was stated in [5], Mizar articles are being published in this periodical using special technology. They are written in High Level Formalized Language for Mathematics called Mizar. Their logical and mathematical correctness is verified by the PC Mizar system (distributed by Mizar Users Group owing to the grant received from the Philippe le Hodey Foundation). Mizar articles are submitted to the Mizar Users Association. For the both addresses see the second page of the cover. Articles form the Main Mizar Library (MML), which is continually enlarged and updated. The power of the PC Mizar system lies in the automatic processing of cross-references between articles. Main Mizar Library together with PC Mizar system constitute a system for collecting, formalizing and verifying mathematical knowledge. MML forms a basis of a Knowledge Management System for Mathematics supplied with Mizar articles.

Programs have being written which process Mizar articles and translate them automatically into English, and generate texts in TEX. Next LaTEX uses a special format called MizTEX to produce them. While we do try to obtain such English which would not resemble a machine language, for instance by making use of the generator of random numbers to improve the style, obtaining good English is not the main objective of our work. What we do aim at is to obtain a readable text owing to which one can watch the development of MML, and also publish new mathematical results (for the time being in a small number). Another interesting problem is how to develop mathematics formal enough to make it verifiable by the computer. This accounts for certain peculiarities of the articles. For instance, the conceptual apparatus is strongly developed, which can best be illustrated by so-called casting functions, which without changing their respective arguments change their types. Other peculiarities include the occurrence of minor propositions resulting from detailed formalization, and repetitions of definitions in the form of so-called definitional theorems (see proposition 1 in [1], page 265).

As was explained in [5], what is translated is not Mizar articles themselves but so-called Mizar abstracts. An article includes certain elements which are transferred to the data base, such as theorems and definitions. But it also contains fragments which are not transferred there, such as proofs of theorems and lemmas. Mizar abstracts contain only those parts of Mizar articles which are transferred to the data base. This has been due to the fact that the material published at first was intended to facilitate to the Mizar users the use of the data base. In the future, as non-trivial proofs are offered, we plan to publish the translations of full Mizar articles.

It must be explained at this point that both PC Mizar system and Main Mizar Library are 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 may by said of the proof-checker, which enables one to write shorter proofs. In the case of MML its development is linked above all to the ever new enhancers of texts, put in operation, which automatically improve the quality of the articles, usually by shortening them. As a simple example one can mention the program Inacc (inaccessible fragments), which removes from the text of the article those fragments which are not transferred to the data base and are not used in the article (for instance, when the author writes a lemma which she/he supposes to use later one but then changes the conception of the proofs and makes no reference to that lemma). Another improvement is that of Chkrprem (relevant premisses): it removes from the text the unnecessary references in justifications of statements. Since both processes interact that causes considerable changes in the articles. For instance, the strengthening of the proof-checker may result in that certain items of information function are understood by default and reference to a sentence A in the justification of a statement B ceases to be necessary. Hence in the successive verifications of the quality of the paper that reference is removed. We have to do with a special case of such interaction when a theorem has become obvious for the proof-checker, which is to say that when checking whether a given sentence is true the proofchecker needs not refer to other theorems and lemmas. The improver Trivtheo (trivial theorems) pin-points such theorems and eliminates them since as a rule they are not useful. But the numbering of the propositions in a given article is retained. Hence a gap in the numbering and the related information in the footnote: ”The propositon (8) became obvious” (see [2], page 670).

Note also that in the present issue the symbol of multiplication of real functions has been changed from · into ⋄ (see [4] and [3]).

Roman Matuszewski

References

[1] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265–267, 1990.

[2] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

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

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

FORMALIZED MATHEMATICS Vol.1, No.4, September–October 1990 Université Catholique de Louvain

Classical Configurations in Affine Planes ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. The classical sequence of implications which hold between Desargues and Pappus Axioms id proved. Formally Minor and Major Desargues Axiom (as suitable properties - predicates - of an affine plane) together with all its indirect forms are introduced; the same procedure is applied to Pappus Axioms. The so called Trapezium Desargues Axiom is also considered.

MML Identifier: AFF 2.

The articles [1], and [2] provide the notation and terminology for this paper. We follow the rules: AP will denote an affine plane, a, a′, b, b′, c, c′, o will denote elements of the points of AP, and A, C, K, M, N, P will denote subsets of the points of AP. Let us consider AP. We say that AP satisfies PPAP if and only if:

Given M, N, a, b, c, a′, b′, c′. Then if M is a line and N is a line and a ∈ M and b ∈ M and c ∈ M and a′ ∈ N and b′ ∈ N and c′ ∈ N and a, b′ ∥ b, a′ and b, c′ ∥ c, b′, then a, c′ ∥ c, a′.

We now state the proposition

(1) Given AP. Then AP satisfies PPAP if and only if for all M, N, a, b, c, a′, b′, c′ such that M is a line and N is a line and a ∈ M and b ∈ M and c ∈ M and a′ ∈ N and b′ ∈ N and c′ ∈ N and a, b′ ∥ b, a′ and b, c′ ∥ c, b′ holds a, c′ ∥ c, a′.

Let us consider AP. We say that AP satisfies PAP if and only if:

Given M, N, o, a, b, c, a′, b′, c′. Suppose that

(i) M is a line, (ii) N is a line, (iii) M ≠ N, (iv) o ∈ M, (v) o ∈ N, (vi) o ≠ a, (vii) o ≠ a′, (viii) o ≠ b, (ix) o ≠ b′,

1Supported by RPBP.III-24.C2

(x) o ≠ c, (xi) o ≠ c′, (xii) a ∈ M, (xiii) b ∈ M, (xiv) c ∈ M, (xv) a′ ∈ N, (xvi) b′ ∈ N, (xvii) c′ ∈ N, (xviii) a, b′ ∥ b, a′, (xix) b, c′ ∥ c, b′.

Then a, c′ ∥ c, a′.

The following proposition is true

(2) Given AP. Then AP satisfies PAP if and only if for all M, N, o, a, b, c, a′, b′, c′ such that M is a line and N is a line and M ≠ N and o ∈ M and o ∈ N and o ≠ a and o ≠ a′ and o ≠ b and o ≠ b′ and o ≠ c and o ≠ c′ and a ∈ M and b ∈ M and c ∈ M and a′ ∈ N and b′ ∈ N and c′ ∈ N and a, b′ ∥ b, a′ and b, c′ ∥ c, b′ holds a, c′ ∥ c, a′.

Let us consider AP. We say that AP satisfies PAP_{1} if and only if:

Given M, N, o, a, b, c, a′, b′, c′. Suppose that

(i) M is a line, (ii) N is a line, (iii) M ≠ N, (iv) o ∈ M,

(v) o ∈ N, (vi) o ≠ a, (vii) o ≠ a′, (viii) o ≠ b, (ix) o ≠ b′,

(x) o ≠ c, (xi) o ≠ c′, (xii) a ∈ M, (xiii) b ∈ M, (xiv) c ∈ M, (xv) b′ ∈ N, (xvi) c′ ∈ N, (xvii) a, b′ ∥ b, a′, (xviii) b, c′ ∥ c, b′, (xix) a, c′ ∥ c, a′, (xx) b ≠ c.

Then a′ ∈ N. One can prove the following proposition

(3) Given AP. Then AP satisfies PAP_{1} if and only if for all M, N, o, a, b, c, a′, b′, c′ such that M is a line and N is a line and M ≠ N and o ∈ M and o ∈ N and o ≠ a and o ≠ a′ and o ≠ b and o ≠ b′ and o ≠ c and o ≠ c′ and a ∈ M and b ∈ M and c ∈ M and b′ ∈ N and c′ ∈ N and a, b′ ∥ b, a′ and b, c′ ∥ c, b′ and a, c′ ∥ c, a′ and b ≠ c holds a′ ∈ N.

Let us consider AP. We say that AP satisfies DES if and only if: Given A, P, C, o, a, b, c, a′, b′, c′. Suppose that

(i) o ∈ A, (ii) o ∈ P, (iii) o ∈ C, (iv) o ≠ a,

(v) o ≠ b, (vi) o ≠ c, (vii) a ∈ A, (viii) a′ ∈ A, (ix) b ∈ P,

(x) b′ ∈ P, (xi) c ∈ C, (xii) c′ ∈ C, (xiii) A is a line, (xiv) P is a line, (xv) C is a line, (xvi) A ≠ P, (xvii) A ≠ C, (xviii) a, b ∥ a′, b′, (xix) a, c ∥ a′, c′.

Then b, c ∥ b′, c′. We now state the proposition

(4) Given AP. Then AP satisfies DES if and only if for all A, P, C, o, a, b, c, a′, b′, c′ such that o ∈ A and o ∈ P and o ∈ C and o ≠ a and o ≠ b and o ≠ c and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and A is a line and P is a line and C is a line and A ≠ P and A ≠ C and a, b ∥ a′, b′ and a, c ∥ a′, c′ holds b, c ∥ b′, c′.

Let us consider AP. We say that AP satisfies DES_{1} if and only if: Given A, P, C, o, a, b, c, a′, b′, c′. Suppose that

(i) o ∈ A, (ii) o ∈ P, (iii) o ≠ a, (iv) o ≠ b,

(v) o ≠ c, (vi) a ∈ A, (vii) a′ ∈ A, (viii) b ∈ P, (ix) b′ ∈ P,

(x) c ∈ C, (xi) c′ ∈ C, (xii) A is a line, (xiii) P is a line, (xiv) C is a line, (xv) A ≠ P, (xvi) A ≠ C, (xvii) a, b ∥ a′, b′, (xviii) a, c ∥ a′, c′, (xix) b, c ∥ b′, c′, (xx) not L(a, b, c), (xxi) c ≠ c′.

Then o ∈ C.

One can prove the following proposition

(5) Given AP. Then AP satisfies DES_{1} if and only if for all A, P, C, o, a, b, c, a′, b′, c′ such that o ∈ A and o ∈ P and o ≠ a and o ≠ b and o ≠ c and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and A is a line and P is a line and C is a line and A ≠ P and A ≠ C and a, b ∥ a′, b′ and a, c ∥ a′, c′ and b, c ∥ b′, c′ and not L(a, b, c) and c ≠ c′ holds o ∈ C.

Let us consider AP. We say that AP satisfies DES_{2} if and only if:

Given A, P, C, o, a, b, c, a′, b′, c′. Suppose that

(i) o ∈ A, (ii) o ∈ P, (iii) o ∈ C, (iv) o ≠ a,

(v) o ≠ b, (vi) o ≠ c, (vii) a ∈ A, (viii) a′ ∈ A, (ix) b ∈ P,

(x) b′ ∈ P, (xi) c ∈ C, (xii) A is a line, (xiii) P is a line, (xiv) C is a line, (xv) A ≠ P, (xvi) A ≠ C, (xvii) a, b ∥ a′, b′, (xviii) a, c ∥ a′, c′, (xix) b, c ∥ b′, c′.

Then c′ ∈ C.

One can prove the following proposition

(6) Given AP. Then AP satisfies DES_{2} if and only if for all A, P, C, o, a, b, c, a′, b′, c′ such that o ∈ A and o ∈ P and o ∈ C and o ≠ a and o ≠ b and o ≠ c and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and A is a line and P is a line and C is a line and A ≠ P and A ≠ C and a, b ∥ a′, b′ and a, c ∥ a′, c′ and b, c ∥ b′, c′ holds c′ ∈ C.

Let us consider AP. We say that AP satisfies TDES if and only if: Given K, o, a, b, c, a′, b′, c′. Suppose that

(i) K is a line, (ii) o ∈ K, (iii) c ∈ K, (iv) c′ ∈ K,

(v) a ∉ K, (vi) o ≠ c, (vii) a ≠ b, (viii) L(o, a, a′), (ix) L(o, b, b′),

(x) a, b ∥ a′, b′, (xi) a, c ∥ a′, c′, (xii) a, b ∥ K.

Then b, c ∥ b′, c′. We now state the proposition

(7) Given AP. Then AP satisfies TDES if and only if for all K, o, a, b, c, a′, b′, c′ such that K is a line and o ∈ K and c ∈ K and c′ ∈ K and a ∉ K and o ≠ c and a ≠ b and L(o, a, a′) and L(o, b, b′) and a, b ∥ a′, b′ and a, c ∥ a′, c′ and a, b ∥ K holds b, c ∥ b′, c′.

Let us consider AP. We say that AP satisfies TDES_{1} if and only if: Given K, o, a, b, c, a′, b′, c′. Suppose that

(i) K is a line, (ii) o ∈ K, (iii) c ∈ K, (iv) c′ ∈ K,

(v) a ∉ K, (vi) o ≠ c, (vii) a ≠ b, (viii) L(o, a, a′), (ix) a, b ∥ a′, b′,

(x) b, c ∥ b′, c′, (xi) a, c ∥ a′, c′, (xii) a, b ∥ K.

Then L(o, b, b′). One can prove the following proposition (8) Given AP. Then AP satisfies TDES_{1} if and only if for all K, o, a, b,

c, a′, b′, c′ such that K is a line and o ∈ K and c ∈ K and c′ ∈ K and a ∉ K and o ≠ c and a ≠ b and L(o, a, a′) and a, b ∥ a′, b′ and b, c ∥ b′, c′ and a, c ∥ a′, c′ and a, b ∥ K holds L(o, b, b′).

Let us consider AP. We say that AP satisfies TDES_{2} if and only if: Given K, o, a, b, c, a′, b′, c′. Suppose that

(i) K is a line, (ii) o ∈ K, (iii) c ∈ K, (iv) c′ ∈ K,

(v) a ∉ K, (vi) o ≠ c, (vii) a ≠ b, (viii) L(o, a, a′), (ix) L(o, b, b′),

(x) b, c ∥ b′, c′, (xi) a, c ∥ a′, c′, (xii) a, b ∥ K.

Then a, b ∥ a′, b′. The following proposition is true

(9) Given AP. Then AP satisfies TDES_{2} if and only if for all K, o, a, b, c, a′, b′, c′ such that K is a line and o ∈ K and c ∈ K and c′ ∈ K and a ∉ K and o ≠ c and a ≠ b and L(o, a, a′) and L(o, b, b′) and b, c ∥ b′, c′ and a, c ∥ a′, c′ and a, b ∥ K holds a, b ∥ a′, b′.

Let us consider AP. We say that AP satisfies TDES_{3} if and only if: Given K, o, a, b, c, a′, b′, c′. Suppose that

(i) K is a line, (ii) o ∈ K, (iii) c ∈ K, (iv) a ∉ K,

(v) o ≠ c, (vi) a ≠ b, (vii) L(o, a, a′), (viii) L(o, b, b′), (ix) a, b ∥ a′, b′,

(x) a, c ∥ a′, c′, (xi) b, c ∥ b′, c′, (xii) a, b ∥ K.

Then c′ ∈ K. We now state the proposition

(10) Given AP. Then AP satisfies TDES_{3} if and only if for all K, o, a, b, c, a′, b′, c′ such that K is a line and o ∈ K and c ∈ K and a ∉ K and o ≠ c and a ≠ b and L(o, a, a′) and L(o, b, b′) and a, b ∥ a′, b′ and a, c ∥ a′, c′ and b, c ∥ b′, c′ and a, b ∥ K holds c′ ∈ K.

Let us consider AP. We say that AP satisfies des if and only if: Given A, P, C, a, b, c, a′, b′, c′. Suppose that

(i) A ∥ P, (ii) A ∥ C, (iii) a ∈ A, (iv) a′ ∈ A,

(v) b ∈ P, (vi) b′ ∈ P, (vii) c ∈ C, (viii) c′ ∈ C, (ix) A is a line,

(x) P is a line, (xi) C is a line, (xii) A ≠ P, (xiii) A ≠ C, (xiv) a, b ∥ a′, b′, (xv) a, c ∥ a′, c′.

Then b, c ∥ b′, c′. The following proposition is true

(11) Given AP. Then AP satisfies des if and only if for all A, P, C, a, b, c, a′, b′, c′ such that A ∥ P and A ∥ C and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and A is a line and P is a line and C is a line and A ≠ P and A ≠ C and a, b ∥ a′, b′ and a, c ∥ a′, c′ holds b, c ∥ b′, c′.

Let us consider AP. We say that AP satisfies des_{1} if and only if: Given A, P, C, a, b, c, a′, b′, c′. Suppose that

(i) A ∥ P, (ii) a ∈ A, (iii) a′ ∈ A, (iv) b ∈ P,

(v) b′ ∈ P, (vi) c ∈ C, (vii) c′ ∈ C, (viii) A is a line, (ix) P is a line,

(x) C is a line, (xi) A ≠ P, (xii) A ≠ C, (xiii) a, b ∥ a′, b′, (xiv) a, c ∥ a′, c′, (xv) b, c ∥ b′, c′, (xvi) not L(a, b, c), (xvii) c ≠ c′.

Then A ∥ C. The following proposition is true (12) Given AP. Then AP satisfies des_{1} if and only if for all A, P, C, a, b,

c, a′, b′, c′ such that A ∥ P and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and A is a line and P is a line and C is a line and A ≠ P and A ≠ C and a, b ∥ a′, b′ and a, c ∥ a′, c′ and b, c ∥ b′, c′ and not L(a, b, c) and c ≠ c′ holds A ∥ C.

Let us consider AP. We say that AP satisfies pap if and only if:

Given M, N, a, b, c, a′, b′, c′. Suppose M is a line and N is a line and a ∈ M and b ∈ M and c ∈ M and M ∥ N and M ≠ N and a′ ∈ N and b′ ∈ N and c′ ∈ N and a, b′ ∥ b, a′ and b, c′ ∥ c, b′. Then a, c′ ∥ c, a′.

The following proposition is true

(13) Given AP. Then AP satisfies pap if and only if for all M, N, a, b, c, a′, b′, c′ such that M is a line and N is a line and a ∈ M and b ∈ M and c ∈ M and M ∥ N and M ≠ N and a′ ∈ N and b′ ∈ N and c′ ∈ N and a, b′ ∥ b, a′ and b, c′ ∥ c, b′ holds a, c′ ∥ c, a′.

Let us consider AP. We say that AP satisfies pap_{1} if and only if:

Given M, N, a, b, c, a′, b′, c′. Suppose that

(i) M is a line, (ii) N is a line, (iii) a ∈ M, (iv) b ∈ M,

(v) c ∈ M, (vi) M ∥ N, (vii) M ≠ N, (viii) a′ ∈ N, (ix) b′ ∈ N,

(x) a, b′ ∥ b, a′, (xi) b, c′ ∥ c, b′, (xii) a, c′ ∥ c, a′, (xiii) a′ ≠ b′.

Then c′ ∈ N.

We now state a number of propositions:

(14) Given AP. Then AP satisfies pap_{1} if and only if for all M, N, a, b, c, a′, b′, c′ such that M is a line and N is a line and a ∈ M and b ∈ M and c ∈ M and M ∥ N and M ≠ N and a′ ∈ N and b′ ∈ N and a, b′ ∥ b, a′ and b, c′ ∥ c, b′ and a, c′ ∥ c, a′ and a′ ≠ b′ holds c′ ∈ N.

(15) AP satisfies PAP if and only if AP satisfies PAP_{1}.

(16) AP satisfies DES if and only if AP satisfies DES_{1}.

(17) If AP satisfies TDES, then AP satisfies TDES_{1}.

(18) If AP satisfies TDES_{1}, then AP satisfies TDES_{2}.

(19) If AP satisfies TDES_{2}, then AP satisfies TDES_{3}.

(20) If AP satisfies TDES_{3}, then AP satisfies TDES.

(21) AP satisfies des if and only if AP satisfies des_{1}.

(22) AP satisfies pap if and only if AP satisfies pap_{1}.

(23) If AP satisfies PAP, then AP satisfies pap.

(24) AP satisfies PPAP if and only if AP satisfies PAP and AP satisfies pap.

(25) If AP satisfies PAP, then AP satisfies DES.

(26) If AP satisfies DES, then AP satisfies TDES.

(27) If AP satisfies TDES_{1}, then AP satisfies des_{1}.

(28) If AP satisfies TDES, then AP satisfies des.

(29) If AP satisfies des, then AP satisfies pap.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

634

Affine Localizations of Desargues Axiom ^{1}

Eugeniusz Kusak Warsaw University Białystok Henryk Oryszczyszyn Warsaw University Białystok

Krzysztof Prażmowski Warsaw University Białystok

Summary. Several affine localizations of Major Desargues Axiom together with its indirect forms are introduced. Logical relationships between these formulas and between them and the classical Desargues Axiom are demonstrated.

MML Identifier: AFF 3.

The articles [1], [3], and [2] provide the notation and terminology for this paper. We follow a convention: AP denotes an affine plane, a, a′, b, b′, c, c′, o, p, q denote elements of the points of AP, and A, C, P denote subsets of the points of AP. Let us consider AP. We say that AP satisfies DES1 if and only if:

Given A, P, C, o, a, a′, b, b′, c, c′, p, q. Suppose that

(i) A is a line, (ii) P is a line, (iii) C is a line, (iv) P ≠ A,

(v) P ≠ C, (vi) A ≠ C, (vii) o ∈ A, (viii) a ∈ A, (ix) a′ ∈ A,

(x) o ∈ P, (xi) b ∈ P, (xii) b′ ∈ P, (xiii) o ∈ C, (xiv) c ∈ C, (xv) c′ ∈ C, (xvi) o ≠ a, (xvii) o ≠ b, (xviii) o ≠ c, (xix) p ≠ q, (xx) not L(b, a, c), (xxi) not L(b′, a′, c′), (xxii) a ≠ a′, (xxiii) L(b, a, p), (xxiv) L(b′, a′, p), (xxv) L(b, c, q), (xxvi) L(b′, c′, q), (xxvii) a, c ∥ a′, c′.

1Supported by RPBP.III-24.C2

Then a, c ∥ p, q.

We now state the proposition

(1) Given AP. Then AP satisfies DES1 if and only if for all A, P, C, o, a, a′, b, b′, c, c′, p, q such that A is a line and P is a line and C is a line and P ≠ A and P ≠ C and A ≠ C and o ∈ A and a ∈ A and a′ ∈ A and o ∈ P and b ∈ P and b′ ∈ P and o ∈ C and c ∈ C and c′ ∈ C and o ≠ a and o ≠ b and o ≠ c and p ≠ q and not L(b, a, c) and not L(b′, a′, c′) and a ≠ a′ and L(b, a, p) and L(b′, a′, p) and L(b, c, q) and L(b′, c′, q) and a, c ∥ a′, c′ holds a, c ∥ p, q.

Let us consider AP. We say that AP satisfies DES1_{1} if and only if:

Given A, P, C, o, a, a′, b, b′, c, c′, p, q. Suppose that

(i) A is a line, (ii) P is a line, (iii) C is a line, (iv) P ≠ A,

(v) P ≠ C, (vi) A ≠ C, (vii) o ∈ A, (viii) a ∈ A, (ix) a′ ∈ A,

(x) o ∈ P, (xi) b ∈ P, (xii) b′ ∈ P, (xiii) o ∈ C, (xiv) c ∈ C, (xv) c′ ∈ C, (xvi) o ≠ a, (xvii) o ≠ b, (xviii) o ≠ c, (xix) p ≠ q, (xx) c ≠ q, (xxi) not L(b, a, c), (xxii) not L(b′, a′, c′), (xxiii) L(b, a, p), (xxiv) L(b′, a′, p), (xxv) L(b, c, q), (xxvi) L(b′, c′, q), (xxvii) a, c ∥ p, q.

Then a, c ∥ a′, c′.

The following proposition is true

(2) Given AP. Then AP satisfies DES1_{1} if and only if for all A, P, C, o, a, a′, b, b′, c, c′, p, q such that A is a line and P is a line and C is a line and P ≠ A and P ≠ C and A ≠ C and o ∈ A and a ∈ A and a′ ∈ A and o ∈ P and b ∈ P and b′ ∈ P and o ∈ C and c ∈ C and c′ ∈ C and o ≠ a and o ≠ b and o ≠ c and p ≠ q and c ≠ q and not L(b, a, c) and not L(b′, a′, c′) and L(b, a, p) and L(b′, a′, p) and L(b, c, q) and L(b′, c′, q) and a, c ∥ p, q holds a, c ∥ a′, c′.

Let us consider AP. We say that AP satisfies DES1_{2} if and only if:

Given A, P, C, o, a, a′, b, b′, c, c′, p, q. Suppose that

(i) A is a line, (ii) P is a line, (iii) C is a line, (iv) P ≠ A,

(v) P ≠ C, (vi) A ≠ C, (vii) o ∈ A, (viii) a ∈ A, (ix) a′ ∈ A,

(x) o ∈ P, (xi) b ∈ P, (xii) b′ ∈ P, (xiii) c ∈ C, (xiv) c′ ∈ C, (xv) o ≠ a, (xvi) o ≠ b, (xvii) o ≠ c, (xviii) p ≠ q, (xix) not L(b, a, c), (xx) not L(b′, a′, c′), (xxi) c ≠ c′, (xxii) L(b, a, p), (xxiii) L(b′, a′, p), (xxiv) L(b, c, q), (xxv) L(b′, c′, q), (xxvi) a, c ∥ a′, c′, (xxvii) a, c ∥ p, q.

Then o ∈ C. Next we state the proposition

(3) Given AP. Then AP satisfies DES1_{2} if and only if for all A, P, C, o, a, a′, b, b′, c, c′, p, q such that A is a line and P is a line and C is a line and P ≠ A and P ≠ C and A ≠ C and o ∈ A and a ∈ A and a′ ∈ A and o ∈ P and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and o ≠ a and o ≠ b and o ≠ c and p ≠ q and not L(b, a, c) and not L(b′, a′, c′) and c ≠ c′ and L(b, a, p) and L(b′, a′, p) and L(b, c, q) and L(b′, c′, q) and a, c ∥ a′, c′ and a, c ∥ p, q holds o ∈ C.

Let us consider AP. We say that AP satisfies DES1_{3} if and only if: Given A, P, C, o, a, a′, b, b′, c, c′, p, q. Suppose that

(i) A is a line, (ii) P is a line, (iii) C is a line, (iv) P ≠ A,

(v) P ≠ C, (vi) A ≠ C, (vii) o ∈ A, (viii) a ∈ A, (ix) a′ ∈ A,

(x) b ∈ P, (xi) b′ ∈ P, (xii) o ∈ C, (xiii) c ∈ C, (xiv) c′ ∈ C, (xv) o ≠ a, (xvi) o ≠ b, (xvii) o ≠ c, (xviii) p ≠ q, (xix) not L(b, a, c), (xx) not L(b′, a′, c′), (xxi) b ≠ b′, (xxii) a ≠ a′, (xxiii) L(b, a, p), (xxiv) L(b′, a′, p), (xxv) L(b, c, q), (xxvi) L(b′, c′, q), (xxvii) a, c ∥ a′, c′, (xxviii) a, c ∥ p, q.

Then o ∈ P. Next we state the proposition (4) Given AP. Then AP satisfies DES1_{3} if and only if for all A, P, C, o,

a, a′, b, b′, c, c′, p, q such that A is a line and P is a line and C is a line and P ≠ A and P ≠ C and A ≠ C and o ∈ A and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and o ∈ C and c ∈ C and c′ ∈ C and o ≠ a and o ≠ b and o ≠ c and p ≠ q and not L(b, a, c) and not L(b′, a′, c′) and b ≠ b′ and a ≠ a′ and L(b, a, p) and L(b′, a′, p) and L(b, c, q) and L(b′, c′, q) and a, c ∥ a′, c′ and a, c ∥ p, q holds o ∈ P.

Let us consider AP. We say that AP satisfies DES2 if and only if: Given A, P, C, a, a′, b, b′, c, c′, p, q. Suppose that

(i) A is a line, (ii) P is a line, (iii) C is a line, (iv) A ≠ P,

(v) A ≠ C, (vi) P ≠ C, (vii) a ∈ A, (viii) a′ ∈ A, (ix) b ∈ P,

(x) b′ ∈ P, (xi) c ∈ C, (xii) c′ ∈ C, (xiii) A ∥ P, (xiv) A ∥ C, (xv) not L(b, a, c), (xvi) not L(b′, a′, c′), (xvii) p ≠ q, (xviii) a ≠ a′, (xix) L(b, a, p), (xx) L(b′, a′, p), (xxi) L(b, c, q), (xxii) L(b′, c′, q), (xxiii) a, c ∥ a′, c′.

Then a, c ∥ p, q. We now state the proposition

(5) Given AP. Then AP satisfies DES2 if and only if for all A, P, C, a, a′, b, b′, c, c′, p, q such that A is a line and P is a line and C is a line and A ≠ P and A ≠ C and P ≠ C and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and A ∥ P and A ∥ C and not L(b, a, c) and not L(b′, a′, c′) and p ≠ q and a ≠ a′ and L(b, a, p) and L(b′, a′, p) and L(b, c, q) and L(b′, c′, q) and a, c ∥ a′, c′ holds a, c ∥ p, q.

Let us consider AP. We say that AP satisfies DES2_{1} if and only if: Given A, P, C, a, a′, b, b′, c, c′, p, q. Suppose that

(i) A is a line, (ii) P is a line, (iii) C is a line, (iv) A ≠ P,

(v) A ≠ C, (vi) P ≠ C, (vii) a ∈ A, (viii) a′ ∈ A, (ix) b ∈ P,

(x) b′ ∈ P, (xi) c ∈ C, (xii) c′ ∈ C, (xiii) A ∥ P, (xiv) A ∥ C, (xv) not L(b, a, c), (xvi) not L(b′, a′, c′), (xvii) p ≠ q, (xviii) L(b, a, p), (xix) L(b′, a′, p), (xx) L(b, c, q), (xxi) L(b′, c′, q), (xxii) a, c ∥ p, q.

Then a, c ∥ a′, c′.

We now state the proposition

(6) Given AP. Then AP satisfies DES2_{1} if and only if for all A, P, C, a, a′, b, b′, c, c′, p, q such that A is a line and P is a line and C is a line and A ≠ P and A ≠ C and P ≠ C and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and A ∥ P and A ∥ C and not L(b, a, c) and not L(b′, a′, c′) and p ≠ q and L(b, a, p) and L(b′, a′, p) and L(b, c, q) and L(b′, c′, q) and a, c ∥ p, q holds a, c ∥ a′, c′.

Let us consider AP. We say that AP satisfies DES2_{2} if and only if:

Given A, P, C, a, a′, b, b′, c, c′, p, q. Suppose that

(i) A is a line, (ii) P is a line, (iii) C is a line, (iv) A ≠ P,

(v) A ≠ C, (vi) P ≠ C, (vii) a ∈ A, (viii) a′ ∈ A, (ix) b ∈ P,

(x) b′ ∈ P, (xi) c ∈ C, (xii) c′ ∈ C, (xiii) A ∥ C, (xiv) not L(b, a, c), (xv) not L(b′, a′, c′), (xvi) p ≠ q, (xvii) a ≠ a′, (xviii) L(b, a, p), (xix) L(b′, a′, p), (xx) L(b, c, q), (xxi) L(b′, c′, q), (xxii) a, c ∥ a′, c′, (xxiii) a, c ∥ p, q.

Then A ∥ P.

Next we state the proposition

(7) Given AP. Then AP satisfies DES2_{2} if and only if for all A, P, C, a, a′, b, b′, c, c′, p, q such that A is a line and P is a line and C is a line and A ≠ P and A ≠ C and P ≠ C and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and A ∥ C and not L(b, a, c) and not L(b′, a′, c′) and p ≠ q and a ≠ a′ and L(b, a, p) and L(b′, a′, p) and L(b, c, q) and L(b′, c′, q) and a, c ∥ a′, c′ and a, c ∥ p, q holds A ∥ P. Let us consider AP. We say that AP satisfies DES2_{3} if and only if: Given A, P, C, a, a′, b, b′, c, c′, p, q. Suppose that

(i) A is a line, (ii) P is a line, (iii) C is a line, (iv) A ≠ P,

(v) A ≠ C, (vi) P ≠ C, (vii) a ∈ A, (viii) a′ ∈ A, (ix) b ∈ P,

(x) b′ ∈ P, (xi) c ∈ C, (xii) c′ ∈ C, (xiii) A ∥ P, (xiv) not L(b, a, c), (xv) not L(b′, a′, c′), (xvi) p ≠ q, (xvii) c ≠ c′, (xviii) L(b, a, p), (xix) L(b′, a′, p), (xx) L(b, c, q), (xxi) L(b′, c′, q), (xxii) a, c ∥ a′, c′, (xxiii) a, c ∥ p, q.

Then A ∥ C.

We now state a number of propositions:

(8) Given AP. Then AP satisfies DES2_{3} if and only if for all A, P, C, a, a′, b, b′, c, c′, p, q such that A is a line and P is a line and C is a line and A ≠ P and A ≠ C and P ≠ C and a ∈ A and a′ ∈ A and b ∈ P and b′ ∈ P and c ∈ C and c′ ∈ C and A ∥ P and not L(b, a, c) and not L(b′, a′, c′) and p ≠ q and c ≠ c′ and L(b, a, p) and L(b′, a′, p) and L(b, c, q) and L(b′, c′, q) and a, c ∥ a′, c′ and a, c ∥ p, q holds A ∥ C.

(9) If AP satisfies DES1, then AP satisfies DES1_{1}.

(10) If AP satisfies DES1_{1}, then AP satisfies DES1.

(11) If AP satisfies DES, then AP satisfies DES1.

(12) If AP satisfies DES, then AP satisfies DES1_{2}.

(13) If AP satisfies DES1_{2}, then AP satisfies DES1_{3}.

(14) If AP satisfies DES1_{2}, then AP satisfies DES.

(15) If AP satisfies DES2_{1}, then AP satisfies DES2.

(16) AP satisfies DES2_{1} if and only if AP satisfies DES2_{3}.

(17) AP satisfies DES2 if and only if AP satisfies DES2_{2}.

(18) If AP satisfies DES1_{3}, then AP satisfies DES2_{1}.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Classical configurations in affine planes. Formalized Mathematics, 1(4):625–633, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

Binary Operations Applied to Finite Sequences

Czesław Byliński^{1} Warsaw University Białystok

Summary. The article contains some propositions and theorems related to [7] and [4]. The notions introduced in [7] are extended to finite sequences. A number additional propositions related to this notions are proved. There are also proved some properties of distributive operations and unary operations. The notation and propositions for inverses are introduced.

MML Identifier: FINSEQOP.

The notation and terminology used in this paper are introduced in the following articles: [9], [1], [5], [3], [2], [6], [7], [4], and [8]. For simplicity we adopt the following convention: x, y will be arbitrary, C, C′, D, D′, E will be non-empty sets, c will be an element of C, c′ will be an element of C′, d, d_{1}, d_{2}, d_{3}, d_{4}, e will be elements of D, and d′ will be an element of D′. Next we state several propositions:

(1) For every function f holds ⟨ , f⟩ = and ⟨f, ⟩ = .

(2) For every function f holds [: , f :] = and [: f, :] = .

(3) (C ⟼ d)(c) = d.

(4) For all functions F, f holds F^{∘}( , f) = and F^{∘}(f, ) = .

(5) For every function F holds F^{∘}( , x) = .

(6) For every function F holds F^{∘}(x, ) = .

(7) For every set X and for arbitrary x_{1}, x_{2} holds ⟨X ⟼ x_{1}, X ⟼ x_{2}⟩ = X ⟼ ⟨⟨x_{1}, x_{2}⟩⟩.

(8) For every function F and for every set X and for arbitrary x_{1}, x_{2} such that ⟨⟨x_{1}, x_{2}⟩⟩ ∈ dom F holds F^{∘}(X ⟼ x_{1}, X ⟼ x_{2}) = X ⟼ F (⟨⟨x_{1}, x_{2}⟩⟩).

1Supported by RPBP.III-24.C1.

For simplicity we adopt the following rules: i, j will denote natural numbers, F will denote a function from [: D, D′:] into E, p, q will denote finite sequences of elements of D, and p′, q′ will denote finite sequences of elements of D′. Let us consider D, D′, E, F, p, p′. Then F^{∘}(p, p′) is a finite sequence of elements of E.

Let us consider D, D′, E, F, p, d′. Then F^{∘}(p, d′) is a finite sequence of elements of E.

Let us consider D, D′, E, F, d, p′. Then F^{∘}(d, p′) is a finite sequence of elements of E.

Let us consider D, i, d. Then i ⟼ d is an element of D^{i}.

In the sequel f, f′ are functions from C into D and h is a function from D into E. Let us consider D, E, p, h. Then h · p is a finite sequence of elements of E.

Next we state two propositions:

(9) h · (p^{}⟨d⟩) = (h · p)^{}⟨h(d)⟩.

(10) h · (p^{}q) = (h · p)^{}(h · q).

For simplicity we follow a convention: T, T_{1}, T_{2}, T_{3} denote elements of D^{i}, T′ denotes an element of D′^{i}, S denotes an element of D^{j}, and S′ denotes an element of D′^{j}. Next we state a number of propositions:

(11) F^{∘}(T ^{}⟨d⟩, T′ ^{}⟨d′⟩) = F^{∘}(T, T′)^{}⟨F (d, d′)⟩.

(12) F^{∘}(T ^{}S, T′ ^{}S′) = F^{∘}(T, T′)^{}F^{∘}(S, S′).

(13) F^{∘}(d, p′ ^{}⟨d′⟩) = F^{∘}(d, p′)^{}⟨F (d, d′)⟩.

(14) F^{∘}(d, p′ ^{}q′) = F^{∘}(d, p′)^{}F^{∘}(d, q′).

(15) F^{∘}(p^{}⟨d⟩, d′) = F^{∘}(p, d′)^{}⟨F (d, d′)⟩.

(16) F^{∘}(p^{}q, d′) = F^{∘}(p, d′)^{}F^{∘}(q, d′).

(17) For every function h from D into E holds h · (i ⟼ d) = i ⟼ h(d).

(18) F^{∘}(i ⟼ d, i ⟼ d′) = i ⟼ F (d, d′).

(19) F^{∘}(d, i ⟼ d′) = i ⟼ F (d, d′).

(20) F^{∘}(i ⟼ d, d′) = i ⟼ F (d, d′).

(21) F^{∘}(i ⟼ d, T′) = F^{∘}(d, T′).

(22) F^{∘}(T, i ⟼ d) = F^{∘}(T, d).

(23) F^{∘}(d, T′) = F^{∘}(d, id_{D′}) · T′.

(24) F^{∘}(T, d) = F^{∘}(id_{D}, d) · T.

In the sequel F, G are binary operations on D, u is a unary operation on D, and H is a binary operation on E. One can prove the following propositions:

(25) If F is associative, then F^{∘}(d, id_{D}) · F^{∘}(f, f′) = F^{∘}(F^{∘}(d, id_{D}) · f, f′).

(26) If F is associative, then F^{∘}(id_{D}, d) · F^{∘}(f, f′) = F^{∘}(f, F^{∘}(id_{D}, d) · f′).

(27) If F is associative, then F^{∘}(d, id_{D}) · F^{∘}(T_{1}, T_{2}) = F^{∘}(F^{∘}(d, id_{D}) · T_{1}, T_{2}).

(28) If F is associative, then F^{∘}(id_{D}, d)·F^{∘}(T_{1}, T_{2}) = F^{∘}(T_{1}, F^{∘}(id_{D}, d)·T_{2}).

(29) If F is associative, then F^{∘}(F^{∘}(T_{1}, T_{2}), T_{3}) = F^{∘}(T_{1}, F^{∘}(T_{2}, T_{3})).

(30) If F is associative, then F^{∘}(F^{∘}(d_{1}, T), d_{2}) = F^{∘}(d_{1}, F^{∘}(T, d_{2})).

(31) If F is associative, then F^{∘}(F^{∘}(T_{1}, d), T_{2}) = F^{∘}(T_{1}, F^{∘}(d, T_{2})).

(32) If F is associative, then F^{∘}(F (d_{1}, d_{2}), T) = F^{∘}(d_{1}, F^{∘}(d_{2}, T)).

(33) If F is associative, then F^{∘}(T, F (d_{1}, d_{2})) = F^{∘}(F^{∘}(T, d_{1}), d_{2}).

(34) If F is commutative, then F^{∘}(T_{1}, T_{2}) = F^{∘}(T_{2}, T_{1}).

(35) If F is commutative, then F^{∘}(d, T) = F^{∘}(T, d).

(36) If F is distributive w.r.t. G, then F^{∘}(G(d_{1}, d_{2}), f) = G^{∘}(F^{∘}(d_{1}, f), F^{∘}(d_{2}, f)).

(37) If F is distributive w.r.t. G, then F^{∘}(f, G(d_{1}, d_{2})) = G^{∘}(F^{∘}(f, d_{1}), F^{∘}(f, d_{2})).

(38) If for all d_{1}, d_{2} holds h(F (d_{1}, d_{2})) = H(h(d_{1}), h(d_{2})), then h · F^{∘}(f, f′) = H^{∘}(h · f, h · f′).

(39) If for all d_{1}, d_{2} holds h(F (d_{1}, d_{2})) = H(h(d_{1}), h(d_{2})), then h·F^{∘}(d, f) = H^{∘}(h(d), h · f).

(40) If for all d_{1}, d_{2} holds h(F (d_{1}, d_{2})) = H(h(d_{1}), h(d_{2})), then h·F^{∘}(f, d) = H^{∘}(h · f, h(d)).

(41) If u is distributive w.r.t. F, then u · F^{∘}(f, f′) = F^{∘}(u · f, u · f′).

(42) If u is distributive w.r.t. F, then u · F^{∘}(d, f) = F^{∘}(u(d), u · f).

(43) If u is distributive w.r.t. F, then u · F^{∘}(f, d) = F^{∘}(u · f, u(d)).

(44) If F has a unity, then F^{∘}(C ⟼ 1_{F} , f) = f and F^{∘}(f, C ⟼ 1_{F}) = f.

(45) If F has a unity, then F^{∘}(1_{F} , f) = f.

(46) If F has a unity, then F^{∘}(f, 1_{F}) = f.

(47) If F is distributive w.r.t. G, then F^{∘}(G(d_{1}, d_{2}), T) = G^{∘}(F^{∘}(d_{1}, T), F^{∘}(d_{2}, T)).

(48) If F is distributive w.r.t. G, then F^{∘}(T, G(d_{1}, d_{2})) = G^{∘}(F^{∘}(T, d_{1}), F^{∘}(T, d_{2})).

(49) If for all d_{1}, d_{2} holds h(F (d_{1}, d_{2})) = H(h(d_{1}), h(d_{2})), then h · F^{∘}(T_{1}, T_{2}) = H^{∘}(h · T_{1}, h · T_{2}).

(50) If for all d_{1}, d_{2} holds h(F (d_{1}, d_{2})) = H(h(d_{1}), h(d_{2})), then h·F^{∘}(d, T) = H^{∘}(h(d), h · T).

(51) If for all d_{1}, d_{2} holds h(F (d_{1}, d_{2})) = H(h(d_{1}), h(d_{2})), then h·F^{∘}(T, d) = H^{∘}(h · T, h(d)).

(52) If u is distributive w.r.t. F, then u · F^{∘}(T_{1}, T_{2}) = F^{∘}(u · T_{1}, u · T_{2}).

(53) If u is distributive w.r.t. F, then u · F^{∘}(d, T) = F^{∘}(u(d), u · T).

(54) If u is distributive w.r.t. F, then u · F^{∘}(T, d) = F^{∘}(u · T, u(d)).

(55) If G is distributive w.r.t. F and u = G^{∘}(d, id_{D}), then u is distributive w.r.t. F.

(56) If G is distributive w.r.t. F and u = G^{∘}(id_{D}, d), then u is distributive w.r.t. F.

(57) If F has a unity, then F^{∘}(i ⟼ 1_{F} , T) = T and F^{∘}(T, i ⟼ 1_{F}) = T.

(58) If F has a unity, then F^{∘}(1_{F} , T) = T.

(59) If F has a unity, then F^{∘}(T, 1_{F}) = T.

Let us consider D, u, F. We say that u is an inverse operation w.r.t. F if and only if:

for every d holds F (d, u(d)) = 1_{F} and F (u(d), d) = 1_{F}. One can prove the following proposition

(60) u is an inverse operation w.r.t. F if and only if for every d holds F (d,

u(d)) = 1_{F} and F (u(d), d) = 1_{F}. Let us consider D, F. We say that F has an inverse operation if and only if: there exists u such that u is an inverse operation w.r.t. F. Next we state the proposition

(61) F has an inverse operation if and only if there exists u such that u is

an inverse operation w.r.t. F. Let us consider D, F. Let us assume that F has a unity and F is associative

and F has an inverse operation. The inverse operation w.r.t.F yields a unary operation on D and is defined as follows:

the inverse operation w.r.t.F is an inverse operation w.r.t. F. We now state a number of propositions:

(62) If F has a unity and F is associative and F has an inverse operation, then for every u holds u = the inverse operation w.r.t.F if and only if u is an inverse operation w.r.t. F.

(63) If F has a unity and F is associative and F has an inverse operation, then F ((the inverse operation w.r.t.F)(d), d) = 1_{F} and F (d, (the inverse operation w.r.t.F)(d)) = 1_{F}.

(64) If F has a unity and F is associative and F has an inverse operation and F (d_{1}, d_{2}) = 1_{F}, then d_{1} = (the inverse operation w.r.t.F)(d_{2}) and (the inverse operation w.r.t.F)(d_{1}) = d_{2}.

(65) If F has a unity and F is associative and F has an inverse operation, then (the inverse operation w.r.t.F)(1_{F}) = 1_{F}.

(66) If F has a unity and F is associative and F has an inverse operation, then (the inverse operation w.r.t.F)((the inverse operation w.r.t.F)(d)) = d.

(67) If F has a unity and F is associative and F is commutative and F has an inverse operation, then the inverse operation w.r.t.F is distributive w.r.t. F.

(68) If F has a unity and F is associative and F has an inverse operation but F (d, d_{1}) = F (d, d_{2}) or F (d_{1}, d) = F (d_{2}, d), then d_{1} = d_{2}.

(69) If F has a unity and F is associative and F has an inverse operation but F (d_{1}, d_{2}) = d_{2} or F (d_{2}, d_{1}) = d_{2}, then d_{1} = 1_{F}.

(70) If F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F and e = 1_{F}, then for every d holds G(e,

d) = e and G(d, e) = e.

(71) If F has a unity and F is associative and F has an inverse operation and u = the inverse operation w.r.t.F and G is distributive w.r.t. F, then u(G(d_{1}, d_{2})) = G(u(d_{1}), d_{2}) and u(G(d_{1}, d_{2})) = G(d_{1}, u(d_{2})).

(72) If F has a unity and F is associative and F has an inverse operation and u = the inverse operation w.r.t.F and G is distributive w.r.t. F and G has a unity, then G^{∘}(u(1_{G}), id_{D}) = u.

(73) If F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F, then (G^{∘}(d, id_{D}))(1_{F}) = 1_{F}.

(74) If F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F, then (G^{∘}(id_{D}, d))(1_{F}) = 1_{F}.

(75) If F has a unity and F is associative and F has an inverse operation, then F^{∘}(f, (the inverse operation w.r.t.F) · f) = C ⟼ 1_{F} and F^{∘}((the inverse operation w.r.t.F) · f, f) = C ⟼ 1_{F}.

(76) If F is associative and F has an inverse operation and F has a unity and F^{∘}(f, f′) = C ⟼ 1_{F}, then f = (the inverse operation w.r.t.F) · f′ and (the inverse operation w.r.t.F) · f = f′.

(77) If F has a unity and F is associative and F has an inverse operation, then F^{∘}(T, (the inverse operation w.r.t.F) · T) = i ⟼ 1_{F} and F^{∘}((the inverse operation w.r.t.F) · T, T) = i ⟼ 1_{F}.

(78) If F is associative and F has an inverse operation and F has a unity and F^{∘}(T_{1}, T_{2}) = i ⟼ 1_{F}, then T_{1} = (the inverse operation w.r.t.F) · T_{2} and (the inverse operation w.r.t.F) · T_{1} = T_{2}.

(79) If F is associative and F has a unity and e = 1_{F} and F has an inverse operation and G is distributive w.r.t. F, then G^{∘}(e, f) = C ⟼ e.

(80) If F is associative and F has a unity and e = 1_{F} and F has an inverse

operation and G is distributive w.r.t. F, then G^{∘}(e, T) = i ⟼ e. Let F, f, g be functions. The functor F ∘ (f, g) yielding a function is defined

by:

F ∘ (f, g) = F · [: f, g:]. Next we state several propositions:

(81) For all functions F, f, g holds F ∘ (f, g) = F · [: f, g:].

(82) For all functions F, f, g such that ⟨⟨x, y⟩⟩ ∈ dom(F ∘ (f, g)) holds (F ∘ (f, g))(⟨⟨x, y⟩⟩) = F (⟨⟨f(x), g(y)⟩⟩).

(83) For all functions F, f, g such that ⟨⟨x, y⟩⟩ ∈ dom(F ∘ (f, g)) holds (F ∘ (f, g))(x, y) = F (f(x), g(y)).

(84) For every function F from [: D, D′:] into E and for every function f from C into D and for every function g from C′ into D′ holds F ∘ (f, g) is a function from [: C, C′:] into E.

(85) For all functions u, u′ from D into D holds F ∘ (u, u′) is a binary

operation on D. Let us consider D, F, and let f, f′ be functions from D into D. Then

F ∘ (f, f′) is a binary operation on D.

The following propositions are true:

(86) For every function F from [: D, D′:] into E and for every function f from C into D and for every function g from C′ into D′ holds (F ∘ (f, g))(c, c′) = F (f(c), g(c′)).

(87) For every function u from D into D holds (F ∘(id_{D}, u))(d_{1}, d_{2}) = F (d_{1}, u(d_{2})) and (F ∘ (u, id_{D}))(d_{1}, d_{2}) = F (u(d_{1}), d_{2}).

(88) (F ∘ (id_{D}, u))^{∘}(f, f′) = F^{∘}(f, u · f′).

(89) (F ∘ (id_{D}, u))^{∘}(T_{1}, T_{2}) = F^{∘}(T_{1}, u · T_{2}).

(90) Suppose F is associative and F has a unity and F is commutative and F has an inverse operation and u = the inverse operation w.r.t.F. Then u((F ∘ (id_{D}, u))(d_{1}, d_{2})) = (F ∘ (u, id_{D}))(d_{1}, d_{2}) and (F ∘ (id_{D}, u))(d_{1}, d_{2}) = u((F ∘ (u, id_{D}))(d_{1}, d_{2})).

(91) If F is associative and F has a unity and F has an inverse operation, then (F ∘ (id_{D}, the inverse operation w.r.t.F))(d, d) = 1_{F}.

(92) If F is associative and F has a unity and F has an inverse operation, then (F ∘ (id_{D}, the inverse operation w.r.t.F))(d, 1_{F}) = d.

(93) If F is associative and F has a unity and F has an inverse operation and u = the inverse operation w.r.t.F, then (F ∘ (id_{D}, u))(1_{F} , d) = u(d).

(94) If F is commutative and F is associative and F has a unity and F has an inverse operation and G = F ∘(id_{D}, the inverse operation w.r.t.F), then for all d_{1}, d_{2}, d_{3}, d_{4} holds F (G(d_{1}, d_{2}), G(d_{3}, d_{4})) = G(F (d_{1}, d_{3}), F (d_{2}, d_{4})).

References

[1] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 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. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

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

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

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

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

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

650

Semigroup operations on finite subsets

Czesław Byliński^{1} Warsaw University Białystok

Summary. A continuation of [10]. The propositions and theorems proved in [10] are extended to finite sequences. Several additional theorems related to semigroup operations of functions not included in [10] are proved. The special notation for operations on finite sequences is introduced.

MML Identifier: SETWOP 2.

The articles [11], [1], [9], [6], [2], [12], [7], [3], [13], [8], [10], [5], and [4] provide the terminology and notation for this paper. For simplicity we adopt the following rules: x will be arbitrary, C, C′, D, E will denote non-empty sets, c, c_{1}, c_{2}, c_{3} will denote elements of C, B, B_{1}, B_{2} will denote elements of Fin C, A will denote an element of Fin C′, d, d_{1}, d_{2}, d_{3}, d_{4}, e will denote elements of D, F, G will denote binary operations on D, u will denote a unary operation on D, f, f′ will denote functions from C into D, g will denote a function from C′ into D, H will denote a binary operation on E, h will denote a function from D into E, i, j will denote natural numbers, s will denote a function, p, p_{1}, p_{2}, q will denote finite sequences of elements of D, and T_{1}, T_{2} will denote elements of D^{i}. We now state a number of propositions:

(1) Seg i is an element of Fin .

(2) i + j ⟼ x = (i ⟼ x)^{}(j ⟼ x).

(3) If F is commutative and F is associative and c_{1} ≠ c_{2}, then F - ∑_{{c_{1},c_{2}}} f = F (f(c_{1}), f(c_{2})).

(4) If F is commutative and F is associative but B ≠ ∅ or F has a unity and c ∉ B, then F - ∑_{B∪{c}} f = F (F - ∑_{B} f, f(c)).

(5) If F is commutative and F is associative and c_{1} ≠ c_{2} and c_{1} ≠ c_{3} and c_{2} ≠ c_{3}, then F - ∑_{{c_{1},c_{2},c_{3}}} f = F (F (f(c_{1}), f(c_{2})), f(c_{3})).

1Supported by RPBP.III-24.C1.

(6) If F is commutative and F is associative but B_{1} ≠ ∅ and B_{2} ≠ ∅ or F has a unity and B_{1}∩B_{2} = ∅, then F - ∑_{B_{1}∪B_{2}} f = F (F - ∑_{B_{1}} f, F - ∑_{B_{2}} f).

(7) If F is commutative and F is associative but A ≠ ∅ or F has a unity and there exists s such that dom s = A and rng s = B and s is one-to-one and g A = f · s, then F - ∑_{A} g = F - ∑_{B} f.

(8) If H is commutative and H is associative but B ≠ ∅ or H has a unity and f is one-to-one, then H- ∑_{f^{∘}B} h = H- ∑_{B}(h · f).

(9) If F is commutative and F is associative but B ≠ ∅ or F has a unity and f B = f′ B, then F - ∑_{B} f = F - ∑_{B} f′.

(10) If F is commutative and F is associative and F has a unity and e = 1_{F} and f ^{∘} B = {e}, then F - ∑_{B} f = e.

(11) Suppose F is commutative and F is associative and F has a unity and e = 1_{F} and G(e, e) = e and for all d_{1}, d_{2}, d_{3}, d_{4} holds F (G(d_{1}, d_{2}), G(d_{3}, d_{4})) = G(F (d_{1}, d_{3}), F (d_{2}, d_{4})). Then G(F - ∑_{B} f, F - ∑_{B} f′) = F - ∑_{B} G^{∘}(f, f′).

(12) If F is commutative and F is associative and F has a unity, then F (F - ∑_{B} f, F - ∑_{B} f′) = F - ∑_{B} F^{∘}(f, f′).

(13) If F is commutative and F is associative and F has a unity and F has an inverse operation and G = F ∘(id_{D}, the inverse operation w.r.t.F), then G(F - ∑_{B} f, F - ∑_{B} f′) = F - ∑_{B} G^{∘}(f, f′).

(14) If F is commutative and F is associative and F has a unity and e = 1_{F} and G is distributive w.r.t. F and G(d, e) = e, then G(d, F - ∑_{B} f) = F - ∑_{B}(G^{∘}(d, f)).

(15) If F is commutative and F is associative and F has a unity and e = 1_{F} and G is distributive w.r.t. F and G(e, d) = e, then G(F - ∑_{B} f, d) = F - ∑_{B}(G^{∘}(f, d)).

(16) If F is commutative and F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F, then G(d, F - ∑_{B} f) = F - ∑_{B}(G^{∘}(d, f)).

(17) If F is commutative and F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F, then G(F - ∑_{B} f,

d) = F - ∑_{B}(G^{∘}(f, d)).

(18) Suppose F is commutative and F is associative and F has a unity and H is commutative and H is associative and H has a unity and h(1_{F}) = 1_{H} and for all d_{1}, d_{2} holds h(F (d_{1}, d_{2})) = H(h(d_{1}), h(d_{2})). Then h(F - ∑_{B} f) = H- ∑_{B}(h · f).

(19) If F is commutative and F is associative and F has a unity and u(1_{F}) = 1_{F} and u is distributive w.r.t. F, then u(F - ∑_{B} f) = F - ∑_{B}(u · f).

(20) If F is commutative and F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F, then (G^{∘}(d, id_{D}))(F - ∑_{B} f) = F - ∑_{B}(G^{∘}(d, id_{D}) · f).

(21) If F is commutative and F is associative and F has a unity and F has an inverse operation, then (the inverse operation w.r.t.F)(F- ∑_{B} f) = F- ∑_{B}((the inverse operation w.r.t.F) · f).

Let us consider D, p, d. The functor Ω_{d}(p) yields a function from into D and is defined by:

if i ∈ Seg(len p), then (Ω_{d}(p))(i) = p(i) but if i ∉ Seg(len p), then (Ω_{d}(p))(i) = d.

Next we state several propositions:

(22) For every function h from into D holds h = Ω_{d}(p) if and only if for every i holds if i ∈ Seg(len p), then h(i) = p(i) but if i ∉ Seg(len p), then h(i) = d.

(23) Ω_{d}(p) Seg(len p) = p.

(24) Ω_{d}((p^{}q)) Seg(len p) = p.

(25) rng(Ω_{d}(p)) = rng p ∪ {d}.

(26) h · Ω_{d}(p) = Ω_{h(d)}((h · p)).

Let us consider i. Then Seg i is an element of Fin .

Let X be a non-empty subset of , and let x be an element of X. Then {x} is an element of Fin X. Let y be an element of X. Then {x, y} is an element of Fin X. Let z be an element of X. Then {x, y, z} is an element of Fin X.

Let us consider D, F, p. The functor F p yielding an element of D is defined by:

F p = F - ∑_{Seg(len p)} Ω_{1_{F}} (p).

Next we state several propositions:

(27) F p = F - ∑_{Seg(len p)} Ω_{1_{F}} (p).

(28) If F is commutative and F is associative and F has a unity, then F ε_{D} = 1_{F}.

(29) If F is commutative and F is associative, then F ⟨d⟩ = d.

(30) If F is commutative and F is associative but len p ≠ 0 or F has a unity, then F (p^{}⟨d⟩) = F (F p, d).

(31) If F is commutative and F is associative but len p_{1} ≠ 0 and len p_{2} ≠ 0 or F has a unity, then F (p_{1} ^{}p_{2}) = F (F p_{1}, F p_{2}).

(32) If F is commutative and F is associative but len p ≠ 0 or F has a unity,

then F (⟨d⟩^{}p) = F (d, F p). Let us consider D, d_{1}, d_{2}. Then ⟨d_{1}, d_{2}⟩ is a finite sequence of elements of D. One can prove the following proposition

(33) If F is commutative and F is associative, then F ⟨d_{1}, d_{2}⟩ = F (d_{1}, d_{2}). Let us consider D, d_{1}, d_{2}, d_{3}. Then ⟨d_{1}, d_{2}, d_{3}⟩ is a finite sequence of elements

of D.

We now state a number of propositions:

(34) If F is commutative and F is associative, then F ⟨d_{1}, d_{2}, d_{3}⟩ = F (F (d_{1}, d_{2}), d_{3}).

(35) If F is commutative and F is associative and F has a unity and e = 1_{F}, then F (i ⟼ e) = e.

(36) If F is commutative and F is associative, then F (1 ⟼ d) = d.

(37) If F is commutative and F is associative but i ≠ 0 and j ≠ 0 or F has a unity, then F (i + j ⟼ d) = F (F (i ⟼ d), F (j ⟼ d)).

(38) If F is commutative and F is associative but i ≠ 0 and j ≠ 0 or F has a unity, then F (i · j ⟼ d) = F (j ⟼ F (i ⟼ d)).

(39) Suppose F is commutative and F is associative and F has a unity and H is commutative and H is associative and H has a unity and h(1_{F}) = 1_{H} and for all d_{1}, d_{2} holds h(F (d_{1}, d_{2})) = H(h(d_{1}), h(d_{2})). Then h(F p) = H (h · p).

(40) If F is commutative and F is associative and F has a unity and u(1_{F}) = 1_{F} and u is distributive w.r.t. F, then u(F p) = F (u · p).

(41) If F is commutative and F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F, then (G^{∘}(d, id_{D}))(F p) = F (G^{∘}(d, id_{D}) · p).

(42) If F is commutative and F is associative and F has a unity and F has an inverse operation, then (the inverse operation w.r.t.F)(F p) = F ((the inverse operation w.r.t.F) · p).

(43) Suppose that (i) F is commutative, (ii) F is associative, (iii) F has a unity, (iv) e = 1_{F},

(v) G(e, e) = e, (vi) for all d_{1}, d_{2}, d_{3}, d_{4} holds F (G(d_{1}, d_{2}), G(d_{3}, d_{4})) = G(F (d_{1}, d_{3}),

F (d_{2}, d_{4})), (vii) len p = len q.

Then G(F p, F q) = F G^{∘}(p, q).

(44) Suppose F is commutative and F is associative and F has a unity and e = 1_{F} and G(e, e) = e and for all d_{1}, d_{2}, d_{3}, d_{4} holds F (G(d_{1}, d_{2}), G(d_{3}, d_{4})) = G(F (d_{1}, d_{3}), F (d_{2}, d_{4})). Then G(F T_{1}, F T_{2}) = F G^{∘}(T_{1}, T_{2}).

(45) If F is commutative and F is associative and F has a unity and len p = len q, then F (F p, F q) = F F^{∘}(p, q).

(46) If F is commutative and F is associative and F has a unity, then F (F T_{1}, F T_{2}) = F F^{∘}(T_{1}, T_{2}).

(47) If F is commutative and F is associative and F has a unity, then F (i ⟼ F (d_{1}, d_{2})) = F (F (i ⟼ d_{1}), F (i ⟼ d_{2})).

(48) If F is commutative and F is associative and F has a unity and F has an inverse operation and G = F ∘(id_{D}, the inverse operation w.r.t.F), then G(F T_{1}, F T_{2}) = F G^{∘}(T_{1}, T_{2}).

(49) If F is commutative and F is associative and F has a unity and e = 1_{F} and G is distributive w.r.t. F and G(d, e) = e, then G(d, F p) = F (G^{∘}(d, p)).

(50) If F is commutative and F is associative and F has a unity and e = 1_{F} and G is distributive w.r.t. F and G(e, d) = e, then G(F p, d) = F (G^{∘}(p, d)).

(51) If F is commutative and F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F, then G(d, F p) = F (G^{∘}(d, p)).

(52) If F is commutative and F is associative and F has a unity and F has an inverse operation and G is distributive w.r.t. F, then G(F p,

d) = F (G^{∘}(p, d)).

References

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

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

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

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

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

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

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

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

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

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

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

[12] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

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

The Collinearity Structure

Wojciech Skaba^{1} Nicolaus Copernicus University Toruń

Summary. The text includes basic axioms and theorems concerning the collinearity structure based on Wanda Szmielew [1], pp. 18-20. Collinearity is defined as a relation on Cartesian product [:S, S, S:] of set S. The basic text is preceeded with a few auxiliary theorems (e.g: ternary relation). Then come the two basic axioms of the collinearity structure: A1.1.1 and A1.1.2 and a few theorems. Another axiom: Aks dim, which states that there exist at least 3 non-collinear points, excludes the trivial structures ( i.e. pairs ⟨⟨S, [:S, S, S:]⟩⟩ ). Following it the notion of a line is included and several additional theorems are appended.

MML Identifier: COLLSP.

The articles [3], and [2] provide the notation and terminology for this paper. In the sequel R, X will denote sets. Let us consider X. A set is said to be a ternary relation on X if:

it ⊆ [: X, X, X:].

Next we state two propositions:

(1) R is a ternary relation on X if and only if R ⊆ [: X, X, X:].

(2) X = ∅ or there exists arbitrary a such that {a} = X or there exist arbitrary a, b such that a ≠ b and a ∈ X and b ∈ X.

We consider collinearity structures which are systems

⟨points, a collinearity relation⟩, where the points constitute a non-empty set and the collinearity relation is a ternary relation on the points. In the sequel CS is a collinearity structure. Let us consider CS. A point of CS is an element of the points of CS.

In the sequel a, b, c denote points of CS. Let us consider CS, a, b, c. We say that a, b and c are collinear if and only if:

⟨⟨a, b, c⟩⟩ ∈ the collinearity relation of CS.

The following proposition is true (5)^{2} a, b and c are collinear if and only if ⟨⟨a, b, c⟩⟩ ∈ the collinearity relation

1Supported by RPBP.III-24.B5.

of CS. A collinearity structure is said to be a collinearity space if: Let a, b, c, p, q, r be points of it . Then

(i) if a = b or a = c or b = c, then ⟨⟨a, b, c⟩⟩ ∈ the collinearity relation of it, (ii) if a ≠ b and ⟨⟨a, b, p⟩⟩ ∈ the collinearity relation of it and ⟨⟨a, b, q⟩⟩ ∈ the collinearity relation of it and ⟨⟨a, b, r⟩⟩ ∈ the collinearity relation of it, then ⟨⟨p, q, r⟩⟩ ∈ the collinearity relation of it.

Next we state the proposition

(6) CS is a collinearity space if and only if for all points a, b, c, p, q, r of CS holds if a = b or a = c or b = c, then ⟨⟨a, b, c⟩⟩ ∈ the collinearity relation of CS but if a ≠ b and ⟨⟨a, b, p⟩⟩ ∈ the collinearity relation of CS and ⟨⟨a, b, q⟩⟩ ∈ the collinearity relation of CS and ⟨⟨a, b, r⟩⟩ ∈ the collinearity relation of CS, then ⟨⟨p, q, r⟩⟩ ∈ the collinearity relation of CS.

We adopt the following rules: CLSP is a collinearity space and a, b, c, d, p, q, r are points of CLSP. We now state several propositions:

(7) If a = b or a = c or b = c, then a, b and c are collinear.

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

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

(10) a, b and a are collinear.

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

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

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

(14) If p ≠ q and a, b and p are collinear and a, b and q are collinear and p,

q and r are collinear, then a, b and r are collinear. Let us consider CLSP, a, b. The functor Line(a, b) yields a set and is defined

as follows:

Line(a, b) = {p : a, b and p are collinear }. One can prove the following propositions:

(15) Line(a, b) = {p : a, b and p are collinear }.

(16) a ∈ Line(a, b) and b ∈ Line(a, b).

(17) a, b and r are collinear if and only if r ∈ Line(a, b).

A collinearity space is said to be a proper collinearity space if: there exist points a, b, c of it such that a, b and c are not collinear. The following proposition is true

(18) CLSP is a proper collinearity space if and only if there exist a, b, c such that a, b and c are not collinear.

2The propositions (3)–(4) became obvious.

We follow a convention: CLSP will be a proper collinearity space and a, b, p, q, r will be points of CLSP. We now state the proposition

(19) For all p, q such that p ≠ q there exists r such that p, q and r are not

collinear. Let us consider CLSP. A set is called a line of CLSP if: there exist a, b such that a ≠ b and it = Line(a, b). The following propositions are true:

(20) For every set X holds X is a line of CLSP if and only if there exist a, b such that a ≠ b and X = Line(a, b).

(21) If a ≠ b, then Line(a, b) is a line of CLSP.

In the sequel P, Q are lines of CLSP. The following propositions are true:

(22) If a = b, then Line(a, b) = the points of CLSP.

(23) For every P there exist a, b such that a ≠ b and a ∈ P and b ∈ P.

(24) If a ≠ b, then there exists P such that a ∈ P and b ∈ P.

(25) If p ∈ P and q ∈ P and r ∈ P, then p, q and r are collinear.

(26) If P ⊆ Q, then P = Q.

(27) If p ≠ q and p ∈ P and q ∈ P, then Line(p, q) ⊆ P.

(28) If p ≠ q and p ∈ P and q ∈ P, then Line(p, q) = P.

(29) If p ≠ q and p ∈ P and q ∈ P and p ∈ Q and q ∈ Q, then P = Q.

(30) P = Q or P ∩ Q = ∅ or there exists p such that P ∩ Q = {p}.

(31) If a ≠ b, then Line(a, b) ≠ the points of CLSP.

References

[1] Wanda Szmielew. From Affine to Euclidean Geometry. Volume 27, PWN – D.Reidel Publ. Co., Warszawa – Dordrecht, 1983.

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

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

660

The Sum and Product of Finite Sequences of Real Numbers

Czesław Byliński^{1} Warsaw University Białystok

Summary. Some operations on the set of n-tuples of real numbers are introduced. Addition, difference of such n-tuples, complement of a n-tuple and multiplication of these by real numbers are defined. In these definitions more general properties of binary operations applied to finite sequences from [3] are used. Then the fact that certain properties are satisfied by those operations is demonstrated directly from [3]. Moreover some properties can be recognized as being those of real vector space. Multiplication of n-tuples of real numbers and square power of n-tuple of real numbers using for notation of some properties of finite sums and products of real numbers are defined, followed by definitions of the finite sum and product of n-tuples of real numbers using notions and properties introduced in [7]. A number of propositions and theorems on sum and product of finite sequences of real numbers are proved. As a additional properties there are proved some properties of real numbers and set representations of binary operations on real numbers.

MML Identifier: RVSUM 1.

The papers [8], [12], [5], [6], [1], [2], [13], [10], [9], [11], [4], [3], and [7] provide the terminology and notation for this paper. For simplicity we follow the rules: i, j, k are natural numbers, r, r′, r_{1}, r_{2}, r_{3} are real numbers, x is an element of , F, F_{1}, F_{2} are finite sequences of elements of , and R, R_{1}, R_{2}, R_{3} are elements of ^{i}. Next we state the proposition

(1) −(r_{1} + r_{2}) = (−r_{1}) + (−r_{2}). Let us consider x. The functor @x yields a real number and is defined by: @x = x. The following propositions are true:

(2) @x = x.

1Supported by RPBP.III-24.C1.

(3) 0 is a unity w.r.t. +_{}.

(4) 1_{+_{}}= 0.

(5) +_{}has a unity.

(6) +_{}is commutative.

(7) +_{}is associative.

The binary operation −_{}on is defined as follows:

−_{}= +_{}∘ (id_{}, −_{}).

We now state two propositions:

(8) −_{}= +_{}∘ (id_{}, −_{}).

(9) −_{}(r_{1}, r_{2}) = r_{1} − r_{2}.

The unary operation sqr_{}on is defined as follows:

for every r holds sqr_{}(r) = r^{2}.

The following propositions are true:

(10) For every unary operation u on holds u = sqr_{}if and only if for every r holds u(r) = r^{2}.

(11) ·_{}is commutative.

(12) ·_{}is associative.

(13) 1 is a unity w.r.t. ·_{}.

(14) 1_{·_{}}= 1.

(15) ·_{}has a unity.

(16) ·_{}is distributive w.r.t. +_{}.

(17) sqr_{}is distributive w.r.t. ·_{}.

Let us consider x. The functor ·x_{}yielding a unary operation on is defined by:

·x_{}= ·_{}^{∘}(x, id_{}).

Next we state several propositions:

(18) ·x_{}= ·_{}^{∘}(x, id_{}).

(19) ·r_{}(x) = r · x.

(20) ·r_{}is distributive w.r.t. +_{}.

(21) −_{}is an inverse operation w.r.t. +_{}.

(22) +_{}has an inverse operation.

(23) The inverse operation w.r.t.+_{}= −_{}.

(24) −_{}is distributive w.r.t. +_{}.

Let us consider F_{1}, F_{2}. The functor F_{1} + F_{2} yields a finite sequence of elements of and is defined by:

F_{1} + F_{2} = +_{}^{∘}(F_{1}, F_{2}).

We now state two propositions:

(25) F_{1} + F_{2} = +_{}^{∘}(F_{1}, F_{2}).

(26) If i ∈ Seg(len(F_{1} + F_{2})) and r_{1} = F_{1}(i) and r_{2} = F_{2}(i), then (F_{1} + F_{2})(i) = r_{1} + r_{2}.

Let us consider i, R_{1}, R_{2}. Then R_{1} + R_{2} is an element of ^{i}.

We now state several propositions:

(27) If j ∈ Seg i and r_{1} = R_{1}(j) and r_{2} = R_{2}(j), then (R_{1} +R_{2})(j) = r_{1} +r_{2}.

(28) ε_{}+ F = ε_{}and F + ε_{}= ε_{}.

(29) ⟨r_{1}⟩ + ⟨r_{2}⟩ = ⟨r_{1} + r_{2}⟩.

(30) (i ⟼ r_{1}) + (i ⟼ r_{2}) = i ⟼ r_{1} + r_{2}.

(31) R_{1} + R_{2} = R_{2} + R_{1}.

(32) R_{1} + (R_{2} + R_{3}) = (R_{1} + R_{2}) + R_{3}.

(33) R + (i ⟼ (0 qua a real number)) = R and

R = (i ⟼ (0 qua a real number)) + R. Let us consider F. The functor −F yields a finite sequence of elements of

and is defined as follows:

−F = −_{}· F.

We now state two propositions:

(34) −F = −_{}· F.

(35) If i ∈ Seg(len(−F)) and r = F (i), then (−F)(i) = −r.

Let us consider i, R. Then −R is an element of ^{i}.

The following propositions are true:

(36) If j ∈ Seg i and r = R(j), then (−R)(j) = −r.

(37) −ε_{}= ε_{}.

(38) −⟨r⟩ = ⟨−r⟩.

(39) −(i ⟼ r) = i ⟼ −r.

(40) R + (−R) = i ⟼ 0 and (−R) + R = i ⟼ 0.

(41) If R_{1} + R_{2} = i ⟼ 0, then R_{1} = −R_{2} and R_{2} = −R_{1}.

(42) −(−R) = R.

(43) If −R_{1} = −R_{2}, then R_{1} = R_{2}.

(44) If R_{1} + R = R_{2} + R or R_{1} + R = R + R_{2}, then R_{1} = R_{2}.

(45) −(R_{1} + R_{2}) = (−R_{1}) + (−R_{2}).

Let us consider F_{1}, F_{2}. The functor F_{1} − F_{2} yielding a finite sequence of elements of is defined as follows:

F_{1} − F_{2} = −_{}^{∘}(F_{1}, F_{2}).

The following two propositions are true:

(46) F_{1} − F_{2} = −_{}^{∘}(F_{1}, F_{2}).

(47) If i ∈ Seg(len(F_{1} − F_{2})) and r_{1} = F_{1}(i) and r_{2} = F_{2}(i), then (F_{1} −

F_{2})(i) = r_{1} − r_{2}. Let us consider i, R_{1}, R_{2}. Then R_{1} − R_{2} is an element of ^{i}. One can prove the following propositions:

(48) If j ∈ Seg i and r_{1} = R_{1}(j) and r_{2} = R_{2}(j), then (R_{1} −R_{2})(j) = r_{1} −r_{2}.

(49) ε_{}− F = ε_{}and F − ε_{}= ε_{}.

(50) ⟨r_{1}⟩ − ⟨r_{2}⟩ = ⟨r_{1} − r_{2}⟩.

(51) (i ⟼ r_{1}) − (i ⟼ r_{2}) = i ⟼ r_{1} − r_{2}.

(52) R_{1} − R_{2} = R_{1} + (−R_{2}).

(53) R − (i ⟼ (0 qua a real number)) = R.

(54) (i ⟼ (0 qua a real number)) − R = −R.

(55) R_{1} − (−R_{2}) = R_{1} + R_{2}.

(56) −(R_{1} − R_{2}) = R_{2} − R_{1}.

(57) −(R_{1} − R_{2}) = (−R_{1}) + R_{2}.

(58) R − R = i ⟼ 0.

(59) If R_{1} − R_{2} = i ⟼ 0, then R_{1} = R_{2}.

(60) (R_{1} − R_{2}) − R_{3} = R_{1} − (R_{2} + R_{3}).

(61) R_{1} + (R_{2} − R_{3}) = (R_{1} + R_{2}) − R_{3}.

(62) R_{1} − (R_{2} − R_{3}) = (R_{1} − R_{2}) + R_{3}.

(63) R_{1} = (R_{1} + R) − R.

(64) R_{1} = (R_{1} − R) + R.

Let us consider r, F. The functor r · F yields a finite sequence of elements of and is defined by:

r · F = ·r_{}·F.

We now state two propositions:

(65) r · F = ·r_{}·F.

(66) If i ∈ Seg(len(r · F)) and r′ = F (i), then (r · F)(i) = r · r′. Let us consider i, r, R. Then r · R is an element of ^{i}. Next we state a number of propositions:

(67) If j ∈ Seg i and r′ = R(j), then (r · R)(j) = r · r′.

(68) r · ε_{}= ε_{}.

(69) r · ⟨r_{1}⟩ = ⟨r · r_{1}⟩.

(70) r_{1} · (i ⟼ r_{2}) = i ⟼ r_{1} · r_{2}.

(71) (r_{1} · r_{2}) · R = r_{1} · (r_{2} · R).

(72) (r_{1} + r_{2}) · R = r_{1} · R + r_{2} · R.

(73) r · (R_{1} + R_{2}) = r · R_{1} + r · R_{2}.

(74) 1 · R = R.

(75) 0 · R = i ⟼ 0.

(76) (−1) · R = −R.

Let us consider F. The functor ^{2}F yielding a finite sequence of elements of is defined as follows:

2F = sqr_{}·F.

Next we state two propositions:

(77) ^{2}F = sqr_{}·F.

(78) If i ∈ Seg(len(^{2}F)) and r = F (i), then ^{2}F (i) = r^{2}.

Let us consider i, R. Then ^{2}R is an element of ^{i}.

Next we state several propositions:

(79) If j ∈ Seg i and r = R(j), then ^{2}R(j) = r^{2}.

(80) ^{2}ε_{}= ε_{}.

(81) ^{2}⟨r⟩ = ⟨r^{2}⟩.

(82) ^{2}(i ⟼ r) = i ⟼ r^{2}.

(83) ^{2}(−R) = ^{2}R.

(84) ^{2}(r · R) = r^{2} ·^{2}R.

Let us consider F_{1}, F_{2}. The functor F_{1}•F_{2} yields a finite sequence of elements of and is defined by:

F_{1} • F_{2} = ·_{}^{∘}(F_{1}, F_{2}).

One can prove the following two propositions:

(85) F_{1} • F_{2} = ·_{}^{∘}(F_{1}, F_{2}).

(86) If i ∈ Seg(len(F_{1} • F_{2})) and r_{1} = F_{1}(i) and r_{2} = F_{2}(i), then F_{1} • F_{2}(i) =

r_{1} · r_{2}. Let us consider i, R_{1}, R_{2}. Then R_{1} • R_{2} is an element of ^{i}. The following propositions are true:

(87) If j ∈ Seg i and r_{1} = R_{1}(j) and r_{2} = R_{2}(j), then R_{1} • R_{2}(j) = r_{1} · r_{2}.

(88) ε_{}• F = ε_{}and F • ε_{}= ε_{}.

(89) ⟨r_{1}⟩ • ⟨r_{2}⟩ = ⟨r_{1} · r_{2}⟩.

(90) R_{1} • R_{2} = R_{2} • R_{1}.

(91) R_{1} • (R_{2} • R_{3}) = (R_{1} • R_{2}) • R_{3}.

(92) (i ⟼ r) • R = r · R and R • (i ⟼ r) = r · R.

(93) (i ⟼ r_{1}) • (i ⟼ r_{2}) = i ⟼ r_{1} · r_{2}.

(94) r · R_{1} • R_{2} = (r · R_{1}) • R_{2}.

(95) r · R_{1} • R_{2} = (r · R_{1}) • R_{2} and r · R_{1} • R_{2} = R_{1} • (r · R_{2}).

(96) r · R = (i ⟼ r) • R.

(97) ^{2}R = R • R.

(98) ^{2}(R_{1} + R_{2}) = (^{2}R_{1} + 2 · R_{1} • R_{2}) +^{2}R_{2}.

(99) ^{2}(R_{1} − R_{2}) = (^{2}R_{1} − 2 · R_{1} • R_{2}) +^{2}R_{2}.

(100) ^{2}(R_{1} • R_{2}) = (^{2}R_{1}) • (^{2}R_{2}). Let F be a finite sequence of elements of . The functor ∑ F yields a real

number and is defined by:

∑ F = +_{}F.

One can prove the following propositions:

(101) ∑ F = +_{}F.

(102) ∑ ε_{}= 0.

(103) ∑⟨r⟩ = r.

(104) ∑(F ^{}⟨r⟩) = ∑ F + r.

(105) ∑(F_{1} ^{}F_{2}) = ∑ F_{1} + ∑ F_{2}.

(106) ∑(⟨r⟩^{}F) = r + ∑ F.

(107) ∑⟨r_{1}, r_{2}⟩ = r_{1} + r_{2}.

(108) ∑⟨r_{1}, r_{2}, r_{3}⟩ = (r_{1} + r_{2}) + r_{3}.

(109) For every element R of ^{0} holds ∑ R = 0.

(110) ∑(i ⟼ r) = i · r.

(111) ∑(i ⟼ (0 qua a real number)) = 0.

(112) If for all j, r_{1}, r_{2} such that j ∈ Seg i and r_{1} = R_{1}(j) and r_{2} = R_{2}(j) holds r_{1} ≤ r_{2}, then ∑ R_{1} ≤ ∑ R_{2}.

(113) Suppose for all j, r_{1}, r_{2} such that j ∈ Seg i and r_{1} = R_{1}(j) and r_{2} = R_{2}(j) holds r_{1} ≤ r_{2} and there exist j, r_{1}, r_{2} such that j ∈ Seg i and r_{1} = R_{1}(j) and r_{2} = R_{2}(j) and r_{1} < r_{2}. Then ∑ R_{1} < ∑ R_{2}.

(114) If for all i, r such that i ∈ Seg(len F) and r = F (i) holds 0 ≤ r, then 0 ≤ ∑ F.

(115) If for all i, r such that i ∈ Seg(len F) and r = F (i) holds 0 ≤ r and there exist i, r such that i ∈ Seg(len F) and r = F (i) and 0 < r, then 0 < ∑ F.

(116) 0 ≤ ∑(^{2}F).

(117) ∑(r · F) = r · ∑ F.

(118) ∑(−F) = − ∑ F.

(119) ∑(R_{1} + R_{2}) = ∑ R_{1} + ∑ R_{2}.

(120) ∑(R_{1} − R_{2}) = ∑ R_{1} − ∑ R_{2}.

(121) If ∑(^{2}R) = 0, then R = i ⟼ 0.

(122) (∑(R_{1} • R_{2}))^{2} ≤ ∑(^{2}R_{1}) · ∑(^{2}R_{2}).

Let F be a finite sequence of elements of . The functor ∏ F yields a real number and is defined as follows:

∏ F = ·_{}F.

Next we state a number of propositions:

(123) ∏ F = ·_{}F.

(124) ∏ ε_{}= 1.

(125) ∏⟨r⟩ = r.

(126) ∏(F ^{}⟨r⟩) = ∏ F · r.

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

(128) ∏(⟨r⟩^{}F) = r · ∏ F.

(129) ∏⟨r_{1}, r_{2}⟩ = r_{1} · r_{2}.

(130) ∏⟨r_{1}, r_{2}, r_{3}⟩ = (r_{1} · r_{2}) · r_{3}.

(131) For every element R of ^{0} holds ∏ R = 1.

(132) ∏(i ⟼ (1 qua a real number)) = 1.

(133) There exists k such that k ∈ Seg(len F) and F (k) = 0 if and only if ∏ F = 0.

(134) ∏(i + j ⟼ r) = ∏(i ⟼ r) · ∏(j ⟼ r).

(135) ∏(i · j ⟼ r) = ∏(j ⟼ ∏(i ⟼ r)).

(136) ∏(i ⟼ r_{1} · r_{2}) = ∏(i ⟼ r_{1}) · ∏(i ⟼ r_{2}).

(137) ∏(R_{1} • R_{2}) = ∏ R_{1} · ∏ R_{2}.

(138) ∏(r · R) = ∏(i ⟼ r) · ∏ R.

(139) ∏(^{2}R) = (∏ R)^{2}.

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] Czesław Byliński. Semigroup operations on finite subsets. Formalized Mathematics, 1(4):651–656, 1990.

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

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

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

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

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

668 Czesław Byliński

[13] Edmund Woronowicz. Relations and their basic properties. Formalized

Mathematics, 1(1):73–83, 1990. Received May 11, 1990

A Classical First Order Language

Czesław Byliński^{1} Warsaw University Białystok

Summary. The aim is to construct a language for the classical predicate calculus. The language is defined as a subset of the language constructed in [8]. Well-formed formulas of this language are defined and some usual connectives and quantifiers of [8,1] are accordingly. We prove inductive and definitional schemes for formulas of our language. Substitution for individual variables in formulas of the introduced language is defined. This definition is borrowed from [7]. For such purpose some auxiliary notation and propositions are introduced.

MML Identifier: CQC LANG.

The articles [10], [3], [4], [5], [9], [2], [8], [1], and [6] provide the notation and terminology for this paper. In the sequel i, j, k will denote natural numbers. One can prove the following proposition

(1) For every non-empty set D and for every finite sequence l of elements of D such that k ∈ Seg(len l) holds l(k) ∈ D.

Let x, y, a, b be arbitrary. The functor (x = y → a, b) is defined as follows: (x = y → a, b) = a if x = y, (x = y → a, b) = b, otherwise. One can prove the following propositions:

(2) For arbitrary x, y, a, b such that x = y holds (x = y → a, b) = a.

(3) For arbitrary x, y, a, b such that x ≠ y holds (x = y → a, b) = b. Let x, y be arbitrary. The functor x⟼ . y yields a function and is defined as

follows:

x⟼ . y = {x} ⟼ y. One can prove the following three propositions:

(4) For arbitrary x, y holds x⟼ . y = {x} ⟼ y.

(5) For arbitrary x, y holds dom(x⟼ . y) = {x} and rng(x⟼ . y) = {y}.

(6) For arbitrary x, y holds (x⟼ . y)(x) = y.

1Supported by RPBP.III-24.C1

For simplicity we follow the rules: x, y are bound variables, a is a free variable, p, q are elements of WFF, l, ll are finite sequences of elements of Var, and P is a predicate symbol. Let F be a function from WFF into WFF, and let us consider p. Then F (p) is an element of WFF.

One can prove the following proposition

(7) For an arbitrary x holds x ∈ Var if and only if x ∈ FixedVar or x ∈ FreeVar or x ∈ BoundVar.

A substitution is a partial function from FreeVar to Var. In the sequel f will be a substitution. Let us consider l, f. The functor l[f]

yielding a finite sequence of elements of Var is defined as follows:

len(l[f]) = len l and for every k such that 1 ≤ k and k ≤ len l holds if l(k) ∈ dom f, then (l[f])(k) = f(l(k)) but if l(k) ∉ dom f, then (l[f])(k) = l(k).

The following proposition is true (9)^{2} ll = l[f] if and only if the following conditions are satisfied:

(i) len ll = len l, (ii) for every k such that 1 ≤ k and k ≤ len l holds if l(k) ∈ dom f, then

ll(k) = f(l(k)) but if l(k) ∉ dom f, then ll(k) = l(k). Let us consider k, and let l be a list of variables of the length k, and let us

consider f. Then l[f] is a list of variables of the length k.

One can prove the following proposition

(10) a⟼ . x is a substitution.

Let us consider a, x. Then a⟼ . x is a substitution. We now state the proposition

(11) If f = a⟼ . x and ll = l[f] and 1 ≤ k and k ≤ len l, then if l(k) = a,

then ll(k) = x but if l(k) ≠ a, then ll(k) = l(k). Let A be a non-empty subset of WFF. We see that it makes sense to consider

the following mode for restricted scopes of arguments. Then all the objests of the mode element of A are a formula.

The non-empty subset WFF_{CQC} of WFF is defined as follows: WFF_{CQC} = {s : Fixed s = ∅ ∧ Free s = ∅}. The following propositions are true:

(12) WFF_{CQC} = {s : Fixed s = ∅ ∧ Free s = ∅}.

(13) p is an element of WFF_{CQC} if and only if Fixed p = ∅ and Free p = ∅. Let us consider k. A list of variables of the length k is said to be a variables

list of k if:

{it(i) : 1 ≤ i ∧ i ≤ len it} ⊆ BoundVar. One can prove the following propositions:

(14) For every list of variables l of the length k holds l is a variables list of k if and only if {l(i) : 1 ≤ i ∧ i ≤ len l} ⊆ BoundVar.

2The proposition (8) became obvious.

(15) Let l be a list of variables of the length k. Then l is a variables list of k if and only if {l(i) : 1 ≤ i ∧ i ≤ len l ∧ l(i) ∈ FreeVar} = ∅ and {l(j) : 1 ≤ j ∧ j ≤ len l ∧ l(j) ∈ FixedVar} = ∅.

In the sequel r, s denote elements of WFF_{CQC}. Next we state two propositions:

(16) VERUM is an element of WFF_{CQC}.

(17) Let P be a k-ary predicate symbol. Let l be a list of variables of the length k. Then P [l] is an element of WFF_{CQC} if and only if {l(i) : 1 ≤ i ∧ i ≤ len l ∧ l(i) ∈ FreeVar} = ∅ and {l(j) : 1 ≤ j ∧ j ≤ len l ∧ l(j) ∈ FixedVar} = ∅.

Let us consider k, and let P be a k-ary predicate symbol, and let l be a variables list of k. Then P [l] is an element of WFF_{CQC}.

We now state two propositions:

(18) ¬p is an element of WFF_{CQC} if and only if p is an element of WFF_{CQC}.

(19) p∧q is an element of WFF_{CQC} if and only if p is an element of WFF_{CQC}

and q is an element of WFF_{CQC}. Let us note that it makes sense to consider the following constant. Then

VERUM is an element of WFF_{CQC}. Let us consider r. Then ¬r is an element of WFF_{CQC}. Let us consider s. Then r ∧ s is an element of WFF_{CQC}.

One can prove the following three propositions:

(20) r ⇒ s is an element of WFF_{CQC}.

(21) r ∨ s is an element of WFF_{CQC}.

(22) r ⇔ s is an element of WFF_{CQC}.

Let us consider r, s. Then r ⇒ s is an element of WFF_{CQC}. Then r ∨ s is an element of WFF_{CQC}. Then r ⇔ s is an element of WFF_{CQC}.

We now state the proposition

(23) ∀_{x}p is an element of WFF_{CQC} if and only if p is an element of WFF_{CQC}. Let us consider x, r. Then ∀_{x}r is an element of WFF_{CQC}. We now state the proposition

(24) ∃_{x}r is an element of WFF_{CQC}.

Let us consider x, r. Then ∃_{x}r is an element of WFF_{CQC}. Let D be a non-empty set, and let F be a function from WFF_{CQC} into D,

and let us consider r. Then F (r) is an element of D.

In this article we present several logical schemes. The scheme CQC Ind concerns a unary predicate P, and states that:

for every r holds P[r] provided the parameter satisfies the following condition:

• for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P holds P[VERUM] and P[P [l]] but if P[r], then P[¬r] but if P[r] and P[s], then P[r ∧ s] but if P[r], then P[∀_{x}r].

The scheme CQC Func Ex concerns a non-empty set A, an element B of A, a ternary functor F yielding an element of A, a unary functor G yielding an element of A, a binary 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_{CQC} into A such that for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = B and F (P [l]) = F(k, P, l) and F (¬r) = G(r′) and F (r ∧ s) = H(r′, s′) and F (∀_{x}r) = I(x, r′) for all values of the parameters.

The scheme CQC Func Uniq concerns a non-empty set A, a function B from WFF_{CQC} into A, a function C from WFF_{CQC} into A, an element D of A, a ternary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, and a binary functor I yielding an element of A and states that:

B = C provided the parameters satisfy the following conditions:

• Given r, s, x, k. Let l be a variables list of k. Let P be a k-ary predicate symbol. Let r′, s′ be elements of A. Suppose r′ = B(r) and s′ = B(s). Then B(VERUM) = D and B(P [l]) = F(k, P, l) and B(¬r) = G(r′) and B(r ∧ s) = H(r′, s′) and B(∀_{x}r) = I(x, r′),

• Given r, s, x, k. Let l be a variables list of k. Let P be a k-ary predicate symbol. Let r′, s′ be elements of A. Suppose r′ = C(r) and s′ = C(s). Then C(VERUM) = D and C(P [l]) = F(k, P, l) and C(¬r) = G(r′) and C(r ∧ s) = H(r′, s′) and C(∀_{x}r) = I(x, r′).

The scheme CQC Def correctn concerns a non-empty set A, an element B of WFF_{CQC}, an element C of A, a ternary functor F yielding an element of A, a unary functor G yielding an element of A, a binary functor H yielding an element of A, and a binary functor I yielding an element of A and states that:

(i) there exists an element d of A and there exists a function F from WFF_{CQC} into A such that d = F (B) and for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = C and F (P [l]) = F(k, P, l) and F (¬r) = G(r′) and F (r ∧ s) = H(r′, s′) and F (∀_{x}r) = I(x, r′), (ii) for all elements d_{1}, d_{2} of A such that there exists a function F from WFF_{CQC} into A such that d_{1} = F (B) and for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = C and F (P [l]) = F(k, P, l) and F (¬r) = G(r′) and F (r ∧ s) = H(r′, s′) and F (∀_{x}r) = I(x, r′) and there exists a function F from WFF_{CQC} into A such that d_{2} = F (B) and for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = C and F (P [l]) = F(k, P, l) and F (¬r) = G(r′) and F (r ∧ s) = H(r′, s′) and F (∀_{x}r) = I(x, r′) holds d_{1} = d_{2} for all values of the parameters.

The scheme CQC Def VERUM concerns a non-empty set A, a unary functor F yielding an element of A, an element B of A, a ternary functor G yielding an element of A, a unary functor H yielding an element of A, a binary functor I yielding an element of A, and a binary functor J yielding an element of A and states that:

F(VERUM) = B provided the parameters satisfy the following condition:

• Let p be an element of WFF_{CQC}. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF_{CQC} into A such that d = F (p) and for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = B and F (P [l]) = G(k, P, l) and F (¬r) = H(r′) and F (r ∧ s) = I(r′, s′) and F (∀_{x}r) = J (x, r′).

The scheme CQC Def atomic concerns a non-empty set A, an element B of A, a unary functor F yielding an element of A, a ternary functor G yielding an element of A, a natural number C, a C-ary predicate symbol D, a variables list E of C, a unary functor H yielding an element of A, a binary functor I yielding an element of A, and a binary functor J yielding an element of A and states that:

F(D[E]) = G(C, D, E) provided the following requirement is met:

• Let p be an element of WFF_{CQC}. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF_{CQC} into A such that d = F (p) and for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = B and F (P [l]) = G(k, P, l) and F (¬r) = H(r′) and F (r ∧ s) = I(r′, s′) and F (∀_{x}r) = J (x, r′).

The scheme CQC Def negative deals with a non-empty set A, a unary functor F yielding an element of A, an element B of A, a ternary functor G yielding an element of A, a unary functor H yielding an element of A, an element C of WFF_{CQC}, a binary functor I yielding an element of A, and a binary functor J yielding an element of A and states that:

F(¬C) = H(F(C)) provided the parameters satisfy the following condition:

• Let p be an element of WFF_{CQC}. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF_{CQC} into A such that d = F (p) and for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = B and F (P [l]) = G(k, P, l) and F (¬r) = H(r′) and F (r ∧ s) = I(r′, s′) and F (∀_{x}r) = J (x, r′).

The scheme QC Def conjuncti concerns a non-empty set A, a unary functor F yielding an element of A, an element B of A, a ternary functor G yielding an element of A, a unary functor H yielding an element of A, a binary functor I yielding an element of A, an element C of WFF_{CQC}, an element D of WFF_{CQC}, and a binary functor J yielding an element of A and states that:

F(C ∧ D) = I(F(C), F(D)) provided the following condition is satisfied:

• Let p be an element of WFF_{CQC}. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF_{CQC} into A such that d = F (p) and for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = B and F (P [l]) = G(k, P, l) and F (¬r) = H(r′) and F (r ∧ s) = I(r′, s′) and F (∀_{x}r) = J (x, r′).

The scheme QC Def universal concerns a non-empty set A, a unary functor F yielding an element of A, an element B of A, a ternary functor G yielding an element of A, a unary functor H yielding an element of A, a binary functor I yielding an element of A, a binary functor J yielding an element of A, a bound variable C, and an element D of WFF_{CQC} and states that:

F(∀_{C}D) = J (C, F(D)) provided the following condition is satisfied:

• Let p be an element of WFF_{CQC}. Let d be an element of A. Then d = F(p) if and only if there exists a function F from WFF_{CQC} into A such that d = F (p) and for all r, s, x, k and for every variables list l of k and for every k-ary predicate symbol P and for all elements r′, s′ of A such that r′ = F (r) and s′ = F (s) holds F (VERUM) = B and F (P [l]) = G(k, P, l) and F (¬r) = H(r′) and F (r ∧ s) = I(r′, s′) and F (∀_{x}r) = J (x, r′).

We now state the proposition

(25) If Arity(P) = len l, then P [l] = ⟨P ⟩^{}l.

Let us consider x, y, p, q. Then (x = y → p, q) is an element of WFF. Let us consider p, x. The functor p(x) yields an element of WFF and is

defined as follows:

there exists a function F from WFF into WFF such that p(x) = F (p) and for every q holds F (VERUM) = VERUM but if q is atomic, then F (q) = PredSym(q)[Args(q)[a_{0}⟼ . x]] but if q is negative, then F (q) = ¬(F (Arg(q))) but if q is conjunctive, then F (q) = (F (LeftArg(q))) ∧ (F (RightArg(q))) but if q is universal, then F (q) = (Bound(q) = x → q, ∀_{Bound(q)}(F (Scope(q)))).

We now state a number of propositions: (27)^{3} Let r be an element of WFF. Then r = p(x) if and only if there

exists a function F from WFF into WFF such that r = F (p) and for every q holds F (VERUM) = VERUM but if q is atomic, then F (q) = PredSym(q)[Args(q)[a_{0}⟼ . x]] but if q is negative, then F (q) = ¬(F (Arg(q))) but if q is conjunctive, then F (q) = (F (LeftArg(q))) ∧ (F (RightArg(q))) but if q is universal, then

3The proposition (26) became obvious.

F (q) = (Bound(q) = x → q, ∀_{Bound(q)}(F (Scope(q)))).

(28) VERUM(x) = VERUM.

(29) If p is atomic, then p(x) = PredSym(p)[Args(p)[a_{0}⟼ . x]].

(30) For every k-ary predicate symbol P and for every list of variables l of the length k holds (P [l])(x) = P [l[a_{0}⟼ . x]].

(31) If p is negative, then p(x) = ¬(Arg(p)(x)).

(32) ¬p(x) = ¬(p(x)).

(33) If p is conjunctive, then p(x) = (LeftArg(p)(x)) ∧ (RightArg(p)(x)).

(34) p ∧ q(x) = (p(x)) ∧ (q(x)).

(35) If p is universal and Bound(p) = x, then p(x) = p.

(36) If p is universal and Bound(p) ≠ x, then p(x) = ∀_{Bound(p)}(Scope(p)(x)).

(37) ∀_{x}p(x) = ∀_{x}p.

(38) If x ≠ y, then ∀_{x}p(y) = ∀_{x}(p(y)).

(39) If Free p = ∅, then p(x) = p.

(40) r(x) = r.

(41) Fixed(p(x)) = Fixed p.

References

[1] Grzegorz Bancerek. Connectives and subformulae of the first order language. Formalized Mathematics, 1(3):451–458, 1990.

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

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

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

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

[6] Czesław Byliński and Grzegorz Bancerek. Variables in formulae of the first order language. Formalized Mathematics, 1(3):459–469, 1990.

[7] Witold A. Pogorzelski and Tadeusz Prucnal. The substitution rule for predicate letters in the first-order predicate calculus. Reports on Mathematical Logic, (5), 1975.

[8] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

676 Czesław Byliński

[10] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathemat-

ics, 1(1):9–11, 1990. Received May 11, 1990

Classical and Non–classical Pasch Configurations in Ordered Affine Planes ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Małgorzata Prażmowska Warsaw University Białystok

Summary. Several configuration axioms, which are commonly called in the literature ”Pasch Axioms” are introduced; three of them were investigated by Szmielew and concern invariantability of the betweenness relation under parallel projections, and two other were introduced by Tarski. It is demonstrated that they all are consequences of the trapezium axiom, adopted to characterize ordered affine spaces.

MML Identifier: PASCH.

The papers [1] and [2] provide the notation and terminology for this paper. We adopt the following rules: OAS will be an ordered affine space and a, a′, b, b′, c, c′, d, d_{1}, d_{2}, p, p′, x, y, z, t, u will be elements of the points of OAS. Let us consider OAS. We say that OAS satisfies inner invariancy of betweenness relation under parallel projections if and only if:

for all a, b, c, d, p such that not L(p, b, c) and B(b, p, a) and L(p, c, d) and b, c |||| d, a holds B(c, p, d).

We now state the proposition

(1) For every OAS holds OAS satisfies inner invariancy of betweenness relation under parallel projections if and only if for all a, b, c, d, p such that not L(p, b, c) and B(b, p, a) and L(p, c, d) and b, c |||| d, a holds B(c, p, d).

1Supported by RPBP.III-24.C2.

Let us consider OAS. We say that OAS satisfies outer invariancy of betweenness relation under parallel projections if and only if:

for all a, b, c, d, p such that B(p, b, c) and L(p, a, d) and a, b |||| c, d and not L(p, a, b) holds B(p, a, d).

We now state the proposition

(2) For every OAS holds OAS satisfies outer invariancy of betweenness relation under parallel projections if and only if for all a, b, c, d, p such that B(p, b, c) and L(p, a, d) and a, b |||| c, d and not L(p, a, b) holds B(p, a, d).

Let us consider OAS. We say that OAS satisfies general invariancy of betweenness relation under parallel projections if and only if:

for all a, b, c, a′, b′, c′ such that not L(a, b, a′) and a, a′ |||| b, b′ and a, a′ |||| c, c′ and B(a, b, c) and L(a′, b′, c′) holds B(a′, b′, c′).

We now state the proposition

(3) For every OAS holds OAS satisfies general invariancy of betweenness relation under parallel projections if and only if for all a, b, c, a′, b′, c′ such that not L(a, b, a′) and a, a′ |||| b, b′ and a, a′ |||| c, c′ and B(a, b, c) and L(a′, b′, c′) holds B(a′, b′, c′).

Let us consider OAS. We say that OAS satisfies outer form of Pasch’ Axiom if and only if:

for all a, b, c, d, x, y such that B(a, b, d) and B(b, x, c) and not L(a, b, c) there exists y such that B(a, y, c) and B(y, x, d).

The following proposition is true

(4) For every OAS holds OAS satisfies outer form of Pasch’ Axiom if and only if for all a, b, c, d, x, y such that B(a, b, d) and B(b, x, c) and not L(a, b, c) there exists y such that B(a, y, c) and B(y, x, d).

Let us consider OAS. We say that OAS satisfies inner form of Pasch’ Axiom if and only if:

for all a, b, c, d, x, y such that B(a, b, d) and B(a, x, c) and not L(a, b, c) there exists y such that B(b, y, c) and B(x, y, d).

The following proposition is true

(5) For every OAS holds OAS satisfies inner form of Pasch’ Axiom if and only if for all a, b, c, d, x, y such that B(a, b, d) and B(a, x, c) and not L(a, b, c) there exists y such that B(b, y, c) and B(x, y, d).

Let us consider OAS. We say that OAS satisfies Fano Axiom if and only if: for all a, b, c, d such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and a, c \begin{matrix} \\ \\ ∥ \end{matrix} b, d and not L(a, b, c) there

exists x such that B(a, x, d) and B(b, x, c).

We now state a number of propositions:

(6) For every OAS holds OAS satisfies Fano Axiom if and only if for all a, b, c, d such that a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and a, c \begin{matrix} \\ \\ ∥ \end{matrix} b, d and not L(a, b, c) there exists x such that B(a, x, d) and B(b, x, c).

(7) If b, p \begin{matrix} \\ \\ ∥ \end{matrix} p, c and p ≠ c and b ≠ p, then there exists d such that a, p \begin{matrix} \\ \\ ∥ \end{matrix} p, d and a, b |||| c, d and c ≠ d and p ≠ d.

(8) If p, b \begin{matrix} \\ \\ ∥ \end{matrix} p, c and p ≠ c and b ≠ p, then there exists d such that p, a \begin{matrix} \\ \\ ∥ \end{matrix} p, d and a, b |||| c, d and c ≠ d.

(9) If p, b |||| p, c and p ≠ b, then there exists d such that p, a |||| p, d and a, b |||| c, d.

(10) If z, x |||| x, t and x ≠ z, then there exists u such that y, x |||| x, u and y, z |||| t, u.

(11) If not L(p, a, b) and L(p, b, c) and L(p, a, d_{1}) and L(p, a, d_{2}) and a, b |||| c, d_{1} and a, b |||| c, d_{2}, then d_{1} = d_{2}.

(12) If not L(a, b, c) and a, b |||| c, d_{1} and a, b |||| c, d_{2} and a, c |||| b, d_{1} and a, c |||| b, d_{2}, then d_{1} = d_{2}.

(13) If not L(p, b, c) and B(b, p, a) and L(p, c, d) and b, c |||| d, a, then B(c, p, d).

(14) OAS satisfies inner invariancy of betweenness relation under parallel projections.

(15) If B(p, b, c) and L(p, a, d) and a, b |||| c, d and not L(p, a, b), then B(p, a, d).

(16) OAS satisfies outer invariancy of betweenness relation under parallel projections.

(17) If not L(a, b, a′) and a, a′ |||| b, b′ and a, a′ |||| c, c′ and B(a, b, c) and L(a′, b′, c′), then B(a′, b′, c′).

(18) OAS satisfies general invariancy of betweenness relation under parallel projections.

(19) If not L(p, a, b) and a, p \begin{matrix} \\ \\ ∥ \end{matrix} p, a′ and b, p \begin{matrix} \\ \\ ∥ \end{matrix} p, b′ and a, b |||| a′, b′, then

a, b \begin{matrix} \\ \\ ∥ \end{matrix} b′, a′.

(20) If not L(p, a, a′) and p, a \begin{matrix} \\ \\ ∥ \end{matrix} p, b and p, a′ \begin{matrix} \\ \\ ∥ \end{matrix} p, b′ and a, a′ |||| b, b′, then

a, a′ \begin{matrix} \\ \\ ∥ \end{matrix} b, b′.

(21) If not L(p, a, b) and p, a |||| b, c and p, b |||| a, c, then p, a \begin{matrix} \\ \\ ∥ \end{matrix} b, c and

p, b \begin{matrix} \\ \\ ∥ \end{matrix} a, c.

(22) If B(p, c, b) and c, d \begin{matrix} \\ \\ ∥ \end{matrix} b, a and p, d \begin{matrix} \\ \\ ∥ \end{matrix} p, a and not L(p, a, b) and p ≠ c, then B(p, d, a).

(23) If B(p, d, a) and c, d \begin{matrix} \\ \\ ∥ \end{matrix} b, a and p, c \begin{matrix} \\ \\ ∥ \end{matrix} p, b and not L(p, a, b) and p ≠ c, then B(p, c, b).

(24) If not L(p, a, b) and p, b \begin{matrix} \\ \\ ∥ \end{matrix} p, c and b, a \begin{matrix} \\ \\ ∥ \end{matrix} c, d and L(a, p, d) and p ≠ d, then not B(a, p, d).

(25) If p, b \begin{matrix} \\ \\ ∥ \end{matrix} p, c and b ≠ p, then there exists x such that p, a \begin{matrix} \\ \\ ∥ \end{matrix} p, x and

b, a \begin{matrix} \\ \\ ∥ \end{matrix} c, x.

(26) If B(p, c, b), then there exists x such that B(p, x, a) and b, a \begin{matrix} \\ \\ ∥ \end{matrix} c, x.

(27) If p ≠ b and B(p, b, c), then there exists x such that B(p, a, x) and

b, a \begin{matrix} \\ \\ ∥ \end{matrix} c, x.

(28) If not L(p, a, b) and B(p, c, b), then there exists x such that B(p, x, a)

and a, b \begin{matrix} \\ \\ ∥ \end{matrix} x, c.

(29) There exists x such that a, x \begin{matrix} \\ \\ ∥ \end{matrix} b, c and a, b \begin{matrix} \\ \\ ∥ \end{matrix} x, c.

(30) If a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and not L(a, b, c), then there exists x such that B(a, x, d) and B(b, x, c).

(31) If a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d and a, c \begin{matrix} \\ \\ ∥ \end{matrix} b, d and not L(a, b, c), then there exists x such that B(a, x, d) and B(b, x, c).

(32) OAS satisfies Fano Axiom.

(33) If a, b |||| c, d and a, c |||| b, d and not L(a, b, c), then there exists x such that L(x, a, d) and L(x, b, c).

(34) If a, b |||| c, d and a, c |||| b, d and not L(a, b, c) and L(p, a, d) and L(p, b, c), then not L(p, a, b).

(35) If B(a, b, d) and B(b, x, c) and not L(a, b, c), then there exists y such that B(a, y, c) and B(y, x, d).

(36) OAS satisfies outer form of Pasch’ Axiom.

(37) If B(a, b, d) and B(a, x, c) and not L(a, b, c), then there exists y such that B(b, y, c) and B(x, y, d).

(38) OAS satisfies inner form of Pasch’ Axiom.

(39) If B(p, a, b) and p, a \begin{matrix} \\ \\ ∥ \end{matrix} p′, a′ and not L(p, a, p′) and L(p′, a′, b′) and

p, p′ \begin{matrix} \\ \\ ∥ \end{matrix} a, a′ and p, p′ \begin{matrix} \\ \\ ∥ \end{matrix} b, b′, then B(p′, a′, b′).

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

The Lattice of Real Numbers. The Lattice of Real Functions

Marek Chmur^{1} Warsaw University Białystok

Summary. A proof of the fact , that ⟨⟨ , max, min⟩⟩ is a lattice (real lattice). Some basic properties (real lattice is distributive and modular) of it are proved. The same is done for the set ^{A} with operations: max(f(A)) and min(f(A)), where ^{A} means the set of all functions from A (being non-empty set) to , f is just such a function.

MML Identifier: REAL LAT.

The articles [4], [1], [3], and [2] provide the terminology and notation for this paper. In the sequel x, y will denote real numbers. Let x be an element of . The functor @x yielding a real number is defined by:

@x = x. We now state the proposition

(1) For every element x of holds @x = x. We now define two new functors. The binary operation min_{}on is defined

by:

min_{}(x, y) = min(x, y). The binary operation max_{}on is defined by:

max_{}(x, y) = max(x, y). The following propositions are true:

(2) min_{}(x, y) = min(x, y).

(3) max_{}(x, y) = max(x, y). In the sequel p, q will denote elements of the carrier of ⟨ , max_{}, min_{}⟩. Let

x be an element of the carrier of ⟨ , max_{}, min_{}⟩. The functor @x yields a real number and is defined by:

@x = x.

1Supported by RPBP.III-24.C1.

Next we state three propositions:

(4) For every element x of the carrier of ⟨ , max_{}, min_{}⟩ holds @x = x.

(5) p ⊔ q = max_{}(p, q).

(6) p ⊓ q = min_{}(p, q). The lattice _{L} is defined as follows: L = ⟨ , max_{}, min_{}⟩. One can prove the following proposition

(7) _{L} = ⟨ , max_{}, min_{}⟩. In the sequel p, q, r denote elements of the carrier of _{L}. One can prove the

following propositions:

(8) max_{}(p, q) = max_{}(q, p).

(9) min_{}(p, q) = min_{}(q, p).

(10) (i) max_{}(p, max_{}(q, r)) = max_{}(max_{}(q, r), p), (ii) max_{}(p, max_{}(q, r)) = max_{}(max_{}(p, q), r), (iii) max_{}(p, max_{}(q, r)) = max_{}(max_{}(q, p), r), (iv) max_{}(p, max_{}(q, r)) = max_{}(max_{}(r, p), q),

(v) max_{}(p, max_{}(q, r)) = max_{}(max_{}(r, q), p), (vi) max_{}(p, max_{}(q, r)) = max_{}(max_{}(p, r), q).

(11) (i) min_{}(p, min_{}(q, r)) = min_{}(min_{}(q, r), p), (ii) min_{}(p, min_{}(q, r)) = min_{}(min_{}(p, q), r), (iii) min_{}(p, min_{}(q, r)) = min_{}(min_{}(q, p), r), (iv) min_{}(p, min_{}(q, r)) = min_{}(min_{}(r, p), q),

(v) min_{}(p, min_{}(q, r)) = min_{}(min_{}(r, q), p), (vi) min_{}(p, min_{}(q, r)) = min_{}(min_{}(p, r), q).

(12) max_{}(min_{}(p, q), q) = q and max_{}(q, min_{}(p, q)) = q and max_{}(q, min_{}(q, p)) = q and max_{}(min_{}(q, p), q) = q.

(13) min_{}(q, max_{}(q, p)) = q and min_{}(max_{}(p, q), q) = q and min_{}(q, max_{}(p, q)) = q and min_{}(max_{}(q, p), q) = q.

(14) min_{}(q, max_{}(p, r)) = max_{}(min_{}(q, p), min_{}(q, r)).

(15) _{L} is a distributive lattice.

In the sequel L will be a distributive lattice. We now state the proposition (16) L is a modular lattice.

In the sequel A will denote a non-empty set and f, g, h will denote elements of ^{A}. Let A be a non-empty set, and let x be an element of ^{A}. The functor @x yielding an element of ^{A} qua a non-empty set is defined as follows:

@x = x. We now state the proposition

(17) For every element f of ^{A} holds @f = f.

We now define two new functors. Let us consider A. The functor max_{^{A}} yielding a binary operation on ^{A} is defined by:

max_{^{A}} (f, g) = max_{}^{∘}(f, g).

The functor min_{^{A}} yields a binary operation on ^{A} and is defined as follows: min_{^{A}} (f, g) = min_{}^{∘}(f, g). Next we state a number of propositions:

(18) max_{^{A}} (f, g) = max_{}^{∘}(f, g).

(19) min_{^{A}} (f, g) = min_{}^{∘}(f, g).

(20) max_{^{A}} (f, g) = max_{^{A}} (g, f).

(21) min_{^{A}} (f, g) = min_{^{A}} (g, f).

(22) max_{^{A}} (max_{^{A}} (f, g), h) = max_{^{A}} (f, max_{^{A}} (g, h)).

(23) min_{^{A}} (min_{^{A}} (f, g), h) = min_{^{A}} (f, min_{^{A}} (g, h)).

(24) max_{^{A}} (f, min_{^{A}} (f, g)) = f.

(25) max_{^{A}} (min_{^{A}} (f, g), f) = f.

(26) max_{^{A}} (min_{^{A}} (g, f), f) = f.

(27) max_{^{A}} (f, min_{^{A}} (g, f)) = f.

(28) min_{^{A}} (f, max_{^{A}} (f, g)) = f.

(29) min_{^{A}} (f, max_{^{A}} (g, f)) = f.

(30) min_{^{A}} (max_{^{A}} (g, f), f) = f.

(31) min_{^{A}} (max_{^{A}} (f, g), f) = f.

(32) min_{^{A}} (f, max_{^{A}} (g, h)) = max_{^{A}} (min_{^{A}} (f, g), min_{^{A}} (f, h)). We now define two new functors. Let us consider A. The functor max_{^{A}}

yields a binary operation on ^{A} and is defined by:

max_{^{A}} (f, g) = max_{^{A}} (f, g). The functor min_{^{A}} yields a binary operation on ^{A} and is defined as follows:

min_{^{A}} (f, g) = min_{^{A}} (f, g).

The following two propositions are true:

(33) max_{^{A}} (f, g) = max_{^{A}} (f, g).

(34) min_{^{A}} (f, g) = min_{^{A}} (f, g).

In the sequel p, q are elements of the carrier of ⟨ ^{A}, max_{^{A}} , min_{^{A}} ⟩. Let us consider A, and let x be an element of the carrier of ⟨ ^{A}, max_{^{A}} , min_{^{A}} ⟩. The functor @x yields an element of ^{A} and is defined as follows:

@x = x.

The following propositions are true:

(35) p ⊔ q = max_{^{A}} (p, q).

(36) p ⊔ q = max_{^{A}} (p, q).

(37) p ⊓ q = min_{^{A}} (p, q).

(38) p ⊓ q = min_{^{A}} (p, q).

Let us consider A. The functor _{L}^{A} yields a lattice and is defined by:

_{L}^{A} = ⟨ ^{A}, max_{^{A}} , min_{^{A}} ⟩.

One can prove the following proposition

(39) _{L}^{A} = ⟨ ^{A}, max_{^{A}} , min_{^{A}} ⟩.

In the sequel p, q, r will denote elements of the carrier of _{L}^{A}. We now state several propositions:

(40) max_{^{A}} (p, q) = max_{^{A}} (q, p).

(41) min_{^{A}} (p, q) = min_{^{A}} (q, p).

(42) (i) max_{^{A}} (p, max_{^{A}} (q, r)) = max_{^{A}} (max_{^{A}} (q, r), p), (ii) max_{^{A}} (p, max_{^{A}} (q, r)) = max_{^{A}} (max_{^{A}} (p, q), r), (iii) max_{^{A}} (p, max_{^{A}} (q, r)) = max_{^{A}} (max_{^{A}} (q, p), r), (iv) max_{^{A}} (p, max_{^{A}} (q, r)) = max_{^{A}} (max_{^{A}} (r, p), q),

(v) max_{^{A}} (p, max_{^{A}} (q, r)) = max_{^{A}} (max_{^{A}} (r, q), p), (vi) max_{^{A}} (p, max_{^{A}} (q, r)) = max_{^{A}} (max_{^{A}} (p, r), q).

(43) (i) min_{^{A}} (p, min_{^{A}} (q, r)) = min_{^{A}} (min_{^{A}} (q, r), p), (ii) min_{^{A}} (p, min_{^{A}} (q, r)) = min_{^{A}} (min_{^{A}} (p, q), r), (iii) min_{^{A}} (p, min_{^{A}} (q, r)) = min_{^{A}} (min_{^{A}} (q, p), r), (iv) min_{^{A}} (p, min_{^{A}} (q, r)) = min_{^{A}} (min_{^{A}} (r, p), q),

(v) min_{^{A}} (p, min_{^{A}} (q, r)) = min_{^{A}} (min_{^{A}} (r, q), p), (vi) min_{^{A}} (p, min_{^{A}} (q, r)) = min_{^{A}} (min_{^{A}} (p, r), q).

(44) max_{^{A}} (min_{^{A}} (p, q), q) = q and max_{^{A}} (q, min_{^{A}} (p, q)) = q and max_{^{A}} (q, min_{^{A}} (q, p)) = q and max_{^{A}} (min_{^{A}} (q, p), q) = q.

(45) min_{^{A}} (q, max_{^{A}} (q, p)) = q and min_{^{A}} (max_{^{A}} (p, q), q) = q and min_{^{A}} (q, max_{^{A}} (p, q)) = q and min_{^{A}} (max_{^{A}} (q, p), q) = q.

(46) min_{^{A}} (q, max_{^{A}} (p, r)) = max_{^{A}} (min_{^{A}} (q, p), min_{^{A}} (q, r)).

(47) _{L}^{A} is a distributive lattice.

In the sequel F will denote a distributive lattice. We now state the proposition

(48) F is a modular lattice.

References

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

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

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

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

A Construction of an Abstract Space of Congruence of Vectors ^{1}

Grzegorz Lewandowski Agriculture and Education School Siedlce Krzysztof Prażmowski Warsaw University Białystok

Summary. In the class of abelian groups a subclass of two-divisible-groups is singled out, and in the latter, a subclass of uniquely-two-divisible-groups. With every such a group a special geometrical structure, more precisely the structure of ”congruence of vectors” is correlated. The notion of ”affine vector space” (denoted by AffVect) is introduced. This term is defined by means of suitable axiom system. It is proved that every structure of the congruence of vectors determined by a non trivial uniquely two divisible group is a affine vector space.

MML Identifier: TDGROUP.

The articles [5], [1], [4], [2], and [3] provide the notation and terminology for this paper. In the sequel AG denotes an Abelian group and G denotes a group structure. One can prove the following propositions:

(1) _{G} is an Abelian group.

(2) If G = _{G}, then for every element a of the carrier of G there exists an element b of the carrier of G such that (the addition of G)(b, b) = a.

(3) If G = _{G}, then for every element a of the carrier of G such that (the addition of G)(a, a) = 0_{G} holds a = 0_{G}.

An Abelian group is called a 2-divisible group if: for every element a of the carrier of it there exists an element b of the carrier

of it such that (the addition of it)(b, b) = a.

The following two propositions are true:

(4) For every AG holds AG is a 2-divisible group if and only if for every element a of the carrier of AG there exists an element b of the carrier of AG such that (the addition of AG)(b, b) = a.

(5) _{G} is a 2-divisible group.

1Supported by RPBP.III-24.C3.

A 2-divisible group is said to be a uniquely 2-divisible group if: for every element a of the carrier of it such that (the addition of it)(a,

a) = 0_{it} holds a = 0_{it}.

One can prove the following three propositions:

(6) For every 2-divisible group AG holds AG is a uniquely 2-divisible group if and only if for every element a of the carrier of AG such that (the addition of AG)(a, a) = 0_{AG} holds a = 0_{AG}.

(7) For every AG holds AG is a uniquely 2-divisible group if and only if for every element a of the carrier of AG there exists an element b of the carrier of AG such that (the addition of AG)(b, b) = a and for every element a of the carrier of AG such that (the addition of AG)(a, a) = 0_{AG} holds a = 0_{AG}.

(8) _{G} is a uniquely 2-divisible group. We adopt the following rules: ADG is a uniquely 2-divisible group and a, b,

c, d, a′, b′, c′, p, q are elements of the carrier of ADG. Let us consider ADG, a, b. The functor a#b yielding an element of the carrier of ADG is defined as follows:

a#b = (the addition of ADG)(a, b). Let us consider ADG. The functor Congr_{ADG} yields a binary relation on [:

the carrier of ADG, the carrier of ADG:] and is defined as follows:

for all a, b, c, d holds ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ Congr_{ADG} if and only if a#d = b#c. Let us consider ADG. The functor Vectors(ADG) yielding an affine structure

is defined by:

Vectors(ADG) = ⟨ the carrier of ADG, Congr_{ADG}⟩. Next we state the proposition

(9) The points of Vectors(ADG) = the carrier of ADG and the congruence of Vectors(ADG) = Congr_{ADG}.

Let us consider ADG, a, b, c, d. The predicate a, b c, d is defined by: ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of Vectors(ADG). Next we state a number of propositions:

(10) a, b c, d if and only if a#d = b#c.

(11) If G = _{G}, then there exist elements a, b of the carrier of G such that a ≠ b.

(12) There exists ADG and there exist a, b such that a ≠ b.

(13) If a, b c, c, then a = b.

(14) If a, b p, q and c, d p, q, then a, b c, d.

(15) There exists d such that a, b c, d.

(16) If a, b a′, b′ and a, c a′, c′, then b, c b′, c′.

(17) There exists b such that a, b b, c.

(18) If a, b b, c and a, b′ b′, c, then b = b′.

(19) If a, b c, d, then a, c b, d.

In the sequel AS denotes an affine structure. Let us consider AS, and let a, b, c, d be elements of the points of AS. The predicate a, b c, d is defined by:

⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of AS. One can prove the following proposition

(20) Suppose there exist elements a, b of the carrier of ADG such that a ≠ b.

Then

(i) there exist elements a, b of the points of Vectors(ADG) such that a ≠ b, (ii) for all elements a, b, c of the points of Vectors(ADG) such that a, b

c, c holds a = b, (iii) for all elements a, b, c, d, p, q of the points of Vectors(ADG) such that

a, b p, q and c, d p, q holds a, b c, d, (iv) for every elements a, b, c of the points of Vectors(ADG) there exists

an element d of the points of Vectors(ADG) such that a, b c, d,

(v) for all elements a, b, c, a′, b′, c′ of the points of Vectors(ADG) such that a, b a′, b′ and a, c a′, c′ holds b, c b′, c′,

(vi) for every elements a, c of the points of Vectors(ADG) there exists an

element b of the points of Vectors(ADG) such that a, b b, c, (vii) for all elements a, b, c, b′ of the points of Vectors(ADG) such that

a, b b, c and a, b′ b′, c holds b = b′, (viii) for all elements a, b, c, d of the points of Vectors(ADG) such that

a, b c, d holds a, c b, d. An affine structure is said to be a space of free vectors if:

(i) there exist elements a, b of the points of it such that a ≠ b, (ii) for all elements a, b, c of the points of it such that a, b c, c holds a = b, (iii) for all elements a, b, c, d, p, q of the points of it such that a, b p, q and c, d p, q holds a, b c, d, (iv) for every elements a, b, c of the points of it there exists an element d of the points of it such that a, b c, d,

(v) for all elements a, b, c, a′, b′, c′ of the points of it such that a, b a′, b′ and a, c a′, c′ holds b, c b′, c′, (vi) for every elements a, c of the points of it there exists an element b of the points of it such that a, b b, c, (vii) for all elements a, b, c, b′ of the points of it such that a, b b, c and a, b′ b′, c holds b = b′, (viii) for all elements a, b, c, d of the points of it such that a, b c, d holds a, c b, d.

We now state several propositions:

(21) Given AS. Then the following conditions are equivalent:

(i) there exist elements a, b of the points of AS such that a ≠ b and for all elements a, b, c of the points of AS such that a, b c, c holds a = b and for all elements a, b, c, d, p, q of the points of AS such that a, b p, q and c, d p, q holds a, b c, d and for every elements a, b, c of the points of AS there exists an element d of the points of AS such that a, b c, d and for all elements a, b, c, a′, b′, c′ of the points of AS such that a, b a′, b′ and a, c a′, c′ holds b, c b′, c′ and for every elements a, c of the points of AS there exists an element b of the points of AS such that a, b b, c and for all elements a, b, c, b′ of the points of AS such that a, b b, c and a, b′ b′, c holds b = b′ and for all elements a, b, c, d of the points of AS such that a, b c, d holds a, c b, d,

(ii) AS is a space of free vectors.

(22) If there exist elements a, b of the carrier of ADG such that a ≠ b, then Vectors(ADG) is a space of free vectors.

(23) For every ADG and for all elements a, b of the carrier of ADG holds a#b = (the addition of ADG)(a, b).

(24) For every ADG and for every binary relation R on [: the carrier of ADG, the carrier of ADG:] holds R = Congr_{ADG} if and only if for all elements a, b, c, d of the carrier of ADG holds ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ R if and only if a#d = b#c.

(25) For every ADG and for every AS being an affine structure holds AS = Vectors(ADG) if and only if AS = ⟨ the carrier of ADG, Congr_{ADG}⟩.

(26) For every ADG and for all elements a, b, c, d of the carrier of ADG holds a, b c, d if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of Vectors(ADG).

(27) For every AS being an affine structure and for all elements a, b, c, d of the points of AS holds a, b c, d if and only if ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of AS.

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

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

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

A First-Order Predicate Calculus

Agata Darmochwał^{1} Warsaw Uniwersity Białystok

Summary. A continuation of [3], with an axiom system of firstorder predicate theory. The consequence Cn of a set of formulas X is defined as the intersection of all theories containing X and some basic properties of it has been proved (monotonicity, idempotency, completness etc.). The notion of a proof of given formula is also introduced and it is shown that CnX = { p : p has a proof w.r.t. X}. First 14 theorems are rather simply facts. I just wanted them to be included in the data base.

MML Identifier: CQC THE1.

The papers [11], [10], [9], [8], [4], [6], [1], [5], [2], [7], and [3] provide the terminology and notation for this paper. In the sequel i, j, n, k, l will be natural numbers. One can prove the following propositions:

(1) If n ≤ 0, then n = 0.

(2) If n ≤ 1, then n = 0 or n = 1.

(3) If n ≤ 2, then n = 0 or n = 1 or n = 2.

(4) If n ≤ 3, then n = 0 or n = 1 or n = 2 or n = 3.

(5) If n ≤ 4, then n = 0 or n = 1 or n = 2 or n = 3 or n = 4.

(6) If n ≤ 5, then n = 0 or n = 1 or n = 2 or n = 3 or n = 4 or n = 5.

(7) If n ≤ 6, then n = 0 or n = 1 or n = 2 or n = 3 or n = 4 or n = 5 or n = 6.

(8) If n ≤ 7, then n = 0 or n = 1 or n = 2 or n = 3 or n = 4 or n = 5 or n = 6 or n = 7.

(9) If n ≤ 8, then n = 0 or n = 1 or n = 2 or n = 3 or n = 4 or n = 5 or n = 6 or n = 7 or n = 8.

(10) If n ≤ 9, then n = 0 or n = 1 or n = 2 or n = 3 or n = 4 or n = 5 or n = 6 or n = 7 or n = 8 or n = 9.

1Supported by RPBP.III-24.C1

Next we state two propositions:

(11) {k : k ≤ n + 1} = {i : i ≤ n} ∪ {n + 1}.

(12) For every n holds {k : k ≤ n} is finite.

In the sequel X, Y, Z denote sets. One can prove the following two propositions:

(13) If X is finite and X ⊆ [: Y, Z:], then there exist sets A, B such that A is finite and A ⊆ Y and B is finite and B ⊆ Z and X ⊆ [: A, B:].

(14) If X is finite and Z is finite and X ⊆ [: Y, Z:], then there exists a set A

such that A is finite and A ⊆ Y and X ⊆ [: A, Z:]. For simplicity we adopt the following convention: T, S, X, Y will be subsets

of WFF_{CQC}, p, q, r, t, F will be elements of WFF_{CQC}, s will be a formula, and x, y will be bound variables. Let us consider T. We say that T is a theory if and only if:

(i) VERUM ∈ T, (ii) for all p, q, r, s, x, y holds (¬p ⇒ p) ⇒ p ∈ T and p ⇒ (¬p ⇒ q) ∈ T and (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)) ∈ T and p ∧ q ⇒ q ∧ p ∈ T but if p ∈ T and p ⇒ q ∈ T, then q ∈ T and ∀_{x}p ⇒ p ∈ T but if p ⇒ q ∈ T and x ∉ snb(p), then p ⇒ ∀_{x}q ∈ T but if s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉ snb(s) and s(x) ∈ T, then s(y) ∈ T.

Next we state a number of propositions:

(15) Suppose that (i) VERUM ∈ T, (ii) for all p, q, r, s, x, y holds (¬p ⇒ p) ⇒ p ∈ T and p ⇒ (¬p ⇒ q) ∈ T

and (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)) ∈ T and p ∧ q ⇒ q ∧ p ∈ T but if p ∈ T and p ⇒ q ∈ T, then q ∈ T and ∀_{x}p ⇒ p ∈ T but if p ⇒ q ∈ T and x ∉ snb(p), then p ⇒ ∀_{x}q ∈ T but if s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉ snb(s) and s(x) ∈ T, then s(y) ∈ T. Then T is a theory.

(16) If T is a theory, then VERUM ∈ T.

(17) If T is a theory, then (¬p ⇒ p) ⇒ p ∈ T.

(18) If T is a theory, then p ⇒ (¬p ⇒ q) ∈ T.

(19) If T is a theory, then (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)) ∈ T.

(20) If T is a theory, then p ∧ q ⇒ q ∧ p ∈ T.

(21) If T is a theory and p ∈ T and p ⇒ q ∈ T, then q ∈ T.

(22) If T is a theory, then ∀_{x}p ⇒ p ∈ T.

(23) If T is a theory and p ⇒ q ∈ T and x ∉ snb(p), then p ⇒ ∀_{x}q ∈ T.

(24) If T is a theory and s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉

snb(s) and s(x) ∈ T, then s(y) ∈ T. Let us consider T, S. Then T ∪ S is a subset of WFF_{CQC}. Then T ∩ S is a

subset of WFF_{CQC}. Then T \ S is a subset of WFF_{CQC}.

Let us consider p. Then {p} is a subset of WFF_{CQC}.

Next we state the proposition (25) If T is a theory and S is a theory, then T ∩ S is a theory.

Let us consider X. The functor Cn X yielding a subset of WFF_{CQC} is defined as follows:

t ∈ Cn X if and only if for every T such that T is a theory and X ⊆ T holds t ∈ T.

We now state a number of propositions:

(26) Y = Cn X if and only if for every t holds t ∈ Y if and only if for every T such that T is a theory and X ⊆ T holds t ∈ T.

(27) VERUM ∈ Cn X.

(28) (¬p ⇒ p) ⇒ p ∈ Cn X.

(29) p ⇒ (¬p ⇒ q) ∈ Cn X.

(30) (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)) ∈ Cn X.

(31) p ∧ q ⇒ q ∧ p ∈ Cn X.

(32) If p ∈ Cn X and p ⇒ q ∈ Cn X, then q ∈ Cn X.

(33) ∀_{x}p ⇒ p ∈ Cn X.

(34) If p ⇒ q ∈ Cn X and x ∉ snb(p), then p ⇒ ∀_{x}q ∈ Cn X.

(35) If s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉ snb(s) and s(x) ∈ Cn X, then s(y) ∈ Cn X.

(36) Cn X is a theory.

(37) If T is a theory and X ⊆ T, then Cn X ⊆ T.

(38) X ⊆ Cn X.

(39) If X ⊆ Y, then Cn X ⊆ Cn Y.

(40) Cn(Cn X) = Cn X.

(41) T is a theory if and only if Cn T = T.

The non-empty set is defined by:

= {k : k ≤ 9}.

Next we state three propositions:

(42) = {k : k ≤ 9}.

(43) 0 ∈ and 1 ∈ and 2 ∈ and 3 ∈ and 4 ∈ and 5 ∈ and 6 ∈ and 7 ∈ and 8 ∈ and 9 ∈ .

(44) is finite.

In the sequel f, g are finite sequences of elements of [: WFF_{CQC}, :]. The following proposition is true

(45) Suppose 1 ≤ n and n ≤ len f. Then (i) (f(n))_{2} = 0, or (ii) (f(n))_{2} = 1, or (iii) (f(n))_{2} = 2, or (iv) (f(n))_{2} = 3, or

(v) (f(n))_{2} = 4, or (vi) (f(n))_{2} = 5, or (vii) (f(n))_{2} = 6, or (viii) (f(n))_{2} = 7, or (ix) (f(n))_{2} = 8, or

(x) (f(n))_{2} = 9. Let P R be a finite sequence of elements of [: WFF_{CQC}, :], and let us consider

n, X. Let us assume that 1 ≤ n and n ≤ len P R. We say that P R(n) is a correct proof step w.r.t. X if and only if:

(P R(n))_{1} ∈ X if (P R(n))_{2} = 0, (P R(n))_{1} = VERUM if (P R(n))_{2} = 1, there exists p such that (P R(n))_{1} = (¬p ⇒ p) ⇒ p if (P R(n))_{2} = 2, there exist p, q such that (P R(n))_{1} = p ⇒ (¬p ⇒ q) if (P R(n))_{2} = 3, there exist p, q, r such that (P R(n))_{1} = (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)) if (P R(n))_{2} = 4, there exist p, q such that (P R(n))_{1} = p ∧ q ⇒ q ∧ p if (P R(n))_{2} = 5, there exist p, x such that (P R(n))_{1} = ∀_{x}p ⇒ p if (P R(n))_{2} = 6, there exist i, j, p, q such that 1 ≤ i and i < n and 1 ≤ j and j < i and p = (P R(j))_{1} and q = (P R(n))_{1} and (P R(i))_{1} = p ⇒ q if (P R(n))_{2} = 7, there exist i, p, q, x such that 1 ≤ i and i < n and (P R(i))_{1} = p ⇒ q and x ∉ snb(p) and (P R(n))_{1} = p ⇒ ∀_{x}q if (P R(n))_{2} = 8, there exist i, x, y, s such that 1 ≤ i and i < n and s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉ snb(s) and s(x) = (P R(i))_{1} and s(y) = (P R(n))_{1} if (P R(n))_{2} = 9.

The following propositions are true:

(46) If 1 ≤ n and n ≤ len f and (f(n))_{2} = 0, then f(n) is a correct proof step w.r.t. X if and only if (f(n))_{1} ∈ X.

(47) If 1 ≤ n and n ≤ len f and (f(n))_{2} = 1, then f(n) is a correct proof step w.r.t. X if and only if (f(n))_{1} = VERUM.

(48) If 1 ≤ n and n ≤ len f and (f(n))_{2} = 2, then f(n) is a correct proof step w.r.t. X if and only if there exists p such that (f(n))_{1} = (¬p ⇒ p) ⇒ p.

(49) If 1 ≤ n and n ≤ len f and (f(n))_{2} = 3, then f(n) is a correct proof step w.r.t. X if and only if there exist p, q such that (f(n))_{1} = p ⇒ (¬p ⇒ q).

(50) If 1 ≤ n and n ≤ len f and (f(n))_{2} = 4, then f(n) is a correct proof step w.r.t. X if and only if there exist p, q, r such that (f(n))_{1} = (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)).

(51) If 1 ≤ n and n ≤ len f and (f(n))_{2} = 5, then f(n) is a correct proof step w.r.t. X if and only if there exist p, q such that (f(n))_{1} = p ∧ q ⇒ q ∧ p.

(52) If 1 ≤ n and n ≤ len f and (f(n))_{2} = 6, then f(n) is a correct proof step w.r.t. X if and only if there exist p, x such that (f(n))_{1} = ∀_{x}p ⇒ p.

(53) Suppose 1 ≤ n and n ≤ len f and (f(n))_{2} = 7. Then f(n) is a correct proof step w.r.t. X if and only if there exist i, j, p, q such that 1 ≤ i and i < n and 1 ≤ j and j < i and p = (f(j))_{1} and q = (f(n))_{1} and (f(i))_{1} = p ⇒ q.

(54) Suppose 1 ≤ n and n ≤ len f and (f(n))_{2} = 8. Then f(n) is a correct proof step w.r.t. X if and only if there exist i, p, q, x such that 1 ≤ i and i < n and (f(i))_{1} = p ⇒ q and x ∉ snb(p) and (f(n))_{1} = p ⇒ ∀_{x}q.

(55) Suppose 1 ≤ n and n ≤ len f and (f(n))_{2} = 9. Then f(n) is a correct proof step w.r.t. X if and only if there exist i, x, y, s such that 1 ≤ i and i < n and s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉ snb(s) and s(x) = (f(i))_{1} and (f(n))_{1} = s(y).

Let us consider X, f. We say that f is a proof w.r.t. X if and only if:

f ≠ ε and for every n such that 1 ≤ n and n ≤ len f holds f(n) is a correct proof step w.r.t. X.

The following propositions are true:

(56) f is a proof w.r.t. X if and only if f ≠ ε and for every n such that 1 ≤ n and n ≤ len f holds f(n) is a correct proof step w.r.t. X.

(57) If f is a proof w.r.t. X, then rng f ≠ ∅.

(58) If f is a proof w.r.t. X, then 1 ≤ len f.

(59) Suppose f is a proof w.r.t. X. Then (f(1))_{2} = 0 or (f(1))_{2} = 1 or (f(1))_{2} = 2 or (f(1))_{2} = 3 or (f(1))_{2} = 4 or (f(1))_{2} = 5 or (f(1))_{2} = 6.

(60) If 1 ≤ n and n ≤ len f, then f(n) is a correct proof step w.r.t. X if and only if f ^{}g(n) is a correct proof step w.r.t. X.

(61) If 1 ≤ n and n ≤ len g and g(n) is a correct proof step w.r.t. X, then f ^{}g(n + len f) is a correct proof step w.r.t. X.

(62) If f is a proof w.r.t. X and g is a proof w.r.t. X, then f ^{}g is a proof w.r.t. X.

(63) If f is a proof w.r.t. X and X ⊆ Y, then f is a proof w.r.t. Y.

(64) If f is a proof w.r.t. X and 1 ≤ l and l ≤ len f, then (f(l))_{1} ∈ Cn X. Let us consider f. Let us assume that f ≠ ε. The functor Effect f yields an

element of WFF_{CQC} and is defined as follows: Effect f = (f(len f))_{1}. The following propositions are true:

(65) If f ≠ ε, then Effect f = (f(len f))_{1}.

(66) If f is a proof w.r.t. X, then Effect f ∈ Cn X.

(67) X ⊆ {F : ⋁_{f}[ f is a proof w.r.t. X ∧ Effect f = F]}.

(68) For every X such that Y = {p : ⋁_{f}[ f is a proof w.r.t. X ∧ Effect f = p]} holds Y is a theory.

(69) For every X holds {p : ⋁_{f}[ f is a proof w.r.t. X ∧ Effect f = p]} = Cn X.

(70) p ∈ Cn X if and only if there exists f such that f is a proof w.r.t. X and Effect f = p.

(71) If p ∈ Cn X, then there exists Y such that Y ⊆ X and Y is finite and

p ∈ Cn Y. The subset ∅_{CQC} of WFF_{CQC} is defined by: ∅_{CQC} = ∅_{WFF_{CQC}}. We now state the proposition

(72) ∅_{CQC} = ∅_{WFF_{CQC}}.

The subset Taut of WFF_{CQC} is defined as follows:

Taut = Cn ∅_{CQC}.

One can prove the following propositions:

(73) Taut = Cn ∅_{CQC}.

(74) If T is a theory, then Taut ⊆ T.

(75) Taut ⊆ Cn X.

(76) Taut is a theory.

(77) VERUM ∈ Taut.

(78) (¬p ⇒ p) ⇒ p ∈ Taut.

(79) p ⇒ (¬p ⇒ q) ∈ Taut.

(80) (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)) ∈ Taut.

(81) p ∧ q ⇒ q ∧ p ∈ Taut.

(82) If p ∈ Taut and p ⇒ q ∈ Taut, then q ∈ Taut.

(83) ∀_{x}p ⇒ p ∈ Taut.

(84) If p ⇒ q ∈ Taut and x ∉ snb(p), then p ⇒ ∀_{x}q ∈ Taut.

(85) If s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉ snb(s) and s(x) ∈

Taut, then s(y) ∈ Taut. Let us consider X, s. The predicate X ⊢ s is defined as follows: s ∈ Cn X. Next we state a number of propositions:

(86) X ⊢ s if and only if s ∈ Cn X.

(87) X ⊢ VERUM.

(88) X ⊢ (¬p ⇒ p) ⇒ p.

(89) X ⊢ p ⇒ (¬p ⇒ q).

(90) X ⊢ (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)).

(91) X ⊢ p ∧ q ⇒ q ∧ p.

(92) If X ⊢ p and X ⊢ p ⇒ q, then X ⊢ q.

(93) X ⊢ ∀_{x}p ⇒ p.

(94) If X ⊢ p ⇒ q and x ∉ snb(p), then X ⊢ p ⇒ ∀_{x}q.

(95) If s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉ snb(s) and X ⊢ s(x),

then X ⊢ s(y). Let us consider s. The predicate ⊢ s is defined as follows: ∅_{CQC} ⊢ s. Next we state two propositions:

(96) ⊢ s if and only if ∅_{CQC} ⊢ s.

(97) ⊢ s if and only if s ∈ Taut.

Let us consider s. Let us note that one can characterize the predicate ⊢ s by the following (equivalent) condition: s ∈ Taut.

We now state a number of propositions:

(98) If ⊢ p, then X ⊢ p.

(99) ⊢ VERUM.

(100) ⊢ (¬p ⇒ p) ⇒ p.

(101) ⊢ p ⇒ (¬p ⇒ q).

(102) ⊢ (p ⇒ q) ⇒ (¬(q ∧ r) ⇒ ¬(p ∧ r)).

(103) ⊢ p ∧ q ⇒ q ∧ p.

(104) If ⊢ p and ⊢ p ⇒ q, then ⊢ q.

(105) ⊢ ∀_{x}p ⇒ p.

(106) If ⊢ p ⇒ q and x ∉ snb(p), then ⊢ p ⇒ ∀_{x}q.

(107) If s(x) ∈ WFF_{CQC} and s(y) ∈ WFF_{CQC} and x ∉ snb(s) and ⊢ s(x), then ⊢ s(y).

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. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

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

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

[7] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

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

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

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

696

Partial Functions from a Domain to a Domain

Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. The value of a partial function from a domain to a domain and a inverse partial function are introduced. The value and inverse function were defined in the article [1], but new definitions are introduced. The basic properties of the value, the inverse partial function, the identity partial function, the composition of partial funtion, the 1−1 partial function, the restriction of a partial function, the image, the inverse image and the graph are proved. Constant partial function are introduced, too.

MML Identifier: PARTFUN2.

The terminology and notation used here are introduced in the following papers: [5], [1], [2], [6], [4], and [3]. For simplicity we follow the rules: x, y are arbitrary, X, Y denote sets, C, D, E denote non-empty sets, SC denotes a subset of C, SD denotes a subset of D, SE denotes a subset of E, c, c_{1}, c_{2} denote elements of C, d denotes an element of D, e denotes an element of E, f, f_{1}, g denote partial functions from C to D, t denotes a partial function from D to C, s denotes a partial function from D to E, h denotes a partial function from C to E, and F denotes a partial function from D to D. The following proposition is true

(1) x is an element of E if and only if x ∈ E.

Let us consider C, D, f, c. Let us assume that c ∈ dom f. The functor f(c) yielding an element of D is defined by:

f(c) = (f qua a function)(c).

Next we state four propositions:

(2) If c ∈ dom f, then f(c) = (f qua a function)(c).

1Supported by RPBP.III-24.C8

(3) If dom f = dom g and for every c such that c ∈ dom f holds f(c) = g(c), then f = g.

(4) y ∈ rng f if and only if there exists c such that c ∈ dom f and y = f(c).

(5) If c ∈ dom f, then f(c) ∈ rng f.

Let us consider D, C, f. Then dom f is a subset of C. Then rng f is a subset of D.

The following propositions are true:

(6) h = s · f if and only if for every c holds c ∈ dom h if and only if c ∈ dom f and f(c) ∈ dom s and for every c such that c ∈ dom h holds h(c) = s(f(c)).

(7) c ∈ dom(s · f) if and only if c ∈ dom f and f(c) ∈ dom s.

(8) If c ∈ dom(s · f), then (s · f)(c) = s(f(c)).

(9) If c ∈ dom f and f(c) ∈ dom s, then (s · f)(c) = s(f(c)).

(10) If rng f ⊆ dom s and c ∈ dom f, then (s · f)(c) = s(f(c)).

(11) If rng f = dom s and c ∈ dom f, then (s · f)(c) = s(f(c)).

Let us consider D, SD. Then id_{SD} is a partial function from D to D.

Next we state several propositions:

(12) F = id_{SD} if and only if dom F = SD and for every d such that d ∈ SD holds F (d) = d.

(13) If d ∈ SD, then id_{SD}(d) = d.

(14) If d ∈ dom F ∩ SD, then F (d) = (F · id_{SD})(d).

(15) d ∈ dom(id_{SD} ·F) if and only if d ∈ dom F and F (d) ∈ SD.

(16) f is one-to-one if and only if for all c_{1}, c_{2} such that c_{1} ∈ dom f and

c_{2} ∈ dom f and f(c_{1}) = f(c_{2}) holds c_{1} = c_{2}. Let us consider C, D, and let f be a partial function from C to D. Let us

assume that f is one-to-one. The functor f^{−1} yields a partial function from D to C and is defined as follows:

f^{−1} = (f qua a function)^{−1}.

One can prove the following propositions:

(17) If f is one-to-one, then for every partial function g from D to C holds g = f^{−1} if and only if g = (f qua a function)^{−1}.

(18) If f is one-to-one, then for every partial function g from D to C holds g = f^{−1} if and only if dom g = rng f and for all d, c holds d ∈ rng f and c = g(d) if and only if c ∈ dom f and d = f(c).

(19) If f is one-to-one, then rng f = dom(f^{−1}) and dom f = rng(f^{−1}).

(20) If f is one-to-one, then dom(f^{−1} ·f) = dom f and rng(f^{−1} ·f) = dom f.

(21) If f is one-to-one, then dom(f · f^{−1}) = rng f and rng(f · f^{−1}) = rng f.

(22) If f is one-to-one and c ∈ dom f, then c = f^{−1}(f(c)) and c = (f^{−1}·f)(c).

(23) If f is one-to-one and d ∈ rng f, then d = f(f^{−1}(d)) and d = (f·f^{−1})(d).

(24) If f is one-to-one and dom f = rng t and rng f = dom t and for all c, d such that c ∈ dom f and d ∈ dom t holds f(c) = d if and only if t(d) = c, then t = f^{−1}.

(25) If f is one-to-one, then f^{−1} · f = id_{dom f} and f · f^{−1} = id_{rng f}.

(26) If f is one-to-one, then f^{−1} is one-to-one.

(27) If f is one-to-one and rng f = dom s and s · f = id_{dom f}, then s = f^{−1}.

(28) If f is one-to-one and rng s = dom f and f · s = id_{rng f}, then s = f^{−1}.

(29) If f is one-to-one, then (f^{−1})^{−1} = f.

(30) If f is one-to-one and s is one-to-one, then (s · f)^{−1} = f^{−1} · s^{−1}.

(31) (id_{SC})^{−1} = id_{SC}.

Let us consider C, D, f, X. Then f X is a partial function from C to D. We now state several propositions:

(32) g = f X if and only if dom g = dom f ∩ X and for every c such that c ∈ dom g holds g(c) = f(c).

(33) If c ∈ dom(f X), then (f X)(c) = f(c).

(34) If c ∈ dom f ∩ X, then (f X)(c) = f(c).

(35) If c ∈ dom f and c ∈ X, then (f X)(c) = f(c).

(36) If c ∈ dom f and c ∈ X, then f(c) ∈ rng(f X).

Let us consider C, D, X, f. Then X f is a partial function from C to D. The following three propositions are true:

(37) g = X f if and only if for every c holds c ∈ dom g if and only if c ∈ dom f and f(c) ∈ X and for every c such that c ∈ dom g holds g(c) = f(c).

(38) c ∈ dom(X f) if and only if c ∈ dom f and f(c) ∈ X.

(39) If c ∈ dom(X f), then (X f)(c) = f(c).

Let us consider C, D, f, X. Then f ^{∘} X is a subset of D.

The following propositions are true:

(40) SD = f ^{∘} X if and only if for every d holds d ∈ SD if and only if there exists c such that c ∈ dom f and c ∈ X and d = f(c).

(41) d ∈ f ^{∘} X if and only if there exists c such that c ∈ dom f and c ∈ X and d = f(c).

(42) If c ∈ dom f, then f ^{∘} {c} = {f(c)}.

(43) If c_{1} ∈ dom f and c_{2} ∈ dom f, then f ^{∘} {c_{1}, c_{2}} = {f(c_{1}), f(c_{2})}. Let us consider C, D, f, X. Then f ^{−1} X is a subset of C. The following propositions are true:

(44) SC = f ^{−1} X if and only if for every c holds c ∈ SC if and only if c ∈ dom f and f(c) ∈ X.

(45) c ∈ f ^{−1} X if and only if c ∈ dom f and f(c) ∈ X.

(46) For every f there exists a function g from C into D such that for every c such that c ∈ dom f holds g(c) = f(c).

(47) f ≈ g if and only if for every c such that c ∈ dom f ∩ dom g holds

f(c) = g(c). In this article we present several logical schemes. The scheme PartFuncExD

deals with a non-empty set A, a non-empty set B, and a binary predicate P, and states that:

there exists a partial function f from A to B such that for every element d of A holds d ∈ dom f if and only if there exists an element c of B such that P[d, c] and for every element d of A such that d ∈ dom f holds P[d, f(d)] provided the following condition is satisfied:

• for every element d of A and for all elements c_{1}, c_{2} of B such that P[d, c_{1}] and P[d, c_{2}] holds c_{1} = c_{2}.

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

there exists a partial function f from A to B such that for every element d of A holds d ∈ dom f if and only if P[d] and for every element d of A such that d ∈ dom f holds f(d) = F(d) for all values of the parameters.

The scheme UnPartFuncD deals with a non-empty set A, a non-empty set B, a set C, and a unary functor F yielding an element of B and states that:

Let f, g be partial functions from A to B. Then if dom f = C and for every element c of A such that c ∈ dom f holds f(c) = F(c) and dom g = C and for every element c of A such that c ∈ dom g holds g(c) = F(c), then f = g for all values of the parameters.

Let us consider C, D, SC, d. Then SC ⟼ d is a partial function from C to D.

The following propositions are true:

(48) If c ∈ SC, then (SC ⟼ d)(c) = d.

(49) If for every c such that c ∈ dom f holds f(c) = d, then f = dom f ⟼ d.

(50) If c ∈ dom f, then f · (SE ⟼ c) = SE ⟼ f(c).

(51) id_{SC} is total if and only if SC = C.

(52) If SC ⟼ d is total, then SC ≠ ∅.

(53) SC ⟼ d is total if and only if SC = C.

Let us consider C, D, f, X. We say that f is a constant on X if and only if: there exists d such that for every c such that c ∈ X ∩ dom f holds f(c) = d. Next we state a number of propositions:

(54) f is a constant on X if and only if there exists d such that for every c such that c ∈ X ∩ dom f holds f(c) = d.

(55) f is a constant on X if and only if for all c_{1}, c_{2} such that c_{1} ∈ X ∩dom f and c_{2} ∈ X ∩ dom f holds f(c_{1}) = f(c_{2}).

(56) If X ∩ dom f ≠ ∅, then f is a constant on X if and only if there exists d such that rng(f X) = {d}.

(57) If f is a constant on X and Y ⊆ X, then f is a constant on Y.

(58) If X ∩ dom f = ∅, then f is a constant on X.

(59) If f SC = dom(f SC) ⟼ d, then f is a constant on SC.

(60) f is a constant on {x}.

(61) If f is a constant on X and f is a constant on Y and (X∩Y)∩dom f ≠ ∅, then f is a constant on X ∪ Y.

(62) If f is a constant on Y, then f X is a constant on Y.

(63) SC ⟼ d is a constant on SC.

(64) graph f ⊆ graph g if and only if dom f ⊆ dom g and for every c such that c ∈ dom f holds f(c) = g(c).

(65) c ∈ dom f and d = f(c) if and only if ⟨⟨c, d⟩⟩ ∈ graph f.

(66) If ⟨⟨c, e⟩⟩ ∈ graph(s · f), then ⟨⟨c, f(c)⟩⟩ ∈ graph f and ⟨⟨f(c), e⟩⟩ ∈ graph s.

(67) If graph f = {⟨⟨c, d⟩⟩}, then f(c) = d.

(68) If dom f = {c}, then graph f = {⟨⟨c, f(c)⟩⟩}.

(69) If graph f_{1} = graph f ∩ graph g and c ∈ dom f_{1}, then f_{1}(c) = f(c) and f_{1}(c) = g(c).

(70) If c ∈ dom f and graph f_{1} = graph f ∪ graph g, then f_{1}(c) = f(c).

(71) If c ∈ dom g and graph f_{1} = graph f ∪ graph g, then f_{1}(c) = g(c).

(72) If c ∈ dom f_{1} and graph f_{1} = graph f ∪ graph g, then f_{1}(c) = f(c) or f_{1}(c) = g(c).

(73) c ∈ dom f and c ∈ SC if and only if ⟨⟨c, f(c)⟩⟩ ∈ graph(f SC).

(74) c ∈ dom f and f(c) ∈ SD if and only if ⟨⟨c, f(c)⟩⟩ ∈ graph(SD f).

(75) c ∈ f ^{−1} SD if and only if ⟨⟨c, f(c)⟩⟩ ∈ graph f and f(c) ∈ SD.

References

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

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

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

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

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

702 Jarosław Kotowicz

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

71, 1990. Received May 31, 1990

Partial Functions from a Domain to the Set of Real Numbers

Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. Basic operations in the set of partial functions which map a domain to the set of all real numbers are introduced. They include adition, substraction, multiplication, division, multipication by a real number and also module. Main properties of these operations are proved. A definition of the partial function bounded on a set (bounded below and bounded above) is presented. There are theorems showing the laws of conservation of totality and boundeness for operations of partial functions. The characteristic function of a subset of a domain as a partial function is redefined and a few properties are proved.

MML Identifier: RFUNCT 1.

The papers [6], [3], [1], [7], [5], [2], and [4] provide the terminology and notation for this paper. For simplicity we follow the rules: X, Y will be sets, C will be a non-empty set, c will be an element of C, f, f_{1}, f_{2}, f_{3}, g, g_{1} will be partial functions from C to , and r, r_{1}, p, p_{1} will be real numbers. We now state two propositions:

(1) (−1)^{−1} = −1.

(2) If 0 ≤ p and 0 ≤ r and p ≤ p_{1} and r ≤ r_{1}, then p · r ≤ p_{1} · r_{1}.

We now define four new functors. Let us consider C, f_{1}, f_{2}. The functor f_{1} + f_{2} yields a partial function from C to and is defined as follows:

dom(f_{1} + f_{2}) = dom f_{1} ∩ dom f_{2} and for every c such that c ∈ dom(f_{1} + f_{2}) holds (f_{1} + f_{2})(c) = f_{1}(c) + f_{2}(c). The functor f_{1} − f_{2} yielding a partial function from C to is defined as follows:

dom(f_{1} − f_{2}) = dom f_{1} ∩ dom f_{2} and for every c such that c ∈ dom(f_{1} − f_{2}) holds (f_{1} − f_{2})(c) = f_{1}(c) − f_{2}(c). The functor f_{1} ⋄ f_{2} yielding a partial function from C to is defined by:

1Supported by RPBP.III-24.C8

dom(f_{1} ⋄ f_{2}) = dom f_{1} ∩ dom f_{2} and for every c such that c ∈ dom(f_{1} ⋄ f_{2}) holds (f_{1} ⋄ f_{2})(c) = f_{1}(c) · f_{2}(c). The functor \frac{f_{1}}{f_{2}} yielding a partial function from C to is defined by:

dom \frac{f_{1}}{f_{2}} = dom f_{1} ∩ (dom f_{2} \ f_{2} ^{−1} \{0\}) and for every c such that c ∈ dom \frac{f_{1}}{f_{2}} holds \frac{f_{1}}{f_{2}} (c) = f_{1}(c) · (f_{2}(c))^{−1}.

Let us consider C, f, r. The functor r ⋄ f yields a partial function from C to and is defined by:

dom(r⋄f) = dom f and for every c such that c ∈ dom(r⋄f) holds (r⋄f)(c) = r · f(c).

We now define three new functors. Let us consider C, f. The functor |f| yields a partial function from C to and is defined by:

dom |f| = dom f and for every c such that c ∈ dom |f| holds |f|(c) = |f(c)|. The functor −f yields a partial function from C to and is defined by:

dom(−f) = dom f and for every c such that c ∈ dom(−f) holds (−f)(c) = −f(c). The functor \frac{1}{f} yielding a partial function from C to is defined by:

dom \frac{1}{f} = dom f \ f ^{−1} \{0\} and for every c such that c ∈ dom \frac{1}{f} holds \frac{1}{f} (c) = (f(c))^{−1}.

One can prove the following propositions:

(3) f = f_{1} + f_{2} if and only if dom f = dom f_{1} ∩ dom f_{2} and for every c such that c ∈ dom f holds f(c) = f_{1}(c) + f_{2}(c).

(4) f = f_{1} − f_{2} if and only if dom f = dom f_{1} ∩ dom f_{2} and for every c such that c ∈ dom f holds f(c) = f_{1}(c) − f_{2}(c).

(5) f = f_{1} ⋄ f_{2} if and only if dom f = dom f_{1} ∩ dom f_{2} and for every c such that c ∈ dom f holds f(c) = f_{1}(c) · f_{2}(c).

(6) f = \frac{f_{1}}{f_{2}} if and only if dom f = dom f_{1} ∩(dom f_{2} \f_{2}^{−1} \{0\}) and for every c such that c ∈ dom f holds f(c) = f_{1}(c) · (f_{2}(c))^{−1}.

(7) f = r ⋄ f_{1} if and only if dom f = dom f_{1} and for every c such that c ∈ dom f holds f(c) = r · f_{1}(c).

(8) f = |f_{1}| if and only if dom f = dom f_{1} and for every c such that c ∈ dom f holds f(c) = |f_{1}(c)|.

(9) f = −f_{1} if and only if dom f = dom f_{1} and for every c such that c ∈ dom f holds f(c) = −f_{1}(c).

(10) f_{1} = \frac{1}{f} if and only if dom f_{1} = dom f \ f ^{−1} \{0\} and for every c such that c ∈ dom f_{1} holds f_{1}(c) = (f(c))^{−1}.

(11) dom \frac{1}{g} ⊆ dom g and dom g ∩ (dom g \ g^{−1} \{0\}) = dom g \ g^{−1} \{0\}.

(12) dom(f_{1} ⋄f_{2})\(f_{1} ⋄f_{2})^{−1} {0} = (dom f_{1} \f_{1}^{−1} {0})∩(dom f_{2} \f_{2}^{−1} {0}).

(13) If c ∈ dom \frac{1}{f}, then f(c) ≠ 0.

(14) \frac{1}{f} ^{−1} \{0\} = ∅.

(15) |f|^{−1} {0} = f ^{−1} {0} and (−f)^{−1} {0} = f ^{−1} {0}.

(16) dom _{\frac{1}{\frac{1}{f}}} = dom(f dom \frac{1}{f}).

(17) If r ≠ 0, then (r ⋄ f)^{−1} {0} = f ^{−1} {0}.

(18) f_{1} + f_{2} = f_{2} + f_{1}.

(19) (f_{1} + f_{2}) + f_{3} = f_{1} + (f_{2} + f_{3}).

(20) f_{1} ⋄ f_{2} = f_{2} ⋄ f_{1}.

(21) (f_{1} ⋄ f_{2}) ⋄ f_{3} = f_{1} ⋄ (f_{2} ⋄ f_{3}).

(22) (f_{1} + f_{2}) ⋄ f_{3} = f_{1} ⋄ f_{3} + f_{2} ⋄ f_{3}.

(23) f_{3} ⋄ (f_{1} + f_{2}) = f_{3} ⋄ f_{1} + f_{3} ⋄ f_{2}.

(24) r ⋄ (f_{1} ⋄ f_{2}) = (r ⋄ f_{1}) ⋄ f_{2}.

(25) r ⋄ (f_{1} ⋄ f_{2}) = f_{1} ⋄ (r ⋄ f_{2}).

(26) (f_{1} − f_{2}) ⋄ f_{3} = f_{1} ⋄ f_{3} − f_{2} ⋄ f_{3}.

(27) f_{3} ⋄ f_{1} − f_{3} ⋄ f_{2} = f_{3} ⋄ (f_{1} − f_{2}).

(28) r ⋄ (f_{1} + f_{2}) = r ⋄ f_{1} + r ⋄ f_{2}.

(29) (r · p) ⋄ f = r ⋄ (p ⋄ f).

(30) r ⋄ (f_{1} − f_{2}) = r ⋄ f_{1} − r ⋄ f_{2}.

(31) f_{1} − f_{2} = (−1) ⋄ (f_{2} − f_{1}).

(32) f_{1} − (f_{2} + f_{3}) = (f_{1} − f_{2}) − f_{3}.

(33) 1 ⋄ f = f.

(34) f_{1} − (f_{2} − f_{3}) = (f_{1} − f_{2}) + f_{3}.

(35) f_{1} + (f_{2} − f_{3}) = (f_{1} + f_{2}) − f_{3}.

(36) |f_{1} ⋄ f_{2}| = |f_{1}| ⋄ |f_{2}|.

(37) |r ⋄ f| = |r| ⋄ |f|.

(38) −f = (−1) ⋄ f.

(39) −(−f) = f.

(40) f_{1} − f_{2} = f_{1} + (−f_{2}).

(41) f_{1} − (−f_{2}) = f_{1} + f_{2}.

(42) _{\frac{1}{\frac{1}{f}}} = f dom \frac{1}{f}.

(43) \frac{1}{f_{1}⋄f_{2}} = \frac{1}{f_{1}} ⋄ \frac{1}{f_{2}} .

(44) If r ≠ 0, then \frac{1}{r⋄f} = r^{−1} ⋄ \frac{1}{f}.

(45) \frac{1}{−f} = (−1) ⋄ \frac{1}{f}.

(46) \frac{1}{|f|} = |\frac{1}{f} |.

(47) \frac{f}{g} = f ⋄ \frac{1}{g}.

(48) r ⋄ \frac{g}{f} = \frac{r⋄g}{f}.

(49) \frac{f}{g} ⋄ g = f dom \frac{1}{g}.

(50) \frac{f}{g} ⋄ \frac{f_{1}}{g_{1}} = \frac{f⋄f_{1}}{g⋄g_{1}} .

(51) _{\frac{1}{\frac{f_{1}}{f_{2}}}} = \frac{f_{2} dom \frac{1}{f_{2}}}{f_{1}} .

(52) g ⋄ \frac{f_{1}}{f_{2}} = \frac{g⋄f_{1}}{f_{2}} .

(53) _{\frac{g}{\frac{f_{1}}{f_{2}}}} = \frac{g⋄f_{2} dom \frac{1}{f_{2}}}{f_{1}} .

(54) −\frac{f}{g} = \frac{−f}{g} and \frac{f}{−g} = −\frac{f}{g}.

(55) \frac{f_{1}}{f} + \frac{f_{2}}{f} = \frac{f_{1}+f_{2}}{f} and \frac{f_{1}}{f} − \frac{f_{2}}{f} = \frac{f_{1}−f_{2}}{f}.

(56) \frac{f_{1}}{f} + \frac{g_{1}}{g} = \frac{f_{1}⋄g+g_{1}⋄f}{f⋄g}.

(57) \frac{\frac{f}{g}}{\frac{f_{1}}{g_{1}}} = \frac{f⋄g_{1} dom \frac{1}{g_{1}}}{g⋄f_{1}} .

(58) \frac{f_{1}}{f} − \frac{g_{1}}{g} = \frac{f_{1}⋄g−g_{1}⋄f}{f⋄g}.

(59) |\frac{f_{1}}{f_{2}} | = \frac{|f_{1}|}{|f_{2}|}.

(60) (f_{1} + f_{2}) X = f_{1} X + f_{2} X and (f_{1} + f_{2}) X = f_{1} X + f_{2} and (f_{1} + f_{2}) X = f_{1} + f_{2} X.

(61) (f_{1} ⋄ f_{2}) X = f_{1} X ⋄ f_{2} X and (f_{1} ⋄ f_{2}) X = f_{1} X ⋄ f_{2} and (f_{1} ⋄ f_{2}) X = f_{1} ⋄ f_{2} X.

(62) (−f) X = −f X and \frac{1}{f} X = \frac{1}{f X} and |f| X = |f X|.

(63) (f_{1} − f_{2}) X = f_{1} X − f_{2} X and (f_{1} − f_{2}) X = f_{1} X − f_{2} and (f_{1} − f_{2}) X = f_{1} − f_{2} X.

(64) \frac{f_{1}}{f_{2}} X = \frac{f_{1} X}{f_{2} X} and \frac{f_{1}}{f_{2}} X = \frac{f_{1} X}{f_{2}} and \frac{f_{1}}{f_{2}} X = \frac{f_{1}}{f_{2} X} .

(65) (r ⋄ f) X = r ⋄ f X.

(66) f_{1} is total and f_{2} is total if and only if f_{1} + f_{2} is total but f_{1} is total and f_{2} is total if and only if f_{1} − f_{2} is total but f_{1} is total and f_{2} is total if and only if f_{1} ⋄ f_{2} is total.

(67) f is total if and only if r ⋄ f is total.

(68) f is total if and only if −f is total.

(69) f is total if and only if |f| is total.

(70) \frac{1}{f} is total if and only if f ^{−1} \{0\} = ∅ and f is total.

(71) f_{1} is total and f_{2} ^{−1} \{0\} = ∅ and f_{2} is total if and only if \frac{f_{1}}{f_{2}} is total.

(72) If f_{1} is total and f_{2} is total, then (f_{1} + f_{2})(c) = f_{1}(c) + f_{2}(c) and (f_{1} − f_{2})(c) = f_{1}(c) − f_{2}(c) and (f_{1} ⋄ f_{2})(c) = f_{1}(c) · f_{2}(c).

(73) If f is total, then (r ⋄ f)(c) = r · f(c).

(74) If f is total, then (−f)(c) = −f(c) and |f|(c) = |f(c)|.

(75) If \frac{1}{f} is total, then \frac{1}{f} (c) = (f(c))^{−1}.

(76) If f_{1} is total and \frac{1}{f_{2}} is total, then \frac{f_{1}}{f_{2}} (c) = f_{1}(c) · (f_{2}(c))^{−1}.

Let us consider X, C. Then χ_{X,C} is a partial function from C to .

Next we state a number of propositions:

(77) f = χ_{X,C} if and only if dom f = C and for every c holds if c ∈ X, then f(c) = 1 but if c ∉ X, then f(c) = 0.

(78) χ_{X,C} is total.

(79) c ∈ X if and only if χ_{X,C}(c) = 1.

(80) c ∉ X if and only if χ_{X,C}(c) = 0.

(81) c ∈ C \ X if and only if χ_{X,C}(c) = 0.

(82) χ_{∅,C}(c) = 0.

(83) χ_{C,C}(c) = 1.

(84) χ_{X,C}(c) ≠ 1 if and only if χ_{X,C}(c) = 0.

(85) If X ∩ Y = ∅, then χ_{X,C} + χ_{Y,C} = χ_{X∪Y,C}.

(86) χ_{X,C} ⋄ χ_{Y,C} = χ_{X∩Y,C}.

We now define two new predicates. Let us consider C, f, Y. We say that f is upper bounded on Y if and only if:

there exists r such that for every c such that c ∈ Y ∩ dom f holds f(c) ≤ r. We say that f is lower bounded on Y if and only if:

there exists r such that for every c such that c ∈ Y ∩ dom f holds r ≤ f(c). Let us consider C, f, Y. We say that f is bounded on Y if and only if: f is upper bounded on Y and f is lower bounded on Y. The following propositions are true:

(87) f is upper bounded on Y if and only if there exists r such that for every c such that c ∈ Y ∩ dom f holds f(c) ≤ r.

(88) f is lower bounded on Y if and only if there exists r such that for every c such that c ∈ Y ∩ dom f holds r ≤ f(c).

(89) f is bounded on Y if and only if f is upper bounded on Y and f is lower bounded on Y.

(90) f is bounded on Y if and only if there exists r such that for every c such that c ∈ Y ∩ dom f holds |f(c)| ≤ r.

(91) If Y ⊆ X and f is upper bounded on X, then f is upper bounded on Y but if Y ⊆ X and f is lower bounded on X, then f is lower bounded on Y but if Y ⊆ X and f is bounded on X, then f is bounded on Y.

(92) If f is upper bounded on X and f is lower bounded on Y, then f is bounded on X ∩ Y.

(93) If X ∩ dom f = ∅, then f is bounded on X.

(94) If 0 = r, then r ⋄ f is bounded on Y.

(95) If f is upper bounded on Y and 0 ≤ r, then r ⋄ f is upper bounded on Y but if f is upper bounded on Y and r ≤ 0, then r ⋄ f is lower bounded on Y.

(96) If f is lower bounded on Y and 0 ≤ r, then r ⋄ f is lower bounded on Y but if f is lower bounded on Y and r ≤ 0, then r ⋄ f is upper bounded on Y.

(97) If f is bounded on Y, then r ⋄ f is bounded on Y.

(98) |f| is lower bounded on X.

(99) If f is bounded on Y, then |f| is bounded on Y and −f is bounded on Y.

(100) If f_{1} is upper bounded on X and f_{2} is upper bounded on Y, then f_{1} +f_{2} is upper bounded on X ∩ Y but if f_{1} is lower bounded on X and f_{2} is lower bounded on Y, then f_{1} + f_{2} is lower bounded on X ∩ Y but if f_{1} is bounded on X and f_{2} is bounded on Y, then f_{1} + f_{2} is bounded on X ∩ Y.

(101) If f_{1} is bounded on X and f_{2} is bounded on Y, then f_{1} ⋄ f_{2} is bounded on X ∩ Y and f_{1} − f_{2} is bounded on X ∩ Y.

(102) If f is upper bounded on X and f is upper bounded on Y, then f is upper bounded on X ∪ Y.

(103) If f is lower bounded on X and f is lower bounded on Y, then f is lower bounded on X ∪ Y.

(104) If f is bounded on X and f is bounded on Y, then f is bounded on X ∪ Y.

(105) If f_{1} is a constant on X and f_{2} is a constant on Y, then f_{1} + f_{2} is a constant on X ∩ Y and f_{1} − f_{2} is a constant on X ∩ Y and f_{1} ⋄ f_{2} is a constant on X ∩ Y.

(106) If f is a constant on Y, then p ⋄ f is a constant on Y.

(107) If f is a constant on Y, then |f| is a constant on Y and −f is a constant on Y.

(108) If f is a constant on Y, then f is bounded on Y.

(109) If f is a constant on Y, then for every r holds r ⋄ f is bounded on Y and −f is bounded on Y and |f| is bounded on Y.

(110) If f_{1} is upper bounded on X and f_{2} is a constant on Y, then f_{1} + f_{2} is upper bounded on X ∩ Y but if f_{1} is lower bounded on X and f_{2} is a constant on Y, then f_{1} + f_{2} is lower bounded on X ∩ Y but if f_{1} is bounded on X and f_{2} is a constant on Y, then f_{1} + f_{2} is bounded on X ∩ Y.

(111) (i) If f_{1} is upper bounded on X and f_{2} is a constant on Y, then f_{1} − f_{2}

is upper bounded on X ∩ Y, (ii) if f_{1} is lower bounded on X and f_{2} is a constant on Y, then f_{1} − f_{2} is

lower bounded on X ∩ Y, (iii) if f_{1} is bounded on X and f_{2} is a constant on Y, then f_{1}−f_{2} is bounded

on X ∩ Y and f_{2} − f_{1} is bounded on X ∩ Y and f_{1} ⋄ f_{2} is bounded on X ∩ Y.

References

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

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

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

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

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

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

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

710

Increasing and Continuous Ordinal Sequences

Grzegorz Bancerek Warsaw University Białystok

Summary. Concatenation of two ordinal sequences, the mode of all ordinals belonging to a universe and the mode of sequences of them with length equal to the rank of the universe are introduced. Besides, the increasing and continuous transfinite sequences, the limes of ordinal sequences and the power of ordinals, and the fact that every increasing and continuous transfinite sequence has critical numbers (fixed points) are discussed.

MML Identifier: ORDINAL4.

The terminology and notation used here have been introduced in the following papers: [6], [4], [2], [3], [1], and [5]. We adopt the following convention: phi, fi, psi are sequences of ordinal numbers and A, B, C are ordinal numbers. The following proposition is true

(1) If dom fi = succ A, then (last fi) (as an ordinal) is the limit of fi and lim fi = last fi.

Let us consider fi, psi. The functor fi ^{}psi yields a sequence of ordinal numbers and is defined as follows:

dom(fi ^{}psi) = dom fi + dom psi and for every A such that A ∈ dom fi holds (fi ^{}psi)(A) = fi(A) and for every A such that A ∈ dom psi holds (fi^{}psi)(dom fi + A) = psi(A).

The following propositions are true:

(2) Let chi be a sequence of ordinal numbers. Then chi = fi^{}psi if and only if dom chi = dom fi + dom psi and for every A such that A ∈ dom fi holds chi(A) = fi(A) and for every A such that A ∈ dom psi holds chi(dom fi + A) = psi(A).

(3) If A is the limit of psi, then A is the limit of fi^{}psi.

(4) If A is the limit of fi, then B + A is the limit of B + fi.

(5) If A is the limit of fi, then A · B is the limit of fi · B.

(6) If dom fi = dom psi and B is the limit of fi and C is the limit of psi but for every A such that A ∈ dom fi holds fi(A) ⊆ psi(A) or for every A such that A ∈ dom fi holds fi(A) ∈ psi(A), then B ⊆ C.

In the sequel f_{1}, f_{2} denote sequences of ordinal numbers. One can prove the following propositions:

(7) If dom f_{1} = dom fi and dom fi = dom f_{2} and A is the limit of f_{1} and A is the limit of f_{2} and for every A such that A ∈ dom fi holds f_{1}(A) ⊆ fi(A) and fi(A) ⊆ f_{2}(A), then A is the limit of fi.

(8) If dom fi ≠ 0 and dom fi is a limit ordinal number and fi is increasing, then sup fi is the limit of fi and lim fi = sup fi.

(9) If fi is increasing and A ⊆ B and B ∈ dom fi, then fi(A) ⊆ fi(B).

(10) If fi is increasing and A ∈ dom fi, then A ⊆ fi(A).

(11) If phi is increasing, then phi^{−1} A is an ordinal number.

(12) If f_{1} is increasing, then f_{2} · f_{1} is a sequence of ordinal numbers.

(13) If f_{1} is increasing and f_{2} is increasing, then there exists phi such that phi = f_{1} · f_{2} and phi is increasing.

(14) If f_{1} is increasing and A is the limit of f_{2} and sup(rng f_{1}) = dom f_{2} and fi = f_{2} · f_{1}, then A is the limit of fi.

(15) If phi is increasing, then phi A is increasing.

(16) If phi is increasing and dom phi is a limit ordinal number, then sup phi is a limit ordinal number.

(17) If fi is increasing and fi is continuous and psi is continuous and phi = psi · fi, then phi is continuous.

(18) If for every A such that A ∈ dom fi holds fi(A) = C + A, then fi is increasing.

(19) If C ≠ 0 and for every A such that A ∈ dom fi holds fi(A) = A · C, then fi is increasing.

(20) If A ≠ 0, then 0^{A} = 0.

(21) If A ≠ 0 and A is a limit ordinal number, then for every fi such that dom fi = A and for every B such that B ∈ A holds fi(B) = C^{B} holds C^{A} is the limit of fi.

(22) If C ≠ 0, then C^{A} ≠ 0.

(23) If 1 ∈ C, then C^{A} ∈ C^{succ A}.

(24) If 1 ∈ C and A ∈ B, then C^{A} ∈ C^{B}.

(25) If 1 ∈ C and for every A such that A ∈ dom fi holds fi(A) = C^{A}, then fi is increasing.

(26) If 1 ∈ C and A ≠ 0 and A is a limit ordinal number, then for every fi such that dom fi = A and for every B such that B ∈ A holds fi(B) = C^{B} holds C^{A} = sup fi.

(27) If C ≠ 0 and A ⊆ B, then C^{A} ⊆ C^{B}.

(28) If A ⊆ B, then A^{C} ⊆ B^{C}.

(29) If 1 ∈ C and A ≠ 0, then 1 ∈ C^{A}.

(30) C^{A+B} = (C^{B}) · (C^{A}).

(31) (C^{A})^{B} = C^{B·A}.

(32) If 1 ∈ C, then A ⊆ C^{A}.

The scheme CriticalNumber concerns a unary functor F yielding an ordinal number and states that:

there exists A such that F(A) = A provided the parameter meets the following conditions:

• for all A, B such that A ∈ B holds F(A) ∈ F(B),

• for every A such that A ≠ 0 and A is a limit ordinal number for every phi such that dom fi = A and for every B such that B ∈ A holds phi(B) = F(B) holds F(A) is the limit of phi.

In the sequel W will be a universal class. We now define two new modes. Let us consider W. An ordinal number is said to be an ordinal of W if:

it ∈ W. A sequence of ordinal numbers is called a transfinite sequence of ordinals of W if:

dom it = On W and rng it ⊆ On W. We now state two propositions:

(33) A is an ordinal of W if and only if A ∈ W.

(34) phi is a transfinite sequence of ordinals of W if and only if dom phi =

On W and rng phi ⊆ On W. In the sequel A_{1}, B_{1} will be ordinals of W and phi will be a transfinite

sequence of ordinals of W. The scheme UOS Lambda concerns a universal class A and a unary functor F yielding an ordinal of A and states that:

there exists a transfinite sequence phi of ordinals of A such that for every ordinal a of A holds phi(a) = F(a) for all values of the parameters.

We now define two new functors. Let us consider W. The functor 0_{W} yielding an ordinal of W is defined as follows:

0_{W} = 0. The functor 1_{W} yields an ordinal of W and is defined by:

1_{W} = 1. Let us consider phi, A_{1}. Then phi(A_{1}) is an ordinal of W.

Let us consider W, and let p_{2}, p_{1} be transfinite sequences of ordinals of W. Then p_{1} · p_{2} is a transfinite sequence of ordinals of W.

We now state the proposition

(35) 0_{W} = 0 and 1_{W} = 1.

Let us consider W, A_{1}. Then succ A_{1} is an ordinal of W. Let us consider B_{1}. Then A_{1} + B_{1} is an ordinal of W.

Let us consider W, A_{1}, B_{1}. Then A_{1} · B_{1} is an ordinal of W.

The following propositions are true:

(36) A_{1} ∈ dom phi.

(37) If dom fi ∈ W and rng fi ⊆ W, then sup fi ∈ W.

We now state the proposition

(38) If phi is increasing and phi is continuous and ω ∈ W, then there exists A such that A ∈ dom phi and phi(A) = A.

References

[1] Grzegorz Bancerek. Ordinal arithmetics. Formalized Mathematics, 1(3):515–519, 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. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[5] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595–600, 1990.

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

Transformations in Affine Spaces ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. Two classes of bijections of its point universe are correlated with every affine structure. The first class consists of the transformations, called formal isometries, which map every segment onto congruent segment, the second class consists of the automorphisms of such a structure. Each of these two classes of bijections forms a group for a given affine structure, if it satisfies a very weak axiom system (models of these axioms are called congruence spaces); formal isometries form a normal subgroup in the group of automorphism. In particular ordered affine spaces and affine spaces are congruence spaces; therefore formal isometries of these structures can be considered. They are called positive dilatations and dilatations, resp. For convenience the class of negative dilatations, transformations which map every ”vector” onto parallel ”vector”, but with opposite sense, is singled out. The class of translations is distinguished as well. Basic facts concerning all these types of transformations are established, like rigidity, decomposition principle, introductory group-theoretical properties. At the end collineations of affine spaces and their properties are investigated; for affine planes it is proved that the class of collineationms coincides with the class of bijections preserving lines.

MML Identifier: TRANSGEO.

The papers [7], [1], [8], [2], [3], [4], [5], and [6] provide the notation and terminology for this paper. We adopt the following convention: A denotes a non-empty set, a, b, x, y, z, t denote elements of A, and f, f_{1}, f_{2}, g, h denote permutations of A. Let us consider A, and let us consider f, and let x be an element of A. Then f(x) is an element of A.

Let us consider A, and let us consider f, and let X be a subset of A. Then f ^{∘} X is a subset of A.

Let us consider A, f, g. Then g · f is a permutation of A. One can prove the following propositions:

1Supported by RPBP.III-24.C2.

(1) For all f_{1}, f_{2} such that for every x holds f_{1}(x) = f_{2}(x) holds f_{1} = f_{2}.

(2) There exists x such that f(x) = y.

(3) If f(x) = f(y), then x = y.

(4) f(x) = y if and only if f^{−1}(y) = x.

(5) (f · g)(x) = f(g(x)).

Let us consider A, f, g. The functor f \ g yields a permutation of A and is defined by:

f \ g = (g · f) · g^{−1}.

One can prove the following proposition

(6) f \ g = (g · f) · g^{−1}.

The scheme EXPermutation deals with a non-empty set A, and a binary predicate P, and states that:

there exists a permutation f of A such that for all elements x, y of A holds f(x) = y if and only if P[x, y] provided the following requirements are met:

• for every element x of A there exists an element y of A such that P[x, y],

• for every element y of A there exists an element x of A such that P[x, y],

• for all elements x, y, x′ of A such that P[x, y] and P[x′, y] holds x = x′,

• for all elements x, y, y′ of A such that P[x, y] and P[x, y′] holds y = y′.

Next we state a number of propositions:

(7) (id_{A})^{−1} = id_{A}.

(8) f · f^{−1} = id_{A} and f^{−1} · f = id_{A}.

(9) f(f^{−1}(x)) = x and f^{−1}(f(x)) = x.

(10) id_{A} ·f = f and f · id_{A} = f.

(11) f · id_{A} = id_{A} ·f.

(12) f · (g · h) = (f · g) · h.

(13) If g · f = h · f or f · g = f · h, then g = h.

(14) (f · g)^{−1} = g^{−1} · f^{−1}.

(15) (f^{−1})^{−1} = f.

(16) f · g \ h = (f \ h) · (g \ h).

(17) f^{−1} \ g = (f \ g)^{−1}.

(18) f \ g · h = (f \ h) \ g.

(19) id_{A} \f = id_{A}.

(20) f \ id_{A} = f.

(21) If f(a) = a, then (f \ g)(g(a)) = g(a). In the sequel R will denote a binary relation on [: A, A:]. Let us consider A,

f, R. We say that f is a formal isometry of R if and only if: for all x, y holds ⟨⟨⟨⟨x, y⟩⟩, ⟨⟨f(x), f(y)⟩⟩⟩⟩ ∈ R. The following propositions are true:

(22) f is a formal isometry of R if and only if for all x, y holds ⟨⟨⟨⟨x, y⟩⟩, ⟨⟨f(x), f(y)⟩⟩⟩⟩ ∈ R.

(23) If R is reflexive in [: A, A:], then id_{A} is a formal isometry of R.

(24) If R is symmetric in [: A, A:] and f is a formal isometry of R, then f^{−1} is a formal isometry of R.

(25) If R is transitive in [: A, A:] and f is a formal isometry of R and g is a formal isometry of R, then f · g is a formal isometry of R.

(26) Suppose that

(i) for all a, b, x, y, z, t such that ⟨⟨⟨⟨x, y⟩⟩, ⟨⟨a, b⟩⟩⟩⟩ ∈ R and ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨z, t⟩⟩⟩⟩ ∈ R and a ≠ b holds ⟨⟨⟨⟨x, y⟩⟩, ⟨⟨z, t⟩⟩⟩⟩ ∈ R,

(ii) for all x, y, z holds ⟨⟨⟨⟨x, x⟩⟩, ⟨⟨y, z⟩⟩⟩⟩ ∈ R, (iii) f is a formal isometry of R, (iv) g is a formal isometry of R.

Then f · g is a formal isometry of R. Let us consider A, f, R. We say that f is an automorphism of R if and only

if:

for all x, y, z, t holds ⟨⟨⟨⟨x, y⟩⟩, ⟨⟨z, t⟩⟩⟩⟩ ∈ R if and only if

⟨⟨⟨⟨f(x), f(y)⟩⟩, ⟨⟨f(z), f(t)⟩⟩⟩⟩ ∈ R.

The following propositions are true:

(27) For all a, f, R holds f is an automorphism of R if and only if for all x, y, z, t holds ⟨⟨⟨⟨x, y⟩⟩, ⟨⟨z, t⟩⟩⟩⟩ ∈ R if and only if ⟨⟨⟨⟨f(x), f(y)⟩⟩, ⟨⟨f(z), f(t)⟩⟩⟩⟩ ∈ R.

(28) id_{A} is an automorphism of R.

(29) If f is an automorphism of R, then f^{−1} is an automorphism of R.

(30) If f is an automorphism of R and g is an automorphism of R, then g · f is an automorphism of R.

(31) If R is symmetric in [: A, A:] and R is transitive in [: A, A:] and f is a formal isometry of R, then f is an automorphism of R.

(32) Suppose that

(i) for all a, b, x, y, z, t such that ⟨⟨⟨⟨x, y⟩⟩, ⟨⟨a, b⟩⟩⟩⟩ ∈ R and ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨z, t⟩⟩⟩⟩ ∈ R and a ≠ b holds ⟨⟨⟨⟨x, y⟩⟩, ⟨⟨z, t⟩⟩⟩⟩ ∈ R,

(ii) for all x, y, z holds ⟨⟨⟨⟨x, x⟩⟩, ⟨⟨y, z⟩⟩⟩⟩ ∈ R, (iii) R is symmetric in [: A, A:], (iv) f is a formal isometry of R.

Then f is an automorphism of R.

(33) If f is a formal isometry of R and g is an automorphism of R, then f \g

is a formal isometry of R. In the sequel AS will be an affine structure. Let us consider AS, and let f

be a permutation of the points of AS. We say that f is a dilatation of AS if and only if:

f is a formal isometry of the congruence of AS.

The following proposition is true

(34) For every permutation f of the points of AS holds f is a dilatation of

AS if and only if f is a formal isometry of the congruence of AS. In the sequel a, b denote elements of the points of AS. Next we state the

proposition

(35) For every permutation f of the points of AS holds f is a dilatation of

AS if and only if for all a, b holds a, b \begin{matrix} \\ \\ ∥ \end{matrix} f(a), f(b). An affine structure is said to be a congruence space if:

(i) for all elements x, y, z, t, a, b of the points of it such that x, y \begin{matrix} \\ \\ ∥ \end{matrix} a, b and

a, b \begin{matrix} \\ \\ ∥ \end{matrix} z, t and a ≠ b holds x, y \begin{matrix} \\ \\ ∥ \end{matrix} z, t,

(ii) for all elements x, y, z of the points of it holds x, x \begin{matrix} \\ \\ ∥ \end{matrix} y, z, (iii) for all elements x, y, z, t of the points of it such that x, y \begin{matrix} \\ \\ ∥ \end{matrix} z, t holds

z, t \begin{matrix} \\ \\ ∥ \end{matrix} x, y,

(iv) for all elements x, y of the points of it holds x, y \begin{matrix} \\ \\ ∥ \end{matrix} x, y.

One can prove the following proposition

(36) Let AS be an affine structure. Then AS is a congruence space if and

only if the following conditions are satisfied:

(i) for all elements x, y, z, t, a, b of the points of AS such that x, y \begin{matrix} \\ \\ ∥ \end{matrix} a, b

and a, b \begin{matrix} \\ \\ ∥ \end{matrix} z, t and a ≠ b holds x, y \begin{matrix} \\ \\ ∥ \end{matrix} z, t,

(ii) for all elements x, y, z of the points of AS holds x, x \begin{matrix} \\ \\ ∥ \end{matrix} y, z, (iii) for all elements x, y, z, t of the points of AS such that x, y \begin{matrix} \\ \\ ∥ \end{matrix} z, t holds

z, t \begin{matrix} \\ \\ ∥ \end{matrix} x, y,

(iv) for all elements x, y of the points of AS holds x, y \begin{matrix} \\ \\ ∥ \end{matrix} x, y. In the sequel CS denotes a congruence space. One can prove the following

three propositions:

(37) id_{the points of CS} is a dilatation of CS.

(38) For every permutation f of the points of CS such that f is a dilatation of CS holds f^{−1} is a dilatation of CS.

(39) For all permutations f, g of the points of CS such that f is a dilatation

of CS and g is a dilatation of CS holds f · g is a dilatation of CS. We follow the rules: OAS denotes an ordered affine space and a, b, c, d, p,

q, x, y, z denote elements of the points of OAS. Next we state the proposition (40) OAS is a congruence space.

In the sequel f, g are permutations of the points of OAS. Let us consider OAS, and let f be a permutation of the points of OAS. We say that f is a positive dilatation if and only if:

f is a dilatation of OAS. We now state two propositions:

(41) For every permutation f of the points of OAS holds f is a positive dilatation if and only if f is a dilatation of OAS.

(42) For every permutation f of the points of OAS holds f is a positive dilatation if and only if for all a, b holds a, b \begin{matrix} \\ \\ ∥ \end{matrix} f(a), f(b).

Let us consider OAS, and let f be a permutation of the points of OAS. We say that f is a negative dilatation if and only if:

for all a, b holds a, b \begin{matrix} \\ \\ ∥ \end{matrix} f(b), f(a).

The following propositions are true:

(43) For every permutation f of the points of OAS holds f is a negative dilatation if and only if for all a, b holds a, b \begin{matrix} \\ \\ ∥ \end{matrix} f(b), f(a).

(44) id_{the points of OAS} is a positive dilatation.

(45) For every permutation f of the points of OAS such that f is a positive dilatation holds f^{−1} is a positive dilatation.

(46) For all permutations f, g of the points of OAS such that f is a positive dilatation and g is a positive dilatation holds f · g is a positive dilatation.

(47) For no f holds f is a negative dilatation and f is a positive dilatation.

(48) If f is a negative dilatation, then f^{−1} is a negative dilatation.

(49) If f is a positive dilatation and g is a negative dilatation, then f · g is

a negative dilatation and g · f is a negative dilatation. Let us consider OAS, and let f be a permutation of the points of OAS. We

say that f is a dilatation if and only if:

f is a formal isometry of λ( the congruence of OAS).

Next we state a number of propositions:

(50) For every permutation f of the points of OAS holds f is a dilatation if and only if f is a formal isometry of λ( the congruence of OAS).

(51) For every permutation f of the points of OAS holds f is a dilatation if and only if for all a, b holds a, b |||| f(a), f(b).

(52) If f is a positive dilatation or f is a negative dilatation, then f is a dilatation.

(53) For every permutation f of the points of OAS such that f is a dilatation there exists a permutation f′ of the points of Λ(OAS) such that f = f′ and f′ is a dilatation of Λ(OAS).

(54) For every permutation f of the points of Λ(OAS) such that f is a dilatation of Λ(OAS) there exists a permutation f′ of the points of OAS such that f = f′ and f′ is a dilatation.

(55) id_{the points of OAS} is a dilatation.

(56) If f is a dilatation, then f^{−1} is a dilatation.

(57) If f is a dilatation and g is a dilatation, then f · g is a dilatation.

(58) If f is a dilatation, then for all a, b, c, d holds a, b |||| c, d if and only if f(a), f(b) |||| f(c), f(d).

(59) If f is a dilatation, then for all a, b, c holds L(a, b, c) if and only if L(f(a), f(b), f(c)).

(60) If f is a dilatation and L(x, f(x), y), then L(x, f(x), f(y)).

(61) If a, b |||| c, d, then a, c |||| b, d or there exists x such that L(a, c, x) and L(b, d, x).

(62) If f is a dilatation, then f = id_{the points of OAS} or for every x holds f(x) ≠ x if and only if for all x, y holds x, f(x) |||| y, f(y).

(63) If f is a dilatation and a ≠ b and f(a) = a and f(b) = b and not L(a, b, x), then f(x) = x.

(64) If f is a dilatation and f(a) = a and f(b) = b and a ≠ b, then f = id_{the points of OAS}.

(65) If f is a dilatation and g is a dilatation and f(a) = g(a) and f(b) = g(b),

then a = b or f = g. Let us consider OAS, and let f be a permutation of the points of OAS. We

say that f is a translation if and only if:

f is a dilatation but f = id_{the points of OAS} or for every a holds a ≠ f(a). One can prove the following propositions:

(66) For every permutation f of the points of OAS holds f is a translation if and only if f is a dilatation but f = id_{the points of OAS} or for every a holds a ≠ f(a).

(67) If f is a dilatation, then f is a translation if and only if for all x, y holds x, f(x) |||| y, f(y).

(68) If f is a translation and f(a) = a, then f = id_{the points of OAS}.

(69) If f is a translation and g is a translation and f(a) = g(a) and f(a) ≠ a and not L(a, f(a), x), then f(x) = g(x).

(70) If f is a translation and g is a translation and f(a) = g(a), then f = g.

(71) If f is a translation, then f^{−1} is a translation.

(72) If f is a translation and g is a translation, then f · g is a translation.

(73) If f is a translation, then f is a positive dilatation.

(74) If f is a dilatation and f(p) = p and B(q, p, f(q)) and not L(p, q, x), then B(x, p, f(x)).

(75) If f is a dilatation and f(p) = p and B(q, p, f(q)) and q ≠ p, then B(x, p, f(x)).

(76) If f is a dilatation and f(p) = p and q ≠ p and B(q, p, f(q)) and not L(p, x, y), then x, y \begin{matrix} \\ \\ ∥ \end{matrix} f(y), f(x).

(77) If f is a dilatation and f(p) = p and q ≠ p and B(q, p, f(q)) and

L(p, x, y), then x, y \begin{matrix} \\ \\ ∥ \end{matrix} f(y), f(x).

(78) If f is a dilatation and f(p) = p and q ≠ p and B(q, p, f(q)), then f is a negative dilatation.

(79) If f is a dilatation and f(p) = p and for every x holds p, x \begin{matrix} \\ \\ ∥ \end{matrix} p, f(x), then for all y, z holds y, z \begin{matrix} \\ \\ ∥ \end{matrix} f(y), f(z).

(80) If f is a dilatation, then f is a positive dilatation or f is a negative

dilatation. We follow the rules: AF S will be an affine space and a, b, c, d, d_{1}, d_{2}, x, y,

z, t will be elements of the points of AF S. The following propositions are true:

(81) For all a, b, c, d holds a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d if and only if a, b ∥ c, d.

(82) AF S is a congruence space.

(83) Λ(OAS) is a congruence space.

In the sequel f, g denote permutations of the points of AF S. Let us consider AF S, f. We say that f is a dilatation if and only if:

f is a dilatation of AF S.

Next we state a number of propositions:

(84) For every f holds f is a dilatation if and only if f is a dilatation of AF S.

(85) f is a dilatation if and only if for all a, b holds a, b ∥ f(a), f(b).

(86) id_{the points of AF S} is a dilatation.

(87) If f is a dilatation, then f^{−1} is a dilatation.

(88) If f is a dilatation and g is a dilatation, then f · g is a dilatation.

(89) If f is a dilatation, then for all a, b, c, d holds a, b ∥ c, d if and only if f(a), f(b) ∥ f(c), f(d).

(90) If f is a dilatation, then for all a, b, c holds L(a, b, c) if and only if L(f(a), f(b), f(c)).

(91) If f is a dilatation and L(x, f(x), y), then L(x, f(x), f(y)).

(92) If a, b ∥ c, d, then a, c ∥ b, d or there exists x such that L(a, c, x) and L(b, d, x).

(93) If f is a dilatation, then f = id_{the points of AF S} or for every x holds f(x) ≠ x if and only if for all x, y holds x, f(x) ∥ y, f(y).

(94) If f is a dilatation and a ≠ b and f(a) = a and f(b) = b and not L(a, b, x), then f(x) = x.

(95) If f is a dilatation and f(a) = a and f(b) = b and a ≠ b, then f = id_{the points of AF S}.

(96) If f is a dilatation and g is a dilatation and f(a) = g(a) and f(b) = g(b), then a = b or f = g.

(97) If not L(a, b, c) and a, b ∥ c, d_{1} and a, b ∥ c, d_{2} and a, c ∥ b, d_{1} and

a, c ∥ b, d_{2}, then d_{1} = d_{2}. Let us consider AF S, f. We say that f is a translation if and only if: f is a dilatation but f = id_{the points of AF S} or for every a holds a ≠ f(a). One can prove the following propositions:

(98) For every f holds f is a translation if and only if f is a dilatation but f = id_{the points of AF S} or for every a holds a ≠ f(a).

(99) id_{the points of AF S} is a translation.

(100) If f is a dilatation, then f is a translation if and only if for all x, y holds x, f(x) ∥ y, f(y).

(101) If f is a translation and f(a) = a, then f = id_{the points of AF S}.

(102) If f is a translation and g is a translation and f(a) = g(a) and f(a) ≠ a and not L(a, f(a), x), then f(x) = g(x).

(103) If f is a translation and g is a translation and f(a) = g(a), then f = g.

(104) If f is a translation, then f^{−1} is a translation.

(105) If f is a translation and g is a translation, then f · g is a translation. Let us consider AF S, f. We say that f is a collineation if and only if: f is an automorphism of the congruence of AF S. Next we state four propositions:

(106) f is a collineation if and only if f is an automorphism of the congruence of AF S.

(107) f is a collineation if and only if for all x, y, z, t holds x, y ∥ z, t if and only if f(x), f(y) ∥ f(z), f(t).

(108) If f is a collineation, then L(x, y, z) if and only if L(f(x), f(y), f(z)).

(109) If f is a collineation and g is a collineation, then f^{−1} is a collineation

and f · g is a collineation and id_{the points of AF S} is a collineation. In the sequel A, C, K will denote subsets of the points of AF S. Next we

state several propositions:

(110) If a ∈ A, then f(a) ∈ f ^{∘} A.

(111) x ∈ f ^{∘} A if and only if there exists y such that y ∈ A and f(y) = x.

(112) If f ^{∘} A = f ^{∘} C, then A = C.

(113) If f is a collineation, then f ^{∘} Line(a, b) = Line(f(a), f(b)).

(114) If f is a collineation and K is a line, then f ^{∘} K is a line.

(115) If f is a collineation and A ∥ C, then f ^{∘} A ∥ f ^{∘} C.

For simplicity we follow the rules: AF P is an affine plane, A, K are subsets of the points of AF P, p, x are elements of the points of AF P, and f is a permutation of the points of AF P. We now state two propositions:

(116) If for every A such that A is a line holds f ^{∘} A is a line, then f is a collineation.

(117) If f is a collineation and K is a line and for every x such that x ∈ K holds f(x) = x and p ∉ K and f(p) = p, then f = id_{the points of AF P}.

References

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

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

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

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[5] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[6] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

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

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

724

Subcategories and Products of Categories

Czesław Byliński^{1} Warsaw University Białystok

Summary. The subcategory of a category and product of categories is defined. The inclusion functor is the injection (inclusion) map {E \atop ↪} which sends each object and each arrow of a Subcategory E of a category C to itself (in C). The inclusion functor is faithful. Full subcategories of C, that is, those subcategories E of C such that Hom_{E}(a, b) = Hom_{C}(b, b) for any objects a, b of E, are defined. A subcategory E of C is full when the inclusion functor {E \atop ↪} is full. The proposition that a full subcategory is determined by giving the set of objects of a category is proved. The product of two categories B and C is constructed in the usual way. Moreover, some simple facts on bifunctors (functors from a product category) are proved. The final notions in this article are that of projection functors and product of two functors (complex functors and product functors).

MML Identifier: CAT 2.

The terminology and notation used in this paper have been introduced in the following articles: [10], [8], [3], [4], [7], [2], [6], [1], [11], [9], and [5]. For simplicity we follow the rules: X denotes a set, C, D, E denote non-empty sets, c denotes an element of C, and d denotes an element of D. Let us consider D, X, E, and let F be a non-empty set of functions from X to E, and let f be a function from D into F, and let d be an element of D. Then f(d) is an element of F.

In the sequel f denotes a function from [: C, D:] into E. The following propositions are true:

(1) curry f is a function from C into E^{D}.

(2) curry′ f is a function from D into E^{C}. Let us consider C, D, E, f. Then curry f is a function from C into E^{D}.

Then curry′ f is a function from D into E^{C}.

The following two propositions are true:

(3) f(⟨⟨c, d⟩⟩) = (curry f(c))(d).

1Supported by RPBP.III-24.C1.

(4) f(⟨⟨c, d⟩⟩) = (curry′ f(d))(c). In the sequel B, C, D, C′, D′ denote categories. Let us consider B, C, and

let c be an object of C. The functor B ⟼ c yielding a functor from B to C is defined as follows:

B ⟼ c = (the morphisms of B) ⟼ id_{c}. One can prove the following propositions:

(5) For every object c of C holds B ⟼ c = (the morphisms of B) ⟼ id_{c}.

(6) For every object c of C and for every morphism f of B holds (B ⟼ c)(f) = id_{c}.

(7) For every object c of C and for every object b of B holds (Obj(B ⟼ c))(b) = c.

Let us consider C, D. The functor Funct(C, D) yields a non-empty set and is defined by:

for an arbitrary x holds x ∈ Funct(C, D) if and only if x is a functor from C to D.

Next we state two propositions:

(8) For every non-empty set F holds F = Funct(C, D) if and only if for an arbitrary x holds x ∈ F if and only if x is a functor from C to D.

(9) For every element T of Funct(C, D) holds T is a functor from C to D. Let us consider C, D. A non-empty set is called a non-empty set of functors

from C into D if:

for every element x of it holds x is a functor from C to D. The following proposition is true

(10) For every non-empty set F holds F is a non-empty set of functors from C into D if and only if for every element x of F holds x is a functor from C to D.

Let us consider C, D, and let F be a non-empty set of functors from C into D. We see that it makes sense to consider the following mode for restricted scopes of arguments. Then all the objests of the mode element of F are a functor from C to D.

Let A be a non-empty set, and let us consider C, D, and let F be a nonempty set of functors from C into D, and let T be a function from A into F, and let x be an element of A. Then T (x) is an element of F.

Let us consider C, D. Then Funct(C, D) is a non-empty set of functors from C into D.

Let us consider C. A category is said to be a subcategory of C if:

(i) the objects of it ⊆ the objects of C, (ii) for all objects a, b of it and for all objects a′, b′ of C such that a = a′ and b = b′ holds hom(a, b) ⊆ hom(a′, b′), (iii) the composition of it ≤ the composition of C, (iv) for every object a of it and for every object a′ of C such that a = a′ holds id_{a} = id_{a′}.

Next we state the proposition

(11) Given C, D. Then D is a subcategory of C if and only if the following

conditions are satisfied:

(i) the objects of D ⊆ the objects of C, (ii) for all objects a, b of D and for all objects a′, b′ of C such that a = a′

and b = b′ holds hom(a, b) ⊆ hom(a′, b′), (iii) the composition of D ≤ the composition of C, (iv) for every object a of D and for every object a′ of C such that a = a′

holds id_{a} = id_{a′}. In the sequel E will be a subcategory of C. We now state several propositions:

(12) For every object e of E holds e is an object of C.

(13) The morphisms of E ⊆ the morphisms of C.

(14) For every morphism f of E holds f is a morphism of C.

(15) For every morphism f of E and for every morphism f′ of C such that f = f′ holds dom f = dom f′ and cod f = cod f′.

(16) For all objects a, b of E and for all objects a′, b′ of C and for every morphism f from a to b such that a = a′ and b = b′ and hom(a, b) ≠ ∅ holds f is a morphism from a′ to b′.

(17) For all morphisms f, g of E and for all morphisms f′, g′ of C such that f = f′ and g = g′ and dom g = cod f holds g · f = g′ · f′.

(18) C is a subcategory of C.

(19) id_{E} is a functor from E to C.

Let us consider C, E. The functor {E \atop ↪} yielding a functor from E to C is defined as follows:

{E \atop ↪} = id_{E}.

The following propositions are true:

(20) {E \atop ↪} = id_{E}.

(21) For every morphism f of E holds {E \atop ↪}(f) = f.

(22) For every object a of E holds (Obj {E \atop ↪})(a) = a.

(23) For every object a of E holds {E \atop ↪}(a) = a.

(24) {E \atop ↪} is faithful.

(25) {E \atop ↪} is full if and only if for all objects a, b of E and for all objects a′, b′

of C such that a = a′ and b = b′ holds hom(a, b) = hom(a′, b′). Let C be a category structure, and let us consider D. We say that C is full

subcategory of D if and only if:

C is a subcategory of D and for all objects a, b of C and for all objects a′, b′ of D such that a = a′ and b = b′ holds hom(a, b) = hom(a′, b′).

The following propositions are true:

(26) For every C being a category structure and for every D holds C is full subcategory of D if and only if C is a subcategory of D and for all objects a, b of C and for all objects a′, b′ of D such that a = a′ and b = b′ holds hom(a, b) = hom(a′, b′).

(27) E is full subcategory of C if and only if {E \atop ↪} is full.

(28) For every non-empty subset O of the objects of C holds ⋃{hom(a, b) : a ∈ O ∧ b ∈ O} is a non-empty subset of the morphisms of C.

(29) Let O be a non-empty subset of the objects of C. Let M be a nonempty set. Suppose M = ⋃{hom(a, b) : a ∈ O ∧ b ∈ O}. Then (the dom-map of C) M is a function from M into O and (the cod-map of C) M is a function from M into O and (the composition of C) [: M, M :] is a partial function from [: M, M :] to M and (the id-map of C) O is a function from O into M.

(30) Let O be a non-empty subset of the objects of C. Let M be a non-empty set. Let d, c be functions from M into O. Let p be a partial function from [: M, M :] to M. Let i be a function from O into M. Suppose M = ⋃{hom(a, b) : a ∈ O ∧ b ∈ O} and d = (the dom-map of C) M and c = (the cod-map of C) M and p = (the composition of C) [: M, M :] and i = (the id-map of C) O. Then ⟨O, M, d, c, p, i⟩ is full subcategory of C.

(31) Let O be a non-empty subset of the objects of C. Let M be a non-empty set. Let d, c be functions from M into O. Let p be a partial function from [: M, M :] to M. Let i be a function from O into M. Suppose ⟨O, M, d, c, p, i⟩ is full subcategory of C. Then M = ⋃{hom(a, b) : a ∈ O ∧ b ∈ O} and d = (the dom-map of C) M and c = (the cod-map of C) M and p = (the composition of C) [: M, M :] and i = (the id-map of C) O.

Let X_{1}, X_{2}, Y_{1}, Y_{2} be non-empty sets, and let f_{1} be a function from X_{1} into Y_{1}, and let f_{2} be a function from X_{2} into Y_{2}. Then [: f_{1}, f_{2} :] is a function from [: X_{1}, X_{2} :] into [: Y_{1}, Y_{2} :].

Let A, B be non-empty sets, and let f be a partial function from [: A, A:] to A, and let g be a partial function from [: B, B:] to B. Then |:f, g:| is a partial function from [: [: A, B:], [: A, B:] :] to [: A, B:].

Let us consider C, D. The functor [: C, D:] yielding a category is defined as follows:

[: C, D:] = ⟨[: the objects of C, the objects of D:], [: the morphisms of C, the morphisms of D:], [: the dom-map of C, the dom-map of D:], [: the cod-map of C, the cod-map of D:], |: the composition of C, the composition of D:|, [: the id-map of C, the id-map of D:]⟩.

Next we state three propositions:

(32) [: C, D:] = ⟨[: the objects of C, the objects of D:], [: the morphisms of C, the morphisms of D:], [: the dom-map of C, the dom-map of D:], [: the cod-map of C, the cod-map of D:], |: the composition of C, the composition of D:|, [: the id-map of C, the id-map of D:]⟩.

(33) (i) The objects of [: C, D:] = [: the objects of C, the objects of D:], (ii) the morphisms of [: C, D:] = [: the morphisms of C, the morphisms of

D:], (iii) the dom-map of [: C, D:] = [: the dom-map of C, the dom-map of D:], (iv) the cod-map of [: C, D:] = [: the cod-map of C, the cod-map of D:],

(v) the composition of [: C, D:] = |: the composition of C, the composition of D:|,

(vi) the id-map of [: C, D:] = [: the id-map of C, the id-map of D:].

(34) For every object c of C and for every object d of D holds ⟨⟨c, d⟩⟩ is an

object of [: C, D:]. Let us consider C, D, and let c be an object of C, and let d be an object of

D. Then ⟨⟨c, d⟩⟩ is an object of [: C, D:].

One can prove the following propositions:

(35) For every object cd of [: C, D:] there exists an object c of C and there exists an object d of D such that cd = ⟨⟨c, d⟩⟩.

(36) For every morphism f of C and for every morphism g of D holds ⟨⟨f, g⟩⟩

is a morphism of [: C, D:]. Let us consider C, D, and let f be a morphism of C, and let g be a morphism

of D. Then ⟨⟨f, g⟩⟩ is a morphism of [: C, D:].

The following propositions are true:

(37) For every morphism fg of [: C, D:] there exists a morphism f of C and there exists a morphism g of D such that fg = ⟨⟨f, g⟩⟩.

(38) For every morphism f of C and for every morphism g of D holds dom ⟨⟨f, g⟩⟩ = ⟨⟨ dom f, dom g⟩⟩ and cod ⟨⟨f, g⟩⟩ = ⟨⟨ cod f, cod g⟩⟩.

(39) For all morphisms f, f′ of C and for all morphisms g, g′ of D such that dom f′ = cod f and dom g′ = cod g holds ⟨⟨f′, g′⟩⟩ · ⟨⟨f, g⟩⟩ = ⟨⟨f′ · f, g′ · g⟩⟩.

(40) For all morphisms f, f′ of C and for all morphisms g, g′ of D such that dom ⟨⟨f′, g′⟩⟩ = cod ⟨⟨f, g⟩⟩ holds ⟨⟨f′, g′⟩⟩ · ⟨⟨f, g⟩⟩ = ⟨⟨f′ · f, g′ · g⟩⟩.

(41) For every object c of C and for every object d of D holds id_{⟨⟨c,d⟩⟩} = ⟨⟨ id_{c}, id_{d} ⟩⟩.

(42) For all objects c, c′ of C and for all objects d, d′ of D holds hom(⟨⟨c, d⟩⟩, ⟨⟨c′, d′⟩⟩) = [: hom(c, c′), hom(d, d′) :].

(43) For all objects c, c′ of C and for every morphism f from c to c′ and for all objects d, d′ of D and for every morphism g from d to d′ such that hom(c, c′) ≠ ∅ and hom(d, d′) ≠ ∅ holds ⟨⟨f, g⟩⟩ is a morphism from ⟨⟨c, d⟩⟩ to ⟨⟨c′, d′⟩⟩.

(44) For every functor S from [: C, C′:] to D and for every object c of C holds curry S(id_{c}) is a functor from C′ to D.

(45) For every functor S from [: C, C′:] to D and for every object c′ of C′

holds curry′ S(id_{c′}) is a functor from C to D. Let us consider C, C′, D, and let S be a functor from [: C, C′:] to D, and let

c be an object of C. The functor S(c, −) yields a functor from C′ to D and is defined as follows:

S(c, −) = curry S(id_{c}). The following three propositions are true:

(46) For every functor S from [: C, C′:] to D and for every object c of C holds S(c, −) = curry S(id_{c}).

(47) For every functor S from [: C, C′:] to D and for every object c of C and for every morphism f of C′ holds S(c, −)(f) = S(⟨⟨ id_{c}, f⟩⟩).

(48) For every functor S from [: C, C′:] to D and for every object c of C and

for every object c′ of C′ holds (Obj S(c, −))(c′) = (Obj S)(⟨⟨c, c′⟩⟩). Let us consider C, C′, D, and let S be a functor from [: C, C′:] to D, and let

c′ be an object of C′. The functor S(−, c′) yielding a functor from C to D is defined by:

S(−, c′) = curry′ S(id_{c′}). We now state several propositions:

(49) For every functor S from [: C, C′:] to D and for every object c′ of C′ holds S(−, c′) = curry′ S(id_{c′}).

(50) For every functor S from [: C, C′:] to D and for every object c′ of C′ and for every morphism f of C holds S(−, c′)(f) = S(⟨⟨f, id_{c′} ⟩⟩).

(51) For every functor S from [: C, C′:] to D and for every object c of C and for every object c′ of C′ holds (Obj S(−, c′))(c) = (Obj S)(⟨⟨c, c′⟩⟩).

(52) Let L be a function from the objects of C into Funct(B, D). Let M be

a function from the objects of B into Funct(C, D). Suppose that

(i) for every object c of C and for every object b of B holds (M(b))(id_{c}) = (L(c))(id_{b}),

(ii) for every morphism f of B and for every morphism g of C holds (M(cod f))(g) · (L(dom g))(f) = (L(cod g))(f) · (M(dom f))(g). Then there exists a functor S from [: B, C:] to D such that for every morphism f of B and for every morphism g of C holds S(⟨⟨f, g⟩⟩) = (L(cod g))(f) · (M(dom f))(g).

(53) Let L be a function from the objects of C into Funct(B, D). Let M be a function from the objects of B into Funct(C, D). Suppose there exists a functor S from [: B, C:] to D such that for every object c of C and for every object b of B holds S(−, c) = L(c) and S(b, −) = M(b). Then for every morphism f of B and for every morphism g of C holds (M(cod f))(g) · (L(dom g))(f) = (L(cod g))(f) · (M(dom f))(g).

(54) π_{1}( (the morphisms of C)× (the morphisms of D)) is a functor from [: C, D:] to C.

(55) π_{2}( (the morphisms of C)× (the morphisms of D)) is a functor from

[: C, D:] to D. We now define two new functors. Let us consider C, D. The functor π_{1}(C ×

D) yields a functor from [: C, D:] to C and is defined as follows:

π_{1}(C × D) = π_{1}( (the morphisms of C)× (the morphisms of D)). The functor π_{2}(C ×D) yielding a functor from [: C, D:] to D is defined as follows:

π_{2}(C × D) = π_{2}( (the morphisms of C)× (the morphisms of D)).

One can prove the following propositions:

(56) π_{1}(C × D) = π_{1}( (the morphisms of C)× (the morphisms of D)).

(57) π_{2}(C × D) = π_{2}( (the morphisms of C)× (the morphisms of D)).

(58) For every morphism f of C and for every morphism g of D holds π_{1}(C × D)(⟨⟨f, g⟩⟩) = f.

(59) For every object c of C and for every object d of D holds (Obj π_{1}(C × D))(⟨⟨c, d⟩⟩) = c.

(60) For every morphism f of C and for every morphism g of D holds π_{2}(C × D)(⟨⟨f, g⟩⟩) = g.

(61) For every object c of C and for every object d of D holds (Obj π_{2}(C × D))(⟨⟨c, d⟩⟩) = d.

(62) For every functor T from C to D and for every functor T′ from C to

D′ holds ⟨T, T′⟩ is a functor from C to [: D, D′:]. Let us consider C, D, D′, and let T be a functor from C to D, and let T′ be

a functor from C to D′. Then ⟨T, T′⟩ is a functor from C to [: D, D′:].

One can prove the following propositions:

(63) For every functor T from C to D and for every functor T′ from C to D′ and for every object c of C holds (Obj⟨T, T′⟩)(c) = ⟨⟨(Obj T)(c), (Obj T′)(c)⟩⟩.

(64) For every functor T from C to D and for every functor T′ from C′ to D′ holds [: T, T′:] = ⟨T · π_{1}(C × C′), T′ · π_{2}(C × C′)⟩.

(65) For every functor T from C to D and for every functor T′ from C′ to

D′ holds [: T, T′:] is a functor from [: C, C′:] to [: D, D′:]. Let us consider C, C′, D, D′, and let T be a functor from C to D, and let

T′ be a functor from C′ to D′. Then [: T, T′:] is a functor from [: C, C′:] to [: D, D′:].

One can prove the following proposition

(66) For every functor T from C to D and for every functor T′ from C′ to D′ and for every object c of C and for every object c′ of C′ holds (Obj[: T, T′:])(⟨⟨c, c′⟩⟩) = ⟨⟨(Obj T)(c), (Obj T′)(c′)⟩⟩.

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

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

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

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

Many-Argument Relations

Edmund Woronowicz^{1} Warsaw University Białystok

Summary. Definitions of relations based on finite sequences. The arity of relation, the set of logical values Boolean consisting of false and true and the operations of negation and conjunction on them are defined.

MML Identifier: MARGREL1.

The notation and terminology used in this paper have been introduced in the following papers: [5], [2], [1], [3], and [4]. In the sequel x, y will be arbitrary, k will denote a natural number, and D will denote a non-empty set. Let B, A be non-empty sets, and let b be an element of B. Then A ⟼ b is an element of B^{A}.

A set is said to be a relation if:

for an arbitrary x such that x ∈ it holds x is a finite sequence and for all finite sequences a, b such that a ∈ it and b ∈ it holds len a = len b.

We follow a convention: X denotes a set, p, r denote relations, and a, b denote finite sequences. We now state several propositions:

(4)^{2} For every X such that for every x such that x ∈ X holds x is a finite sequence and for all a, b such that a ∈ X and b ∈ X holds len a = len b holds X is a relation.

(5) If x ∈ p, then x is a finite sequence.

(6) If a ∈ p and b ∈ p, then len a = len b.

(7) If X ⊆ p, then X is a relation.

(8) {a} is a relation.

(9) {⟨x, y⟩} is a relation.

The scheme rel exist concerns a set A, and a unary predicate P, and states that:

1Supported by RPBP III.24 C1

2The propositions (1)–(3) became obvious.

there exists r such that for every a holds a ∈ r if and only if a ∈ A and P[a] provided the parameters satisfy the following condition:

• for all a, b such that P[a] and P[b] holds len a = len b. Let us consider p, r. Let us note that one can characterize the predicate

p = r by the following (equivalent) condition: for every a holds a ∈ p if and only if a ∈ r.

We now state the proposition

(10) p = r if and only if for every a holds a ∈ p if and only if a ∈ r. The relation is defined by: a ∉ . One can prove the following propositions:

(11) a ∉ .

(12) p = if and only if for no a holds a ∈ p.

(13) = ∅.

Let us consider p. Let us assume that p ≠ . The functor Arity(p) yielding a natural number is defined by:

for every a such that a ∈ p holds Arity(p) = len a. We now state two propositions:

(14) If p ≠ , then for every k holds k = Arity(p) if and only if for every a such that a ∈ p holds k = len a.

(15) If a ∈ p and p ≠ , then Arity(p) = len a.

Let us consider k. A relation is called a k-ary relation if: for every a such that a ∈ it holds len a = k. One can prove the following two propositions:

(16) For all k, r such that for every a such that a ∈ r holds len a = k holds r is a k-ary relation.

(17) For every k-ary relation r such that a ∈ r holds len a = k.

Let X be a set. A relation is called a relation on X if: for every a such that a ∈ it holds rng a ⊆ X. In the sequel X denotes a set. Next we state four propositions:

(18) For all X, r such that for every a such that a ∈ r holds rng a ⊆ X holds r is a relation on X.

(19) For every relation r on X such that a ∈ r holds rng a ⊆ X.

(20) is a relation on X.

(21) is a k-ary relation.

Let us consider X, k. A relation is called a k-ary relation of X if: it is a relation on X and it is a k-ary relation. Next we state two propositions:

(22) For every relation r holds r is a k-ary relation of X if and only if r is a relation on X and r is a k-ary relation.

(23) For every k-ary relation R of X holds R is a relation on X and R is a

k-ary relation. Let us consider D. The functor Rel(D) yielding a non-empty family of sets

is defined as follows:

for every X holds X ∈ Rel(D) if and only if X ⊆ D^{∗} and for all finite sequences a, b of elements of D such that a ∈ X and b ∈ X holds len a = len b.

The following propositions are true:

(24) For every non-empty set D and for every non-empty family S of sets holds S = Rel(D) if and only if for every X holds X ∈ S if and only if X ⊆ D^{∗} and for all finite sequences a, b of elements of D such that a ∈ X and b ∈ X holds len a = len b.

(25) X ∈ Rel(D) if and only if X ⊆ D^{∗} and for all finite sequences a, b of

elements of D such that a ∈ X and b ∈ X holds len a = len b. Let D be a non-empty set. A relation on D is an element of Rel(D). In the sequel a will denote a finite sequence of elements of D and p, r will

denote elements of Rel(D). Next we state three propositions:

(26) If X ⊆ r, then X is an element of Rel(D).

(27) {a} is an element of Rel(D).

(28) For all elements x, y of D holds {⟨x, y⟩} is an element of Rel(D). Let us consider D, p, r. Let us note that one can characterize the predicate

p = r by the following (equivalent) condition: for every a holds a ∈ p if and only if a ∈ r.

One can prove the following proposition

(29) p = r if and only if for every a holds a ∈ p if and only if a ∈ r. The scheme rel D exist deals with a non-empty set A, and a unary predicate

P, and states that:

there exists an element r of Rel(A) such that for every finite sequence a of elements of A holds a ∈ r if and only if P[a] provided the parameters satisfy the following condition:

• for all finite sequences a, b of elements of A such that P[a] and P[b] holds len a = len b.

Let us consider D. The functor _{D} yielding an element of Rel(D) is defined as follows:

a ∉ _{D}. The following three propositions are true:

(30) r = _{D} if and only if for no a holds a ∈ r.

(31) a ∉ _{D}.

(32) _{D} = ∅.

Let us consider D, p. Let us assume that p ≠ _{D}. The functor Arity(p) yielding a natural number is defined by:

if a ∈ p, then Arity(p) = len a. Next we state two propositions:

(33) If p ≠ _{D}, then for every k holds k = Arity(p) if and only if for every a such that a ∈ p holds k = len a.

(34) If a ∈ p and p ≠ _{D}, then Arity(p) = len a.

The scheme rel D exist2 concerns a non-empty set A, a natural number B, and a unary predicate P, and states that:

there exists an element r of Rel(A) such that for every finite sequence a of elements of A such that len a = B holds a ∈ r if and only if P[a] for all values of the parameters.

The non-empty set Boolean is defined by:

Boolean = {0, 1}.

We now define two new functors. The element false of Boolean is defined by: false = 0.

The element true of Boolean is defined as follows:

true = 1.

The following four propositions are true:

(35) Boolean = {0, 1}.

(36) false = 0 and true = 1.

(37) Boolean = {false, true}.

(38) false ≠ true.

In the sequel u, v, w will denote elements of Boolean. Next we state the proposition

(39) v = false or v = true.

We now define two new functors. Let us consider v. The functor ¬v yielding an element of Boolean is defined by:

¬v = true if v = false, ¬v = false if v = true. Let us consider w. The functor v ∧ w yielding an element of Boolean is defined by:

v ∧ w = true if v = true and w = true, v ∧ w = false, otherwise.

The following propositions are true:

(40) ¬(¬v) = v.

(41) v = false if and only if ¬v = true but v = true if and only if ¬v = false.

(42) If v ≠ false, then v = true but if v ≠ true, then v = false.

(43) v ≠ true if and only if v = false.

(44) It is not true that: v = true and w = true if and only if v = false or w = false.

(45) v ∧ w = true if and only if v = true and w = true but v ∧ w = false if and only if v = false or w = false.

(46) v ∧ ¬v = false.

(47) ¬(v ∧ ¬v) = true.

(48) v ∧ w = w ∧ v.

(49) false ∧ v = false.

(50) true ∧ v = v.

(51) If v ∧ v = false, then v = false.

(52) v ∧ (w ∧ u) = (v ∧ w) ∧ u.

Let us consider X. The functor Boolean(false ∉ X) yields an element of Boolean and is defined as follows:

Boolean(false ∉ X) = true if false ∉ X, Boolean(false ∉ X) = false, otherwise.

One can prove the following proposition

(53) false ∉ X if and only if Boolean(false ∉ X) = true but false ∈ X if and only if Boolean(false ∉ X) = false.

References

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

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

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

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

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

738

Interpretation and Satisfiability in the First Order Logic

Edmund Woronowicz^{1} Warsaw University Białystok

Summary. The main notion discussed is satisfiability. Interpretation and some auxiliary concepts are also introduced.

MML Identifier: VALUAT 1.

The articles [6], [3], [1], [5], [4], [2], and [7] provide the notation and terminology for this paper. In the sequel i, k are natural numbers and A, D are non-empty sets. Let us consider A. The functor V(A) yields a non-empty set of functions and is defined by:

V(A) = A^{BoundVar}.

The following propositions are true:

(1) V(A) = A^{BoundVar}.

(2) For an arbitrary x such that x is an element of V(A) holds x is a function from BoundVar into A.

Let us consider A. Then V(A) is a non-empty set of functions from BoundVar to A.

In the sequel x, y will be bound variables and v, v_{1} will be elements of V(A). Let us consider A, v, x. Then v(x) is an element of A.

We now define two new functors. Let us consider A, and let p be an element of Boolean^{A}. The functor ¬p yields an element of Boolean^{A} and is defined by:

for every element x of A holds (¬p)(x) = ¬(p(x)). Let q be an element of Boolean^{A}. The functor p ∧ q yielding an element of Boolean^{A} is defined as follows:

for every element x of A holds (p ∧ q)(x) = (p(x)) ∧ (q(x)).

We now state two propositions:

1Supported by RPBP III.24 C1

(4)^{2} For every element p of Boolean^{A} and for every element x of A holds

¬p(x) = ¬(p(x)).

(5) For all elements p, q of Boolean^{A} and for every element x of A holds p ∧ q(x) = (p(x)) ∧ (q(x)).

Let us consider A, and let f be an element of Boolean^{V(A)}, and let us consider v. Then f(v) is an element of Boolean.

Let us consider A, x, and let p be an element of Boolean^{V(A)}. The functor ⋀_{x} p yields an element of Boolean^{V(A)} and is defined as follows:

for every v holds (⋀_{x} p)(v) = Boolean(false ∉ {p(v′) : ⋀_{y}[x ≠ y ⇒ v′(y) = v(y)]}).

Next we state three propositions:

(6) For all x, v and for every element p of Boolean^{V(A)} holds (⋀_{x} p)(v) = Boolean(false ∉ {p(v′) : ⋀[x ≠ y ⇒ v′(y) = v(y)]}).

(7) For every element p of Boolean^{V(A)} holds (⋀_{x} p)(v) = false if and only if there exists v_{1} such that p(v_{1}) = false and for every y such that x ≠ y holds v_{1}(y) = v(y).

(8) For every element p of Boolean^{V(A)} holds (⋀_{x} p)(v) = true if and only if for every v_{1} such that for every y such that x ≠ y holds v_{1}(y) = v(y) holds p(v_{1}) = true.

In the sequel ll is a variables list of k. Let us consider A, v, k, ll. The functor ll[v] yielding a finite sequence of elements of A is defined as follows:

len(ll[v]) = k and for every i such that 1 ≤ i and i ≤ k holds (ll[v])(i) = v(ll(i)).

We now state the proposition

(9) For all v, k, ll holds len(ll[v]) = k and for every natural number i such that 1 ≤ i and i ≤ k holds ll[v](i) = v(ll(i)).

Let us consider A, k, ll, and let r be an element of Rel(A). The functor llϵr yields an element of Boolean^{V(A)} and is defined by:

for every element v of V(A) holds if ll[v] ∈ r, then (llϵr)(v) = true but if ll[v] ∉ r, then (llϵr)(v) = false.

Next we state the proposition

(10) For all k, ll, v and for every element r of Rel(A) holds if ll[v] ∈ r, then llϵr(v) = true but if ll[v] ∉ r, then llϵr(v) = false.

Let us consider A, and let F be a function from WFF_{CQC} into Boolean^{V(A)}, and let p be an element of WFF_{CQC}. Then F (p) is an element of Boolean^{V(A)}.

Let us consider D. A function from PredSym into Rel(D) is called an interpretation of D if:

for every element P of PredSym and for every element r of Rel(D) such that it(P) = r holds r = _{D} or Arity(P) = Arity(r).

2The proposition (3) became obvious.

Next we state two propositions:

(11) For every non-empty set D and for every function F from PredSym into Rel(D) such that for every element P of PredSym and for every element r of Rel(D) such that F (P) = r holds r = _{D} or Arity(P) = Arity(r) holds F is an interpretation of D.

(12) For every D and for every interpretation J of D and for every element P of PredSym and for every element r of Rel(D) such that J(P) = r holds r = _{D} or Arity(P) = Arity(r).

Let us consider A, and let J be an interpretation of A, and let p be an element of PredSym. Then J(p) is a set.

For simplicity we adopt the following rules: p, q, t will be elements of WFF_{CQC}, J will be an interpretation of A, P will be a k-ary predicate symbol, and r will be an element of Rel(A). Let us consider A, k, J, P. Then J(P) is an element of Rel(A).

Let us consider A, J, p. The functor Valid(p, J) yielding an element of Boolean^{V(A)} is defined by:

there exists a function F from WFF_{CQC} into Boolean^{V(A)} such that Valid(p, J) = F (p) and for all elements p, q of WFF_{CQC} and for every bound variable x and for every natural number k and for every variables list ll of k and for every k-ary predicate symbol P and for all elements p′, q′ of Boolean^{V(A)} such that p′ = F (p) and q′ = F (q) holds

F (VERUM) = V(A) ⟼ true

and F (P [ll]) = llϵ(J(P)) and F (¬p) = ¬p′ and F (p ∧ q) = p′ ∧ q′ and F (∀_{x}p) = ⋀_{x} p′.

We now state a number of propositions:

(13) Valid(VERUM, J) = V(A) ⟼ true.

(14) Valid(VERUM, J)(v) = true.

(15) Valid(P [ll], J) = llϵ(J(P)).

(16) If p = P [ll] and r = J(P), then ll[v] ∈ r if and only if Valid(p, J)(v) = true.

(17) If p = P [ll] and r = J(P), then ll[v] ∉ r if and only if Valid(p, J)(v) = false.

(18) If p = P [ll] and r = J(P), then ll[v] ∉ r if and only if Valid(p, J)(v) = false.

(19) Valid(¬p, J) = ¬ Valid(p, J).

(20) Valid(¬p, J)(v) = ¬(Valid(p, J)(v)).

(21) Valid(p ∧ q, J) = Valid(p, J) ∧ Valid(q, J).

(22) Valid(p ∧ q, J)(v) = (Valid(p, J)(v)) ∧ (Valid(q, J)(v)).

(23) Valid(∀_{x}p, J) = ⋀_{x} Valid(p, J).

(24) Valid(p ∧ ¬p, J)(v) = false.

(25) Valid(¬(p ∧ ¬p), J)(v) = true.

Let us consider A, p, J, v. The predicate J, v ⊧ p is defined by:

Valid(p, J)(v) = true.

The following propositions are true:

(26) J, v ⊧ p if and only if Valid(p, J)(v) = true.

(27) J, v ⊧ P [ll] if and only if llϵ(J(P))(v) = true.

(28) J, v ⊧ ¬p if and only if J, v |̸= p.

(29) J, v ⊧ p ∧ q if and only if J, v ⊧ p and J, v ⊧ q.

(30) J, v ⊧ ∀_{x}p if and only if (⋀_{x} Valid(p, J))(v) = true.

(31) J, v ⊧ ∀_{x}p if and only if for every v_{1} such that for every y such that x ≠ y holds v_{1}(y) = v(y) holds Valid(p, J)(v_{1}) = true.

(32) Valid(¬(¬p), J) = Valid(p, J).

(33) Valid(p ∧ p, J) = Valid(p, J).

(34) Valid(p ∧ p, J)(v) = Valid(p, J)(v).

(35) J, v ⊧ p ⇒ q if and only if Valid(p, J)(v) = false or Valid(q, J)(v) = true.

(36) J, v ⊧ p ⇒ q if and only if if J, v ⊧ p, then J, v ⊧ q.

(37) For every element p of Boolean^{V(A)} such that (⋀_{x} p)(v) = true holds

p(v) = true. Let us consider A, J, p. The predicate J ⊧ p is defined by: for every v holds J, v ⊧ p. One can prove the following proposition

(38) J ⊧ p if and only if for every v holds J, v ⊧ p.

In the sequel w denotes an element of V(A). The scheme Lambda Val deals with a non-empty set A, a bound variable B, a bound variable C, an element D of V(A), and an element E of V(A) and states that:

there exists an element v of V(A) such that for every bound variable x such that x ≠ B holds v(x) = D(x) and v(B) = E(C) for all values of the parameters.

One can prove the following three propositions:

(39) If x ∉ snb(p), then for all v, w such that for every y such that x ≠ y holds w(y) = v(y) holds Valid(p, J)(v) = Valid(p, J)(w).

(40) If J, v ⊧ p and x ∉ snb(p), then for every w such that for every y such that x ≠ y holds w(y) = v(y) holds J, w ⊧ p.

(41) J, v ⊧ ∀_{x}p if and only if for every w such that for every y such that

x ≠ y holds w(y) = v(y) holds J, w ⊧ p. In the sequel s′ will be a formula. We now state a number of propositions:

(42) If x ≠ y and p = s′(x) and q = s′(y), then for every v such that v(x) = v(y) holds Valid(p, J)(v) = Valid(q, J)(v).

(43) If x ≠ y and x ∉ snb(s′), then x ∉ snb(s′(y)).

(44) J, v ⊧ VERUM.

(45) J, v ⊧ p ∧ q ⇒ q ∧ p.

(46) J, v ⊧ (¬p ⇒ p) ⇒ p.

(47) J, v ⊧ p ⇒ (¬p ⇒ q).

(48) J, v ⊧ (p ⇒ q) ⇒ (¬(q ∧ t) ⇒ ¬(p ∧ t)).

(49) If J, v ⊧ p and J, v ⊧ p ⇒ q, then J, v ⊧ q.

(50) J, v ⊧ (∀_{x}p) ⇒ p.

(51) J ⊧ VERUM.

(52) J ⊧ p ∧ q ⇒ q ∧ p.

(53) J ⊧ (¬p ⇒ p) ⇒ p.

(54) J ⊧ p ⇒ (¬p ⇒ q).

(55) J ⊧ (p ⇒ q) ⇒ (¬(q ∧ t) ⇒ ¬(p ∧ t)).

(56) If J ⊧ p and J ⊧ p ⇒ q, then J ⊧ q.

(57) J ⊧ (∀_{x}p) ⇒ p.

(58) If J ⊧ p ⇒ q and x ∉ snb(p), then J ⊧ p ⇒ (∀_{x}q).

(59) For every formula s such that p = s(x) and q = s(y) and x ∉ snb(s) and J ⊧ p holds J ⊧ q.

References

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

[2] Czesław Byliński. A classical first order language. Formalized Mathematics, 1(4):669–676, 1990.

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

[4] Piotr Rudnicki and Andrzej Trybulec. A first order language. Formalized Mathematics, 1(2):303–311, 1990.

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

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

[7] Edmund Woronowicz. Many–argument relations. Formalized Mathematics, 1(4):733–737, 1990.

744

Probability

Andrzej Ne‘dzusiak^{1} Warsaw University Białystok

Summary. Some further theorems concerning probability, among them the equivalent definition of probability are discussed, followed by notions of independence of events and conditional probability and basic theorems on them.

MML Identifier: PROB 2.

The notation and terminology used in this paper have been introduced in the following papers: [8], [2], [4], [3], [6], [5], [9], [7], and [1]. For simplicity we adopt the following convention: Omega denotes a non-empty set, f denotes a function, m, n denote natural numbers, r, r_{1}, r_{2}, r_{3} denote real numbers, seq, seq_{1} denote sequences of real numbers, Sigma denotes a σ-field of subsets of Omega, ASeq, BSeq denote sequences of subsets of Sigma, P, P_{1} denote probabilities on Sigma, and A, B, C, A_{1}, A_{2}, A_{3} denote events of Sigma. One can prove the following propositions:

(1) (r − r_{1}) + r_{2} = (r + r_{2}) − r_{1}.

(2) r ≤ r_{1} if and only if r < r_{1} or r = r_{1}.

(3) For all r, r_{1}, r_{2} such that 0 < r and r_{1} ≤ r_{2} holds \frac{r_{1}}{r} ≤ \frac{r_{2}}{r}.

(4) For all r, r_{1}, r_{2}, r_{3} such that r ≠ 0 and r_{1} ≠ 0 holds \frac{r_{3}}{r_{1}} = \frac{r_{2}}{r} if and only if r_{3} · r = r_{2} · r_{1}.

(5) If seq is convergent and for every n holds seq_{1}(n) = r − seq(n), then seq_{1} is convergent and lim seq_{1} = r − lim seq.

(6) A∩Omega = A and Omega∩A = A and A∩Ω_{Sigma} = A and Ω_{Sigma} ∩ A = A.

(7) If B misses C, then A ∩ B misses A ∩ C and B ∩ A misses C ∩ A.

The scheme SeqEx concerns a unary functor F and states that:

746 Andrzej Ne‘dzusiak

there exists f such that dom f = and for every n holds f(n) = F(n) for all values of the parameter.

1Supported by RPBP III.24

Let us consider Omega, Sigma, ASeq, n. Then ASeq(n) is an event of Sigma.

Let us consider Omega, Sigma, ASeq. The functor ⋂ ASeq yielding an event of Sigma is defined by:

⋂ ASeq = Intersection ASeq. One can prove the following propositions:

(8) ⋂ ASeq = Intersection ASeq.

(9) For every B, ASeq there exists BSeq such that for every n holds BSeq(n) = ASeq(n) ∩ B.

(10) For all B, ASeq, BSeq such that ASeq is nonincreasing and for every n holds BSeq(n) = ASeq(n) ∩ B holds BSeq is nonincreasing.

(11) For every function f from Sigma into and for all ASeq, n holds (f · ASeq)(n) = f(ASeq(n)).

(12) For all ASeq, BSeq, B such that for every n holds BSeq(n) = ASeq(n)∩ B holds (Intersection ASeq) ∩ B = Intersection BSeq.

(13) For all P, P_{1} such that for every A holds P (A) = P_{1}(A) holds P = P_{1}.

(14) For every Omega and for every sequence ASeq of subsets of Omega holds ASeq is nonincreasing if and only if for every n holds ASeq(n+1) ⊆ ASeq(n).

(15) For every sequence ASeq of subsets of Omega holds ASeq is nondecreasing if and only if for every n holds ASeq(n) ⊆ ASeq(n + 1).

(16) For all sequences ASeq, BSeq of subsets of Omega such that for every n holds ASeq(n) = BSeq(n) holds ASeq = BSeq.

(17) For every sequence ASeq of subsets of Omega holds ASeq is nonin-

creasing if and only if Complement ASeq is nondecreasing. Let us consider Omega, Sigma, ASeq. The functor ASeq^{c} yields a sequence

of subsets of Sigma and is defined by: ASeq^{c} = Complement ASeq. The following proposition is true

(18) ASeq^{c} = Complement ASeq.

Let us consider Omega, Sigma, ASeq. We say that ASeq is pairwise disjoint if and only if:

for all m, n such that m ≠ n holds ASeq(m) misses ASeq(n). We now state a number of propositions:

(19) ASeq is pairwise disjoint if and only if for all m, n such that m ≠ n holds ASeq(m) misses ASeq(n).

(20) Let P be a function from Sigma into . Then P is a probability on

Sigma if and only if the following conditions are satisfied:

(i) for every A holds 0 ≤ P (A), (ii) P (Omega) = 1, (iii) for all A, B such that A misses B holds P (A ∪ B) = P (A) + P (B), (iv) for every ASeq such that ASeq is nondecreasing holds P · ASeq is

convergent and lim(P · ASeq) = P (Union ASeq).

(21) P ((A ∪ B) ∪ C) = (((P (A) + P (B)) + P (C)) − ((P (A ∩ B) + P (B ∩ C)) + P (A ∩ C))) + P ((A ∩ B) ∩ C).

(22) P (A \ A ∩ B) = P (A) − P (A ∩ B).

(23) For all P, A, B holds P (A ∩ B) ≤ P (B) and P (A ∩ B) ≤ P (A).

(24) For all P, A, B, C such that C = B^{c} holds P (A) = P (A∩B)+P (A∩C).

(25) For all P, A, B holds (P (A) + P (B)) − 1 ≤ P (A ∩ B).

(26) For all P, A holds P (A) = 1 − P (Ω_{Sigma} \ A).

(27) For all P, A holds P (A) < 1 if and only if 0 < P (Ω_{Sigma} \ A).

(28) For all P, A holds P (Ω_{Sigma} \ A) < 1 if and only if 0 < P (A). We now define two new predicates. Let us consider Omega, Sigma, P, A,

B. We say that A and B are independent w.r.t P if and only if:

P (A ∩ B) = P (A) · P (B). Let us consider C. We say that A, B and C are independent w.r.t P if and only if:

(i) P ((A ∩ B) ∩ C) = (P (A) · P (B)) · P (C), (ii) P (A ∩ B) = P (A) · P (B), (iii) P (A ∩ C) = P (A) · P (C), (iv) P (B ∩ C) = P (B) · P (C).

We now state a number of propositions:

(29) A and B are independent w.r.t P if and only if P (A∩B) = P (A)·P (B).

(30) A, B and C are independent w.r.t P if and only if the following condi-

tions are satisfied:

(i) P ((A ∩ B) ∩ C) = (P (A) · P (B)) · P (C), (ii) P (A ∩ B) = P (A) · P (B), (iii) P (A ∩ C) = P (A) · P (C), (iv) P (B ∩ C) = P (B) · P (C).

(31) For all A, B, P holds A and B are independent w.r.t P if and only if B and A are independent w.r.t P.

(32) For all A, B, C, P holds A, B and C are independent w.r.t P if and only if P ((A∩B)∩C) = (P (A)·P (B))·P (C) and A and B are independent w.r.t P and B and C are independent w.r.t P and A and C are independent w.r.t P.

(33) For all A, B, C, P such that A, B and C are independent w.r.t P holds B, A and C are independent w.r.t P.

(34) For all A, B, C, P such that A, B and C are independent w.r.t P holds A, C and B are independent w.r.t P.

(35) A and ∅_{Sigma} are independent w.r.t P.

(36) A and Ω_{Sigma} are independent w.r.t P.

(37) For all A, B, P such that A and B are independent w.r.t P holds A and Ω_{Sigma} \ B are independent w.r.t P.

(38) For all A, B, P such that A and B are independent w.r.t P holds Ω_{Sigma} \ A and Ω_{Sigma} \ B are independent w.r.t P.

(39) For all A, B, C, P such that A and B are independent w.r.t P and A and C are independent w.r.t P and B misses C holds A and B ∪ C are independent w.r.t P.

(40) For all P, A, B such that A and B are independent w.r.t P and P (A) <

1 and P (B) < 1 holds P (A ∪ B) < 1. Let us consider Omega, Sigma, P, B. Let us assume that 0 < P (B). The

functor P(P/B) yielding a probability on Sigma is defined by: for every A holds (P(P/B))(A) = \frac{P (A∩B)}{P (B)}. Next we state a number of propositions:

(41) For all P, B such that 0 < P (B) for every A holds P(P/B)(A) = \frac{P (A∩B)}{P (B)}.

(42) For all P, B, A such that 0 < P (B) holds P (A∩B) = P(P/B)(A)·P (B).

(43) For all P, A, B, C such that 0 < P (A ∩ B) holds P ((A ∩ B) ∩ C) = (P (A) · P(P/A)(B)) · P(P/(A ∩ B))(C).

(44) For all P, A, B, C such that C = B^{c} and 0 < P (B) and 0 < P (C) holds P (A) = P(P/B)(A) · P (B) + P(P/C)(A) · P (C).

(45) Given P, A, A_{1}, A_{2}, A_{3}. Suppose A_{1} misses A_{2} and A_{3} = (A_{1}∪A_{2})^{c} and 0 < P (A_{1}) and 0 < P (A_{2}) and 0 < P (A_{3}). Then P (A) = (P(P/A_{1})(A) · P (A_{1}) + P(P/A_{2})(A) · P (A_{2})) + P(P/A_{3})(A) · P (A_{3}).

(46) For all P, A, B such that 0 < P (B) holds P(P/B)(A) = P (A) if and only if A and B are independent w.r.t P.

(47) For all P, A, B such that 0 < P (B) and P (B) < 1 and P(P/B)(A) = P(P/(Ω_{Sigma} \ B))(A) holds A and B are independent w.r.t P.

(48) For all P, A, B such that 0 < P (B) holds \frac{(P (A)+P (B))−1}{P (B)} ≤ P(P/B)(A).

(49) For all A, B, P such that 0 < P (A) and 0 < P (B) holds P(P/B)(A) = \frac{P(P/A)(B)·P (A)}{P (B)}.

(50) Given B, A_{1}, A_{2}, P. Suppose 0 < P (B) and A_{2} = A_{1}^{c} and 0 < P (A_{1})

and 0 < P (A_{2}). Then

(i) P(P/B)(A_{1}) = \frac{P(P/A_{1})(B)·P (A_{1})}{P(P/A_{1})(B)·P (A_{1})+P(P/A_{2})(B)·P (A_{2})},

(ii) P(P/B)(A_{2}) = \frac{P(P/A_{2})(B)·P (A_{2})}{P(P/A_{1})(B)·P (A_{1})+P(P/A_{2})(B)·P (A_{2})}.

(51) Given B, A_{1}, A_{2}, A_{3}, P. Suppose 0 < P (B) and 0 < P (A_{1}) and

0 < P (A_{2}) and 0 < P (A_{3}) and A_{1} misses A_{2} and A_{3} = (A_{1} ∪ A_{2})^{c}. Then (i) P(P/B)(A_{1}) = \frac{P(P/A_{1})(B)·P (A_{1})}{(P(P/A_{1})(B)·P (A_{1})+P(P/A_{2})(B)·P (A_{2}))+P(P/A_{3})(B)·P (A_{3})}, (ii) P(P/B)(A_{2}) = \frac{P(P/A_{2})(B)·P (A_{2})}{(P(P/A_{1})(B)·P (A_{1})+P(P/A_{2})(B)·P (A_{2}))+P(P/A_{3})(B)·P (A_{3})}, (iii) P(P/B)(A_{3}) = \frac{P(P/A_{3})(B)·P (A_{3})}{(P(P/A_{1})(B)·P (A_{1})+P(P/A_{2})(B)·P (A_{2}))+P(P/A_{3})(B)·P (A_{3})}.

(52) For all A, B, P such that 0 < P (B) holds 1− \frac{P (Ω_{Sigma}\A)}{P (B)} ≤ P(P/B)(A).

References

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

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

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

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

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

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

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

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

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

750

Translations in Affine Planes ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. Connections between Minor Desargues Axiom and the transitivity of translation groups are investigated. A formal proof of the theorem which establishes the equivalence of these two properties of affine planes is given. We also prove that, under additional requirement, the plane in question satisfies Fano Axiom; its translation group is uniquely two-divisible.

MML Identifier: TRANSLAC.

The terminology and notation used in this paper are introduced in the following papers: [1], [3], [4], [2], and [5]. We adopt the following rules: AS is an affine space and a, b, c, d, p, q, r, x are elements of the points of AS. Let us consider AS. We say that AS satisfies Fano Axiom if and only if:

for all a, b, c, d such that a, b ∥ c, d and a, c ∥ b, d and a, d ∥ b, c holds L(a, b, c).

The following propositions are true:

(1) AS satisfies Fano Axiom if and only if for all a, b, c, d such that a, b ∥ c, d and a, c ∥ b, d and a, d ∥ b, c holds L(a, b, c).

(2) If there exist a, b, c such that L(a, b, c) and a ≠ b and a ≠ c and b ≠ c, then for all p, q such that p ≠ q there exists r such that L(p, q, r) and p ≠ r and q ≠ r.

(3) If there exist a, b such that a ≠ b and for every x such that L(a, b, x) holds x = a or x = b, then for all p, q, r such that p ≠ q and L(p, q, r) holds r = p or r = q.

We follow a convention: AF P is an affine plane, a, a′, b, b′, c, c′, d, p, q, r, x, y are elements of the points of AF P, and f, g, f_{1}, f_{2} are permutations of the points of AF P. We now state a number of propositions:

1Supported by RPBP.III-24.C2.

(4) If AF P satisfies Fano Axiom and a, b ∥ c, d and a, c ∥ b, d and not L(a, b, c), then there exists p such that L(b, c, p) and L(a, d, p).

(5) If f is a translation and not L(a, f(a), x) and a, f(a) ∥ x, y and a, x ∥ f(a), y, then y = f(x).

(6) AF P satisfies des if and only if for all a, a′, b, c, b′, c′ such that not L(a, a′, b) and not L(a, a′, c) and a, a′ ∥ b, b′ and a, a′ ∥ c, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′ holds b, c ∥ b′, c′.

(7) There exists f such that f is a translation and f(a) = a.

(8) If for all p, q, r such that p ≠ q and L(p, q, r) holds r = p or r = q and a, b ∥ p, q and a, p ∥ b, q and not L(a, b, p), then a, q ∥ b, p.

(9) If AF P satisfies des, then there exists f such that f is a translation and f(a) = b.

(10) If for every a, b there exists f such that f is a translation and f(a) = b, then AF P satisfies des.

(11) If f is a translation and g is a translation and not L(a, f(a), g(a)), then f · g = g · f.

(12) If AF P satisfies des and f is a translation and g is a translation, then f · g = g · f.

(13) If f is a translation and g is a translation and p, f(p) ∥ p, g(p), then p, f(p) ∥ p, (f · g)(p).

(14) If AF P satisfies Fano Axiom and AF P satisfies des and f is a translation, then there exists g such that g is a translation and g · g = f.

(15) If AF P satisfies Fano Axiom and f is a translation and f · f = id_{the points of AF P}, then f = id_{the points of AF P}.

(16) If AF P satisfies des and AF P satisfies Fano Axiom and g is a translation and f_{1} is a translation and f_{2} is a translation and g = f_{1} · f_{1} and g = f_{2} · f_{2}, then f_{1} = f_{2}.

References

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

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Classical configurations in affine planes. Formalized Mathematics, 1(4):625–633, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

Translations in Affine Planes 753

[5] Henryk Oryszczyszyn and Krzysztof Prażmowski. Transformations in affine

spaces. Formalized Mathematics, 1(4):715–723, 1990. Received June 12, 1990 754

Introduction to Probability

Jan Popiołek Warsaw University Białystok

Summary. Definitions of Elementary Event and Event in any sample space E are given. Next, the probability of an Event when E is finite is introduced and some properties of this function are investigated. Last part of the paper is devoted to the conditional probability and essential properties of this function (Bayes Theorem).

MML Identifier: RPR 1.

The articles [7], [8], [3], [6], [5], [2], [4], and [1] provide the terminology and notation for this paper. For simplicity we follow the rules: E will denote a non-empty set, a will denote an element of E, A, B, B_{1}, B_{2}, B_{3}, C will denote subsets of E, X, Y will denote sets, and p will denote a finite sequence. Let us consider E. A subset of E is called an elementary event of E if:

it ⊆ E and it ≠ ∅ but Y ⊆ it if and only if Y = ∅ or Y = it. In the sequel e, e_{1}, e_{2} will denote elementary events of E. One can prove the

following propositions:

(1) If e is an elementary event of E, then e ⊆ E.

(2) If e is an elementary event of E, then e ≠ ∅.

(3) For every e such that e is an elementary event of E holds Y ⊆ e if and only if Y = ∅ or Y = e.

(4) e is an elementary event of E if and only if e ⊆ E and e ≠ ∅ but Y ⊆ e if and only if Y = ∅ or Y = e.

(5) If e is an elementary event of E and e = A ∪ B and A ≠ B, then A = ∅ and B = e or A = e and B = ∅.

(6) If e is an elementary event of E and e = A ∪ B, then A = e and B = e or A = e and B = ∅ or A = ∅ and B = e.

(7) If a ∈ E, then {a} is an elementary event of E.

(8) If {a} is an elementary event of E, then a ∈ E.

(9) a ∈ E if and only if {a} is an elementary event of E.

(10) If e_{1} is an elementary event of E and e_{2} is an elementary event of E and e_{1} ⊆ e_{2}, then e_{1} = e_{2}.

(11) If e is an elementary event of E, then there exists a such that a ∈ E and e = {a}.

(12) For every E there exists e such that e is an elementary event of E.

(13) For every E such that e is an elementary event of E holds e is finite.

(14) If e is an elementary event of E, then there exists p such that p is a

finite sequence of elements of E and rng p = e and len p = 1. Let us consider E. An event of E is a subset of E. The following propositions are true:

(15) For every subset X of E holds X is an event of E.

(16) ∅ is an event of E.

(17) E is an event of E.

(18) If A is an event of E and B is an event of E, then A ∩ B is an event of E.

(19) If A is an event of E and B is an event of E, then A ∪ B is an event of E.

(20) If A ⊆ B and B is an event of E, then A is an event of E.

(21) If A is an event of E, then A^{c} is an event of E.

(22) If e is an elementary event of E and A is an event of E, then e ∩ A = ∅ or e ∩ A = e.

(23) If A is an event of E and B is an event of E, then A \ B is an event of E.

(24) If e is an elementary event of E, then e is an event of E.

(25) If A is an event of E and A ≠ ∅, then there exists e such that e is an elementary event of E and e ⊆ A.

(26) If e is an elementary event of E and A is an event of E and e ⊆ A ∪ A^{c}, then e ⊆ A or e ⊆ A^{c}.

(27) If e_{1} is an elementary event of E and e_{2} is an elementary event of E,

then e_{1} = e_{2} or e_{1} ∩ e_{2} = ∅. Let us consider X, Y. We say that X exclude Y if and only if: X ∩ Y = ∅. Next we state several propositions:

(28) X exclude Y if and only if X ∩ Y = ∅.

(29) If X exclude Y, then Y exclude X.

(30) A exclude A^{c}.

(31) For every A holds A exclude ∅.

(32) A exclude B if and only if A \ B = A.

(33) A ∩ B exclude A \ B.

(34) A ∩ B exclude A ∩ B^{c}.

(35) If A exclude B, then A exclude B ∩ C.

(36) If A exclude B, then A ∩ C exclude B ∩ C.

Let us consider E. Let us assume that E is finite. Let us consider A. The functor P(A) yields a real number and is defined as follows:

P(A) = \frac{card A}{card E}.

Let us consider E. Then Ω_{E} is an event of E. Then ∅_{E} is an event of E. The following propositions are true:

(37) If E is finite and A is an event of E, then P(A) = \frac{card A}{card E}.

(38) If E is finite and e is an elementary event of E, then P(e) = \frac{1}{card E}.

(39) If E is finite, then P(Ω_{E}) = 1.

(40) If E is finite, then P(∅_{E}) = 0.

(41) If E is finite and A is an event of E and B is an event of E and A exclude B, then P(A ∩ B) = 0.

(42) If E is finite and A is an event of E, then P(A) ≤ 1.

(43) If E is finite and A is an event of E, then 0 ≤ P(A).

(44) If E is finite and A is an event of E and B is an event of E and A ⊆ B, then P(A) ≤ P(B).

(46)^{1} If E is finite and A is an event of E and B is an event of E, then

P(A ∪ B) = (P(A) + P(B)) − P(A ∩ B).

(47) If E is finite and A is an event of E and B is an event of E and A exclude B, then P(A ∪ B) = P(A) + P(B).

(48) If E is finite and A is an event of E, then P(A) = 1 − P(A^{c}) and P(A^{c}) = 1 − P(A).

(49) If E is finite and A is an event of E and B is an event of E, then P(A \ B) = P(A) − P(A ∩ B).

(50) If E is finite and A is an event of E and B is an event of E and B ⊆ A, then P(A \ B) = P(A) − P(B).

(51) If E is finite and A is an event of E and B is an event of E, then P(A ∪ B) ≤ P(A) + P(B).

(52) If E is finite and A is an event of E and B is an event of E, then P(A \ B) = P(A ∩ B^{c}).

(53) If E is finite and A is an event of E and B is an event of E, then P(A) = P(A ∩ B) + P(A ∩ B^{c}).

(54) If E is finite and A is an event of E and B is an event of E, then P(A) = P(A ∪ B) − P(B \ A).

(55) If E is finite and A is an event of E and B is an event of E, then P(A) + P(A^{c} ∩ B) = P(B) + P(B^{c} ∩ A).

(56) Suppose E is finite and A is an event of E and B is an event of E and C is an event of E. Then P((A ∪ B) ∪ C) = (((P(A) + P(B)) + P(C)) − ((P(A ∩ B) + P(A ∩ C)) + P(B ∩ C))) + P((A ∩ B) ∩ C).

1The proposition (45) became obvious.

(57) If E is finite and A is an event of E and B is an event of E and C is an event of E and A exclude B and A exclude C and B exclude C, then P((A ∪ B) ∪ C) = (P(A) + P(B)) + P(C).

(58) If E is finite and A is an event of E and B is an event of E, then

P(A) − P(B) ≤ P(A \ B). Let us consider E. Let us assume that E is finite. Let us consider B. Let us

assume that 0 < P(B). Let us consider A. The functor P(A/B) yielding a real number is defined by:

P(A/B) = \frac{P(A∩B)}{P(B)}.

One can prove the following propositions:

(59) If E is finite and A is an event of E and B is an event of E and 0 < P(B), then P(A/B) = \frac{P(A∩B)}{P(B)}.

(60) If E is finite and A is an event of E and B is an event of E and 0 < P(B), then P(A ∩ B) = P(A/B) · P(B).

(61) If E is finite and A is an event of E, then P(A/Ω_{E}) = P(A).

(62) If E is finite, then P(Ω_{E}/Ω_{E}) = 1.

(63) If E is finite, then P(∅_{E}/Ω_{E}) = 0.

(64) If E is finite and A is an event of E and B is an event of E and 0 < P(B), then P(A/B) ≤ 1.

(65) If E is finite and A is an event of E and B is an event of E and 0 < P(B), then 0 ≤ P(A/B).

(66) If E is finite and A is an event of E and B is an event of E and 0 < P(B), then P(A/B) = 1 − \frac{P(B\A)}{P(B)}.

(67) If E is finite and A is an event of E and B is an event of E and 0 < P(B) and A ⊆ B, then P(A/B) = \frac{P(A)}{P(B)}.

(68) If E is finite and A is an event of E and B is an event of E and 0 < P(B) and A exclude B, then P(A/B) = 0.

(69) If E is finite and A is an event of E and B is an event of E and 0 < P(A) and 0 < P(B), then P(A) · P(B/A) = P(B) · P(A/B).

(70) If E is finite and A is an event of E and B is an event of E and 0 < P(B), then P(A/B) = 1 − P(A^{c}/B) and P(A^{c}/B) = 1 − P(A/B).

(71) If E is finite and A is an event of E and B is an event of E and 0 < P(B) and B ⊆ A, then P(A/B) = 1.

(72) If E is finite and B is an event of E and 0 < P(B), then P(Ω_{E}/B) = 1.

(73) If E is finite and A is an event of E and 0 < P(A), then P(A^{c}/A) = 0.

(74) If E is finite and A is an event of E and P(A) < 1, then P(A/A^{c}) = 0.

(75) If E is finite and A is an event of E and B is an event of E and 0 < P(B) and A exclude B, then P(A^{c}/B) = 1.

(76) If E is finite and A is an event of E and B is an event of E and 0 < P(A) and P(B) < 1 and A exclude B, then P(A/B^{c}) = \frac{P(A)}{1−P(B)}.

(77) If E is finite and A is an event of E and B is an event of E and 0 < P(A) and P(B) < 1 and A exclude B, then P(A^{c}/B^{c}) = 1 − \frac{P(A)}{1−P(B)}.

(78) If E is finite and A is an event of E and B is an event of E and C is an event of E and 0 < P(B ∩ C) and 0 < P(C), then P((A ∩ B) ∩ C) = (P(A/(B ∩ C)) · P(B/C)) · P(C).

(79) If E is finite and A is an event of E and B is an event of E and 0 < P(B) and P(B) < 1, then P(A) = P(A/B) · P(B) + P(A/B^{c}) · P(B^{c}).

(80) Suppose E is finite and A is an event of E and B_{1} is an event of E and B_{2} is an event of E and 0 < P(B_{1}) and 0 < P(B_{2}) and B_{1} ∪ B_{2} = E and B_{1} ∩ B_{2} = ∅. Then P(A) = P(A/B_{1}) · P(B_{1}) + P(A/B_{2}) · P(B_{2}).

(81) Suppose that (i) E is finite, (ii) A is an event of E, (iii) B_{1} is an event of E, (iv) B_{2} is an event of E,

(v) B_{3} is an event of E, (vi) 0 < P(B_{1}), (vii) 0 < P(B_{2}), (viii) 0 < P(B_{3}), (ix) (B_{1} ∪ B_{2}) ∪ B_{3} = E,

(x) B_{1} ∩ B_{2} = ∅, (xi) B_{1} ∩ B_{3} = ∅, (xii) B_{2} ∩ B_{3} = ∅.

Then P(A) = (P(A/B_{1}) · P(B_{1}) + P(A/B_{2}) · P(B_{2})) + P(A/B_{3}) · P(B_{3}).

(82) Suppose E is finite and A is an event of E and B_{1} is an event of E and B_{2} is an event of E and 0 < P(A) and 0 < P(B_{1}) and 0 < P(B_{2}) and B_{1} ∪ B_{2} = E and B_{1} ∩ B_{2} = ∅. Then P(B_{1}/A) = \frac{P(A/B_{1})·P(B_{1})}{P(A/B_{1})·P(B_{1})+P(A/B_{2})·P(B_{2})}.

(83) Suppose that (i) E is finite, (ii) A is an event of E, (iii) B_{1} is an event of E, (iv) B_{2} is an event of E,

(v) B_{3} is an event of E, (vi) 0 < P(A), (vii) 0 < P(B_{1}), (viii) 0 < P(B_{2}), (ix) 0 < P(B_{3}),

(x) (B_{1} ∪ B_{2}) ∪ B_{3} = E, (xi) B_{1} ∩ B_{2} = ∅, (xii) B_{1} ∩ B_{3} = ∅, (xiii) B_{2} ∩ B_{3} = ∅.

Then P(B_{1}/A) = \frac{P(A/B_{1})·P(B_{1})}{(P(A/B_{1})·P(B_{1})+P(A/B_{2})·P(B_{2}))+P(A/B_{3})·P(B_{3})}.

Let us consider E, A, B. We say that A and B are independent if and only if:

P(A ∩ B) = P(A) · P(B).

The following propositions are true:

(84) A and B are independent if and only if P(A ∩ B) = P(A) · P(B).

(85) If A and B are independent, then B and A are independent.

(86) If E is finite and A is an event of E and B is an event of E and 0 < P(B) and A and B are independent, then P(A/B) = P(A).

(87) If E is finite and A is an event of E and B is an event of E and P(B) = 0, then A and B are independent.

(88) If E is finite and A is an event of E and B is an event of E and A and B are independent, then A^{c} and B are independent.

(89) If E is finite and A is an event of E and B is an event of E and A exclude B and A and B are independent, then P(A) = 0 or P(B) = 0.

References

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

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

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

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

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

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

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

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

A Construction of Analytical Projective Space

Wojciech Leończuk^{1} Warsaw University Białystok Krzysztof Prażmowski^{2} Warsaw University Białystok

Summary. The collinearity structure denoted by ProjectiveSpace(V) is correlated with a given vector space V (over the field of Reals). It is a formalization of the standard construction of a projective space, where points are interpreted as equivalence classes of the relation of proportionality considered in the set of all non-zero vectors. Then the relation of collinearity corresponds to the relation of linear dependence of vectors. Several facts concerning vectors are proved, which correspond in this language to some classical axioms of projective geometry.

MML Identifier: ANPROJ 1.

The notation and terminology used here are introduced in the following articles: [7], [8], [6], [2], [3], [4], [5], [1], and [9]. We adopt the following rules: V is a real linear space, p, q, r, u, v, w, y, u_{1}, v_{1}, w_{1} are vectors of V, and a, b, c, a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} are real numbers. Let us consider V, p. We say that p is a proper vector if and only if:

p ≠ 0_{V}.

The following proposition is true

(1) p is a proper vector if and only if p ≠ 0_{V}.

Let us consider V, p, q. We say that p and q are proportional if and only if: there exist a, b such that a · p = b · q and a ≠ 0 and b ≠ 0. One can prove the following propositions:

(2) p and q are proportional if and only if there exist a, b such that a·p = b·q and a ≠ 0 and b ≠ 0.

(3) p and p are proportional.

(4) If p and q are proportional, then q and p are proportional.

1Supported by RPBP.III-24.C6.

2Supported by RPBP.III-24.C2.

(5) p and q are proportional if and only if there exists a such that a ≠ 0 and p = a · q.

(6) If p and u are proportional and u and q are proportional, then p and q are proportional.

(7) p and 0_{V} are proportional if and only if p = 0_{V}. Let us consider V, u, v, w. We say that u, v and w are lineary dependent if

and only if:

there exist a, b, c such that (a · u + b · v) + c · w = 0_{V} but a ≠ 0 or b ≠ 0 or c ≠ 0.

We now state a number of propositions:

(8) u, v and w are lineary dependent if and only if there exist a, b, c such that (a · u + b · v) + c · w = 0_{V} but a ≠ 0 or b ≠ 0 or c ≠ 0.

(9) If u and u_{1} are proportional and v and v_{1} are proportional and w and w_{1} are proportional and u, v and w are lineary dependent, then u_{1}, v_{1} and w_{1} are lineary dependent.

(10) If u, v and w are lineary dependent, then u, w and v are lineary dependent and v, u and w are lineary dependent and w, v and u are lineary dependent and w, u and v are lineary dependent and v, w and u are lineary dependent.

(11) If v is a proper vector and w is a proper vector and v and w are not proportional, then v, w and u are lineary dependent if and only if there exist a, b such that u = a · v + b · w.

(12) If p and q are not proportional and a_{1} · p + b_{1} · q = a_{2} · p + b_{2} · q and p is a proper vector and q is a proper vector, then a_{1} = a_{2} and b_{1} = b_{2}.

(13) If u, v and w are not lineary dependent and (a_{1} · u + b_{1} · v) + c_{1} · w = (a_{2} · u + b_{2} · v) + c_{2} · w, then a_{1} = a_{2} and b_{1} = b_{2} and c_{1} = c_{2}.

(14) Suppose p and q are not proportional and u = a_{1} · p + b_{1} · q and v = a_{2} · p + b_{2} · q and a_{1} · b_{2} − a_{2} · b_{1} = 0 and p is a proper vector and q is a proper vector. Then u and v are proportional or u = 0_{V} or v = 0_{V}.

(15) If u = 0_{V} or v = 0_{V} or w = 0_{V}, then u, v and w are lineary dependent.

(16) If u and v are proportional or w and u are proportional or v and w are proportional, then w, u and v are lineary dependent.

(17) If u, v and w are not lineary dependent, then u is a proper vector and v is a proper vector and w is a proper vector and u and v are not proportional and v and w are not proportional and w and u are not proportional.

(18) If p + q = 0_{V}, then p and q are proportional.

(19) If p and q are not proportional and p, q and u are lineary dependent and p, q and v are lineary dependent and p, q and w are lineary dependent and p is a proper vector and q is a proper vector, then u, v and w are lineary dependent.

(20) If u, v and w are not lineary dependent and u, v and p are lineary dependent and v, w and q are lineary dependent, then there exists y such that u, w and y are lineary dependent and p, q and y are lineary dependent and y is a proper vector.

(21) If p and q are not proportional and p is a proper vector and q is a proper vector, then for every u, v there exists y such that y is a proper vector and u, v and y are lineary dependent and u and y are not proportional and v and y are not proportional.

(22) If p, q and r are not lineary dependent, then for all u, v such that u is a proper vector and v is a proper vector and u and v are not proportional there exists y such that y is a proper vector and u, v and y are not lineary dependent.

(23) Suppose u, v and q are lineary dependent and w, y and q are lineary dependent and u, w and p are lineary dependent and v, y and p are lineary dependent and u, y and r are lineary dependent and v, w and r are lineary dependent and p, q and r are lineary dependent and p is a proper vector and q is a proper vector and r is a proper vector. Then u, v and y are lineary dependent or u, v and w are lineary dependent or u, w and y are lineary dependent or v, w and y are lineary dependent.

In the sequel x, y, z are arbitrary and X denotes a set. Let us consider V. The proper vectors ofV yields a set and is defined as follows:

for an arbitrary u holds u ∈ the proper vectors ofV if and only if u ≠ 0_{V} and u is a vector of V.

Next we state three propositions:

(24) For every X holds X = the proper vectors ofV if and only if for an arbitrary u holds u ∈ X if and only if u ≠ 0_{V} and u is a vector of V.

(25) For an arbitrary u such that u ∈ the proper vectors ofV holds u is a vector of V.

(26) For every u holds u ∈ the proper vectors ofV if and only if u is a proper

vector. Let us consider V. The proportionality in V yields an equivalence relation of

the proper vectors ofV and is defined as follows:

for all x, y holds ⟨⟨x, y⟩⟩ ∈ the proportionality in V if and only if x ∈ the proper vectors ofV and y ∈ the proper vectors ofV and there exist vectors u, v of V such that x = u and y = v and u and v are proportional.

We now state three propositions:

(27) For every equivalence relation R of the proper vectors ofV holds R = the proportionality in V if and only if for all x, y holds ⟨⟨x, y⟩⟩ ∈ R if and only if x ∈ the proper vectors ofV and y ∈ the proper vectors ofV and there exist vectors u, v of V such that x = u and y = v and u and v are proportional.

(28) If ⟨⟨x, y⟩⟩ ∈ the proportionality in V, then x is a vector of V and y is a vector of V.

(29) ⟨⟨u, v⟩⟩ ∈ the proportionality in V if and only if u is a proper vector and v is a proper vector and u and v are proportional.

Let us consider V, v. Let us assume that v is a proper vector. The direction of v yields a subset of the proper vectors ofV and is defined by:

the direction of v = [v]_{the proportionality in V}. We now state the proposition

(30) If v is a proper vector, then the direction of v = [v]_{the proportionality in V}. Let us consider V. The projective points overV yields a set and is defined as

follows:

there exists a family Y of subsets of the proper vectors ofV such that Y = Classes(the proportionality in V) and the projective points overV = Y.

The following proposition is true

(31) For every X holds X = the projective points overV if and only if there exists a family Y of subsets of the proper vectors ofV such that Y = Classes(the proportionality in V) and X = Y.

A real linear space is said to be a non-trivial real linear space if: there exists a vector u of it such that u ≠ 0_{it}. The following two propositions are true:

(32) For every real linear space V holds V is a non-trivial real linear space if and only if there exists a vector u of V such that u ≠ 0_{V}.

(33) For every real linear space V holds V is a non-trivial real linear space

if and only if there exists u such that u ∈ the proper vectors ofV. We follow the rules: V will denote a non-trivial real linear space, p, q, r, u,

v, w will denote vectors of V, and y will be arbitrary. Let us consider V. Then the proper vectors ofV is a non-empty set.

Let us consider V. Then the projective points overV is a non-empty set. Next we state two propositions:

(34) If p is a proper vector, then the direction of p is an element of the projective points overV.

(35) If p is a proper vector and q is a proper vector, then the direction of p =

the direction of q if and only if p and q are proportional. Let us consider V. The projective collinearity overV yielding a ternary rela-

tion on the projective points overV is defined by:

for arbitrary x, y, z holds ⟨⟨x, y, z⟩⟩ ∈ the projective collinearity overV if and only if there exist p, q, r such that x = the direction of p and y = the direction of q and z = the direction of r and p is a proper vector and q is a proper vector and r is a proper vector and p, q and r are lineary dependent.

We now state the proposition

(36) Let R be a ternary relation on the projective points overV. Then R = the projective collinearity overV if and only if for arbitrary x, y, z holds ⟨⟨x, y, z⟩⟩ ∈ R if and only if there exist p, q, r such that x = the direction of p and y = the direction of q and z = the direction of r and p is a proper vector and q is a proper vector and r is a proper vector and p, q and r are lineary dependent.

Let us consider V. The projective space over V yields a collinearity structure and is defined by:

the projective space over V = ⟨the projective points overV, the projective collinearity overV⟩.

In the sequel CS will be a collinearity structure. One can prove the following propositions:

(37) For every CS holds CS = the projective space over V if and only if CS = ⟨the projective points overV, the projective collinearity overV⟩.

(38) The projective space over V = ⟨the projective points overV, the projective collinearity overV⟩.

(39) For every V holds the points of the projective space over V = the projective points overV and the collinearity relation of the projective space over V = the projective collinearity overV.

(40) If ⟨⟨x, y, z⟩⟩ ∈ the collinearity relation of the projective space over V, then there exist p, q, r such that x = the direction of p and y = the direction of q and z = the direction of r and p is a proper vector and q is a proper vector and r is a proper vector and p, q and r are lineary dependent.

(41) If u is a proper vector and v is a proper vector and w is a proper vector, then ⟨⟨ the direction of u, the direction of v, the direction of w⟩⟩ ∈ the collinearity relation of the projective space over V if and only if u, v and w are lineary dependent.

(42) x is an element of the points of the projective space over V if and only if there exists u such that u is a proper vector and x = the direction of u.

(43) For every real linear space V and for every vector v of V such that v is a proper vector for every subset X of the proper vectors ofV holds X = the direction of v if and only if X = [v]_{the proportionality in V}.

References

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

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

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

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

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

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

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

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

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

Projective Spaces - Part I

Wojciech Leończuk^{1} Warsaw University Białystok Krzysztof Prażmowski^{2} Warsaw University Białystok

Summary. In the class of all collinearity structures a subclass of (dimension free) projective spaces, defined by means of a suitable axiom system, is singled out. Whenever a real vector space V is at least 3dimensional, the structure ProjectiveSpace(V) is a projective space in the above meaning. Some narrower classes of projective spaces are defined: Fano projective spaces, projective planes, and Fano projective planes. For any of the above classes an explicit axiom system is given, as well as an analytical example. There is also a construction a of 3-dimensional and a 4-dimensional real vector space; these are needed to show appropriate examples of projective spaces.

MML Identifier: ANPROJ 2.

The notation and terminology used here are introduced in the following papers: [1], [5], [7], [6], [3], [4], and [2]. For simplicity we adopt the following rules: V will denote a real linear space, p, q, r, u, v, w, y, u_{1}, v_{1} will denote vectors of V, a, b, c, d, a_{1}, b_{1} will denote real numbers, and z will be arbitrary. We now state three propositions:

(1) Suppose for all a, b, c such that (a · u + b · v) + c · w = 0_{V} holds a = 0 and b = 0 and c = 0. Then u is a proper vector and v is a proper vector and w is a proper vector and u, v and w are not lineary dependent and u and v are not proportional.

(2) Given u, v, u_{1}, v_{1}. Suppose for all a, b, a_{1}, b_{1} such that ((a · u + b · v) + a_{1} · u_{1}) + b_{1} · v_{1} = 0_{V} holds a = 0 and b = 0 and a_{1} = 0 and b_{1} = 0. Then u is a proper vector and v is a proper vector and u and v are not proportional and u_{1} is a proper vector and v_{1} is a proper vector and u_{1} and v_{1} are not proportional and u, v and u_{1} are not lineary dependent and u_{1}, v_{1} and u are not lineary dependent.

1Supported by RPBP.III-24.C6.

2Supported by RPBP.III-24.C2.

(3) Suppose for every w there exist a, b, c such that w = (a · p + b · q) + c · r and for all a, b, c such that (a · p + b · q) + c · r = 0_{V} holds a = 0 and b = 0 and c = 0. Then for every u, u_{1} there exists y such that p, q and y are lineary dependent and u, u_{1} and y are lineary dependent and y is a proper vector.

We follow a convention: A is a non-empty set, f, g, h, f_{1} are elements of ^{A}, and x_{1}, x_{2}, x_{3}, x_{4} are elements of A. We now state a number of propositions:

(4) Suppose x_{1} ∈ A and x_{2} ∈ A and x_{3} ∈ A and x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{2} ≠ x_{3}. Then there exist f, g, h such that for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1 but if z ≠ x_{1}, then f(z) = 0 and for every z such that z ∈ A holds if z = x_{2}, then g(z) = 1 but if z ≠ x_{2}, then g(z) = 0 and for every z such that z ∈ A holds if z = x_{3}, then h(z) = 1 but if z ≠ x_{3}, then h(z) = 0.

(5) Suppose that

(i) x_{1} ∈ A, (ii) x_{2} ∈ A, (iii) x_{3} ∈ A, (iv) x_{1} ≠ x_{2},

(v) x_{1} ≠ x_{3}, (vi) x_{2} ≠ x_{3}, (vii) for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1 but if z ≠ x_{1},

then f(z) = 0, (viii) for every z such that z ∈ A holds if z = x_{2}, then g(z) = 1 but if z ≠ x_{2},

then g(z) = 0, (ix) for every z such that z ∈ A holds if z = x_{3}, then h(z) = 1 but if z ≠ x_{3},

then h(z) = 0. Then for all a, b, c such that +_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)) = 0_{^{A}} holds a = 0 and b = 0 and c = 0.

(6) Suppose x_{1} ∈ A and x_{2} ∈ A and x_{3} ∈ A and x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{2} ≠ x_{3}. Then there exist f, g, h such that for all a, b, c such that +_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)) = 0_{^{A}} holds a = 0 and b = 0 and c = 0.

(7) Suppose that

(i) A = {x_{1}, x_{2}, x_{3}}, (ii) x_{1} ≠ x_{2}, (iii) x_{1} ≠ x_{3}, (iv) x_{2} ≠ x_{3},

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

(vi) for every z such that z ∈ A holds if z = x_{2}, then g(z) = 1 but if z ≠ x_{2},

then g(z) = 0, (vii) for every z such that z ∈ A holds if z = x_{3}, then h(z) = 1 but if z ≠ x_{3},

then h(z) = 0.

Then for every element h′ of ^{A} there exist a, b, c such that h′ = +_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)).

(8) Suppose A = {x_{1}, x_{2}, x_{3}} and x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{2} ≠ x_{3}. Then there exist f, g, h such that for every element h′ of ^{A} there exist a, b, c such that h′ = +_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)).

(9) Suppose A = {x_{1}, x_{2}, x_{3}} and x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{2} ≠ x_{3}. Then there exist f, g, h such that for all a, b, c such that +_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)) = 0_{^{A}} holds a = 0 and b = 0 and c = 0 and for every element h′ of ^{A} there exist a, b, c such that h′ = +_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)).

(10) There exists a non-trivial real linear space V and there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a·u+b·v)+c·w.

(11) Suppose x_{1} ∈ A and x_{2} ∈ A and x_{3} ∈ A and x_{4} ∈ A and x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4} and x_{3} ≠ x_{4}. Then there exist f, g, h, f_{1} such that for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1 but if z ≠ x_{1}, then f(z) = 0 and for every z such that z ∈ A holds if z = x_{2}, then g(z) = 1 but if z ≠ x_{2}, then g(z) = 0 and for every z such that z ∈ A holds if z = x_{3}, then h(z) = 1 but if z ≠ x_{3}, then h(z) = 0 and for every z such that z ∈ A holds if z = x_{4}, then f_{1}(z) = 1 but if z ≠ x_{4}, then f_{1}(z) = 0.

(12) Suppose that (i) x_{1} ∈ A, (ii) x_{2} ∈ A, (iii) x_{3} ∈ A, (iv) x_{4} ∈ A,

(v) x_{1} ≠ x_{2}, (vi) x_{1} ≠ x_{3}, (vii) x_{1} ≠ x_{4}, (viii) x_{2} ≠ x_{3}, (ix) x_{2} ≠ x_{4},

(x) x_{3} ≠ x_{4}, (xi) for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1 but if z ≠ x_{1},

then f(z) = 0, (xii) for every z such that z ∈ A holds if z = x_{2}, then g(z) = 1 but if z ≠ x_{2},

then g(z) = 0, (xiii) for every z such that z ∈ A holds if z = x_{3}, then h(z) = 1 but if z ≠ x_{3},

then h(z) = 0, (xiv) for every z such that z ∈ A holds if z = x_{4}, then f_{1}(z) = 1 but if

z ≠ x_{4}, then f_{1}(z) = 0. Given a, b, c, d. Suppose +_{^{A}} (+_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)), ·_{^{A}}^{}(⟨⟨d, f_{1}⟩⟩)) = 0_{^{A}} . Then a = 0 and b = 0 and c = 0 and d = 0.

(13) Suppose x_{1} ∈ A and x_{2} ∈ A and x_{3} ∈ A and x_{4} ∈ A and x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4} and x_{3} ≠ x_{4}. Then there exist f, g, h, f_{1} such that for all a, b, c, d such that +_{^{A}} (+_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)), ·_{^{A}}^{}(⟨⟨d, f_{1}⟩⟩)) = 0_{^{A}} holds a = 0 and b = 0 and c = 0 and d = 0.

(14) Suppose that (i) A = {x_{1}, x_{2}, x_{3}, x_{4}}, (ii) x_{1} ≠ x_{2}, (iii) x_{1} ≠ x_{3}, (iv) x_{1} ≠ x_{4},

(v) x_{2} ≠ x_{3}, (vi) x_{2} ≠ x_{4}, (vii) x_{3} ≠ x_{4}, (viii) for every z such that z ∈ A holds if z = x_{1}, then f(z) = 1 but if z ≠ x_{1},

then f(z) = 0, (ix) for every z such that z ∈ A holds if z = x_{2}, then g(z) = 1 but if z ≠ x_{2},

then g(z) = 0,

(x) for every z such that z ∈ A holds if z = x_{3}, then h(z) = 1 but if z ≠ x_{3}, then h(z) = 0,

(xi) for every z such that z ∈ A holds if z = x_{4}, then f_{1}(z) = 1 but if z ≠ x_{4}, then f_{1}(z) = 0. Then for every element h′ of ^{A} there exist a, b, c, d such that h′ = +_{^{A}} (+_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)), ·_{^{A}}^{}(⟨⟨d, f_{1}⟩⟩)).

(15) Suppose A = {x_{1}, x_{2}, x_{3}, x_{4}} and x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4} and x_{3} ≠ x_{4}. Then there exist f, g, h, f_{1} such that for every element h′ of ^{A} there exist a, b, c, d such that h′ = +_{^{A}} (+_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)), ·_{^{A}}^{}(⟨⟨d, f_{1}⟩⟩)).

(16) Suppose A = {x_{1}, x_{2}, x_{3}, x_{4}} and x_{1} ≠ x_{2} and x_{1} ≠ x_{3} and x_{1} ≠ x_{4} and x_{2} ≠ x_{3} and x_{2} ≠ x_{4} and x_{3} ≠ x_{4}. Then there exist f, g, h, f_{1} such that for all a, b, c, d such that +_{^{A}} (+_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)), ·_{^{A}}^{}(⟨⟨d, f_{1}⟩⟩)) = 0_{^{A}} holds a = 0 and b = 0 and c = 0 and d = 0 and for every element h′ of ^{A} there exist a, b, c, d such that h′ = +_{^{A}} (+_{^{A}} (+_{^{A}} (·_{^{A}}^{}(⟨⟨a, f⟩⟩), ·_{^{A}}^{}(⟨⟨b, g⟩⟩)), ·_{^{A}}^{}(⟨⟨c, h⟩⟩)), ·_{^{A}}^{}(⟨⟨d, f_{1}⟩⟩)).

(17) There exists a non-trivial real linear space V and there exist u, v, w, u_{1} such that for all a, b, c, d such that ((a · u + b · v) + c · w) + d · u_{1} = 0_{V} holds a = 0 and b = 0 and c = 0 and d = 0 and for every y there exist a, b, c, d such that y = ((a · u + b · v) + c · w) + d · u_{1}.

We follow the rules: V is a non-trivial real linear space, u, v, w, y, w_{1} are vectors of V, and p, p_{1}, p_{2}, q, q_{1}, r, r_{1}, r_{2} are elements of the points of the projective space over V. The following propositions are true:

(18) p, q and r are collinear if and only if there exist u, v, w such that p = the direction of u and q = the direction of v and r = the direction of w and u is a proper vector and v is a proper vector and w is a proper vector and u, v and w are lineary dependent.

(19) p, q and p are collinear and p, p and q are collinear and p, q and q are collinear.

(20) If p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear, then r, r_{1} and r_{2} are collinear.

(21) If p, q and r are collinear, then p, r and q are collinear and q, p and r are collinear and r, q and p are collinear and r, p and q are collinear and q, r and p are collinear.

(22) If p, p_{1} and p_{2} are collinear and p, p_{1} and r are collinear and p_{1}, p_{2} and r_{1} are collinear, then there exists r_{2} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear.

(23) If p, p_{1} and p_{2} are not collinear and p, p_{1} and r are collinear and p_{1}, p_{2} and r_{1} are collinear, then there exists r_{2} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear.

(24) If p, p_{1} and r are collinear and p_{1}, p_{2} and r_{1} are collinear, then there exists r_{2} such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear.

(25) Suppose p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear. Then p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(26) If there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0, then there exist p, q, r such that p, q and r are not collinear.

(27) Suppose there exist u, v, w_{1} such that for all a, b, c such that (a · u + b · v) + c · w_{1} = 0_{V} holds a = 0 and b = 0 and c = 0. Then for every p, q there exists r such that p ≠ r and q ≠ r and p, q and r are collinear.

(28) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w. Then there exist elements x_{1}, x_{2} of the points of the projective space over V such that x_{1} ≠ x_{2} and for every r_{1}, r_{2} there exists q such that x_{1}, x_{2} and q are collinear and r_{1}, r_{2} and q are collinear.

(29) If there exist elements x_{1}, x_{2} of the points of the projective space over V such that x_{1} ≠ x_{2} and for every r_{1}, r_{2} there exists q such that x_{1}, x_{2} and q are collinear and r_{1}, r_{2} and q are collinear, then for every p, p_{1}, q, q_{1} there exists r such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

(30) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w.

Then for every p, p_{1}, q, q_{1} there exists r such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

(31) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w. Then there exist p, q, r such that p, q and r are not collinear and for every p, p_{1}, q, q_{1} there exists r such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

A collinearity structure is said to be a projective space defined in terms of collinearity if:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of it such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear, (ii) for all elements p, q, r of the points of it holds p, q and p are collinear and p, p and q are collinear and p, q and q are collinear, (iii) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of it such that p, p_{1} and r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of it such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear, (iv) for every elements p, q of the points of it there exists an element r of the points of it such that p ≠ r and q ≠ r and p, q and r are collinear,

(v) there exist elements p, q, r of the points of it such that p, q and r are not collinear.

Next we state three propositions:

(32) Let CS be a collinearity structure. Then CS is a projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of CS such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of CS holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of CS such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of CS such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(iv) for every elements p, q of the points of CS there exists an element r of

the points of CS such that p ≠ r and q ≠ r and p, q and r are collinear,

(v) there exist elements p, q, r of the points of CS such that p, q and r

are not collinear.

(33) For every projective space CS defined in terms of collinearity holds CS is a proper collinearity space.

(34) If there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0, then the projective space over V is a

projective space defined in terms of collinearity. A projective space defined in terms of collinearity is called a Fanoian projec-

tive space defined in terms of collinearity if:

Let p_{1}, r_{2}, q, r_{1}, q_{1}, p, r be elements of the points of it . Suppose p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear. Then p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

The following propositions are true:

(35) Let CS be a projective space defined in terms of collinearity. Then CS is a Fanoian projective space defined in terms of collinearity if and only if for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of CS such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(36) Let CS be a collinearity structure. Then CS is a Fanoian projective space defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of CS such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of CS holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for all elements p, p_{1}, p_{2}, r, r_{1} of the points of CS such that p, p_{1} and

r are collinear and p_{1}, p_{2} and r_{1} are collinear there exists an element r_{2} of the points of CS such that p, p_{2} and r_{2} are collinear and r, r_{1} and r_{2} are collinear,

(iv) for every elements p, q of the points of CS there exists an element r of

the points of CS such that p ≠ r and q ≠ r and p, q and r are collinear,

(v) there exist elements p, q, r of the points of CS such that p, q and r are not collinear,

(vi) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of CS such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(37) If there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0, then the projective space over V is a Fanoian projective space defined in terms of collinearity.

A projective space defined in terms of collinearity is called a projective plane defined in terms of collinearity if:

for every elements p, p_{1}, q, q_{1} of the points of it there exists an element r of the points of it such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

We now state three propositions:

(38) For every projective space CP S defined in terms of collinearity holds CP S is a projective plane defined in terms of collinearity if and only if for every elements p, p_{1}, q, q_{1} of the points of CP S there exists an element r of the points of CP S such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

(39) Let CS be a collinearity structure. Then CS is a projective plane defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of CS such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of CS holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for every elements p, q of the points of CS there exists an element r of

the points of CS such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) there exist elements p, q, r of the points of CS such that p, q and r

are not collinear,

(v) for every elements p, p_{1}, q, q_{1} of the points of CS there exists an element r of the points of CS such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

(40) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w. Then the projective space over V is a projective plane defined in terms of collinearity.

A projective plane defined in terms of collinearity is said to be a Fanoian projective plane defined in terms of collinearity if:

Let p_{1}, r_{2}, q, r_{1}, q_{1}, p, r be elements of the points of it . Suppose p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear. Then p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

Next we state four propositions:

(41) Let CS be a projective plane defined in terms of collinearity. Then CS is a Fanoian projective plane defined in terms of collinearity if and only if for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of CS such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(42) Let CS be a collinearity structure. Then CS is a Fanoian projective plane defined in terms of collinearity if and only if the following conditions are satisfied:

(i) for all elements p, q, r, r_{1}, r_{2} of the points of CS such that p ≠ q and p, q and r are collinear and p, q and r_{1} are collinear and p, q and r_{2} are collinear holds r, r_{1} and r_{2} are collinear,

(ii) for all elements p, q, r of the points of CS holds p, q and p are collinear

and p, p and q are collinear and p, q and q are collinear, (iii) for every elements p, q of the points of CS there exists an element r of

the points of CS such that p ≠ r and q ≠ r and p, q and r are collinear, (iv) there exist elements p, q, r of the points of CS such that p, q and r

are not collinear,

(v) for every elements p, p_{1}, q, q_{1} of the points of CS there exists an element r of the points of CS such that p, p_{1} and r are collinear and q, q_{1} and r are collinear,

(vi) for all elements p_{1}, r_{2}, q, r_{1}, q_{1}, p, r of the points of CS such that p_{1}, r_{2} and q are collinear and r_{1}, q_{1} and q are collinear and p_{1}, r_{1} and p are collinear and r_{2}, q_{1} and p are collinear and p_{1}, q_{1} and r are collinear and r_{2}, r_{1} and r are collinear and p, q and r are collinear holds p_{1}, r_{2} and q_{1} are collinear or p_{1}, r_{2} and r_{1} are collinear or p_{1}, r_{1} and q_{1} are collinear or r_{2}, r_{1} and q_{1} are collinear.

(43) Suppose that

(i) there exist u, v, w such that for all a, b, c such that (a·u+b·v)+c·w = 0_{V} holds a = 0 and b = 0 and c = 0 and for every y there exist a, b, c such that y = (a · u + b · v) + c · w. Then the projective space over V is a Fanoian projective plane defined in terms of collinearity.

(44) For every CS being a collinearity structure holds CS is a Fanoian projective plane defined in terms of collinearity if and only if CS is a Fanoian projective space defined in terms of collinearity and for every elements p, p_{1}, q, q_{1} of the points of CS there exists an element r of the points of CS such that p, p_{1} and r are collinear and q, q_{1} and r are collinear.

References

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

[2] Wojciech Leończuk and Krzysztof Prażmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.

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

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

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

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

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

Topological Properties of Subsets in Real Numbers ^{1}

Konrad Raczkowski Warsaw University Białystok Paweł Sadowski Warsaw University Białystok

Summary. The following notions for real subsets are defined: open set, closed set, compact set, intervals and neighbourhoods. In the sequel some theorems involving above mentioned notions are proved.

MML Identifier: RCOMP 1.

The notation and terminology used in this paper have been introduced in the following articles: [9], [3], [10], [1], [2], [7], [5], [6], [4], and [8]. For simplicity we adopt the following convention: n, m are natural numbers, x is arbitrary, s, g, g_{1}, g_{2}, r, p, q are real numbers, s_{1}, s_{2} are sequences of real numbers, and X, Y, Y_{1} are subsets of . In this article we present several logical schemes. The scheme SeqChoice concerns a non-empty set A, and a binary predicate P, and states that:

there exists a function f from into A such that for every element t of holds P[t, f(t)] provided the following requirement is met:

• for every element t of there exists an element ff of A such that P[t, ff].

The scheme RealSeqChoice concerns a binary predicate P, and states that: there exists s_{1} such that for every n holds P[n, s_{1}(n)]

provided the parameter meets the following requirement:

• for every n there exists r such that P[n, r].

We now state several propositions:

(1) X ⊆ Y if and only if for every r such that r ∈ X holds r ∈ Y.

(2) r ∈ X if and only if r ∉ X^{c}.

1Supported by RPBP.III-24.C8.

(3) If there exists x such that x ∈ Y_{1} and Y_{1} ⊆ Y and Y is lower bounded, then Y_{1} is lower bounded.

(4) If there exists x such that x ∈ Y_{1} and Y_{1} ⊆ Y and Y is upper bounded, then Y_{1} is upper bounded.

(5) If there exists x such that x ∈ Y_{1} and Y_{1} ⊆ Y and Y is bounded, then Y_{1} is bounded.

Let us consider g, s. The functor [g, s] yields a subset of and is defined by: [g, s] = {r : g ≤ r ∧ r ≤ s}. Next we state the proposition

(6) [g, s] = {r : g ≤ r ∧ r ≤ s}. Let us consider g, s. The functor ]g, s[ yields a subset of and is defined as

follows:

]g, s[ = {r : g < r ∧ r < s}. Next we state a number of propositions:

(7) ]g, s[ = {r : g < r ∧ r < s}.

(8) r ∈ ]p − g, p + g[ if and only if |r − p| < g.

(9) r ∈ [p, g] if and only if |(p + g) − 2 · r| ≤ g − p.

(10) r ∈ ]p, g[ if and only if |(p + g) − 2 · r| < g − p.

(11) For all g, s such that g ≤ s holds [g, s] = ]g, s[ ∪ {g, s}.

(12) If p ≤ g, then ]g, p[ = ∅.

(13) If p < g, then [g, p] = ∅.

(14) If p = g, then [p, g] = {p} and [g, p] = {p} and ]p, g[ = ∅.

(15) If p < g, then ]p, g[ ≠ ∅ but if p ≤ g, then p ∈ [p, g] and g ∈ [p, g] and [p, g] ≠ ∅ and ]p, g[ ⊆ [p, g].

(16) If r ∈ [p, g] and s ∈ [p, g], then [r, s] ⊆ [p, g].

(17) If r ∈ ]p, g[ and s ∈ ]p, g[, then [r, s] ⊆ ]p, g[.

(18) If p ≤ g, then [p, g] = [p, g] ∪ [g, p].

Let us consider X. We say that X is compact if and only if: for every s_{1} such that rng s_{1} ⊆ X there exists s_{2} such that s_{2} is a subsequence

of s_{1} and s_{2} is convergent and lim s_{2} ∈ X.

Next we state the proposition

(19) X is compact if and only if for every s_{1} such that rng s_{1} ⊆ X there exists s_{2} such that s_{2} is a subsequence of s_{1} and s_{2} is convergent and lim s_{2} ∈ X.

Let us consider X. We say that X is closed if and only if: for every s_{1} such that rng s_{1} ⊆ X and s_{1} is convergent holds lim s_{1} ∈ X. The following proposition is true

(20) X is closed if and only if for every s_{1} such that rng s_{1} ⊆ X and s_{1} is

convergent holds lim s_{1} ∈ X. Let A be a non-empty set, and let X be a subset of A. Then X^{c} is a subset

of A.

Let us consider X. We say that X is open if and only if:

X^{c} is closed.

One can prove the following propositions:

(21) X is open if and only if X^{c} is closed.

(22) For all s, g such that s ≤ g for every s_{1} such that rng s_{1} ⊆ [s, g] holds s_{1} is bounded.

(23) For all s, g such that s ≤ g holds [s, g] is closed.

(24) For all s, g such that s ≤ g holds [s, g] is compact.

(25) For all p, q such that p < q holds ]p, q[ is open.

(26) If X is compact, then X is closed.

(27) Given X, s_{1}. Suppose X ≠ ∅ and rng s_{1} ⊆ X and for every p such that p ∈ X there exist r, n such that 0 < r and for every m such that n < m holds r < |s_{1}(m) − p|. Then for every s_{2} such that s_{2} is a subsequence of s_{1} holds it is not true that: s_{2} is convergent and lim s_{2} ∈ X.

(28) If there exists r such that r ∈ X and X is compact, then X is bounded.

(29) If there exists r such that r ∈ X, then X is compact if and only if X is bounded and X is closed.

(30) For every X such that X ≠ ∅ and X is closed and X is upper bounded holds sup X ∈ X.

(31) For every X such that X ≠ ∅ and X is closed and X is lower bounded holds inf X ∈ X.

(32) For every X such that X ≠ ∅ and X is compact holds sup X ∈ X and inf X ∈ X.

(33) If X is compact and for all g_{1}, g_{2} such that g_{1} ∈ X and g_{2} ∈ X holds

[g_{1}, g_{2}] ⊆ X, then there exist p, g such that X = [p, g]. A subset of is called a real open subset if: it is open. We now state the proposition

(34) For every subset X of holds X is a real open subset if and only if X

is open. Let us consider r. A real open subset is said to be a neighbourhood of r if: there exists g such that 0 < g and it = ]r − g, r + g[. One can prove the following propositions:

(35) For every r and for every real open subset X holds X is a neighbourhood of r if and only if there exists g such that 0 < g and X = ]r − g, r + g[.

(36) For all r, X holds X is a neighbourhood of r if and only if there exists g such that 0 < g and X = ]r − g, r + g[.

(37) For every r and for every neighbourhood N of r holds r ∈ N.

(38) For every r and for every neighbourhoods N_{1}, N_{2} of r there exists a neighbourhood N of r such that N ⊆ N_{1} and N ⊆ N_{2}.

(39) For every real open subset X and for every r such that r ∈ X there exists a neighbourhood N of r such that N ⊆ X.

(40) For every real open subset X and for every r such that r ∈ X there exists g such that 0 < g and ]r − g, r + g[ ⊆ X.

(41) For every X such that for every r such that r ∈ X there exists a neighbourhood N of r such that N ⊆ X holds X is open.

(42) For every X holds for every r such that r ∈ X there exists a neighbourhood N of r such that N ⊆ X if and only if X is open.

(43) If X ≠ ∅ and X is open and X is upper bounded, then sup X ∉ X.

(44) If X ≠ ∅ and X is open and X is lower bounded, then inf X ∉ X.

(45) If X is open and X is bounded and for all g_{1}, g_{2} such that g_{1} ∈ X and g_{2} ∈ X holds [g_{1}, g_{2}] ⊆ X, then there exist p, g such that X = ]p, g[.

References

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

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

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

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

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

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

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

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

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

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

Properties of Real Functions

Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. The list of theorems concerning properties of real sequences and functions is enlarged. (See e.g. [9], [4], [8]). The monotone real functions are introduced and their properties are discussed.

MML Identifier: RFUNCT 2.

The papers [11], [3], [1], [9], [5], [6], [4], [2], [7], [10], and [8] provide the terminology and notation for this paper. For simplicity we follow a convention: x is arbitrary, X, X_{1}, Y denote sets, g, r, r_{1}, r_{2}, p denote real numbers, R denotes a subset of , seq, seq_{1}, seq_{2}, seq_{3} denote sequences of real numbers, Ns denotes an increasing sequence of naturals, n denotes a natural number, and h, h_{1}, h_{2} denote partial functions from to . The following propositions are true:

(1) For all functions F, G and for every X such that X ⊆ dom F and F ^{∘} X ⊆ dom G holds X ⊆ dom(G · F).

(2) For all functions F, G and for every X holds G (F ^{∘} X) · F X = (G · F) X.

(3) For all functions F, G and for all X, X_{1} holds G X_{1}·F X = (G·F) (X ∩ F ^{−1} X_{1}).

(4) For all functions F, G and for every X holds X ⊆ dom(G · F) if and only if X ⊆ dom F and F ^{∘} X ⊆ dom G.

(5) For every function F and for every X holds (F X)^{∘} X = F ^{∘} X.

Let us consider seq. Then rng seq is a subset of .

One can prove the following propositions:

(6) seq_{1} = seq_{2} − seq_{3} if and only if for every n holds seq_{1}(n) = seq_{2}(n) − seq_{3}(n).

(7) rng(seq^{}n) ⊆ rng seq.

1Supported by RPBP.III-24.C.8.

(8) If rng seq ⊆ dom h, then seq(n) ∈ dom h.

(9) x ∈ rng seq if and only if there exists n such that x = seq(n).

(10) seq(n) ∈ rng seq.

(11) If seq_{1} is a subsequence of seq, then rng seq_{1} ⊆ rng seq.

(12) If seq_{1} is a subsequence of seq and seq is non-zero, then seq_{1} is non-zero.

(13) (seq_{1} + seq_{2}) · Ns = seq_{1} · Ns + seq_{2} · Ns and (seq_{1} − seq_{2}) · Ns = seq_{1} · Ns − seq_{2} · Ns and (seq_{1} ⋄ seq_{2}) · Ns = (seq_{1} · Ns) ⋄ (seq_{2} · Ns).

(14) (p ⋄ seq) · Ns = p ⋄ (seq · Ns).

(15) (−seq) · Ns = −seq · Ns and |seq| · Ns = |seq · Ns|.

(16) If seq is non-zero, then (seq · Ns)^{−1} = seq^{−1} · Ns.

(17) If seq is non-zero, then \frac{seq_{1}}{seq} · Ns = \frac{seq_{1}·Ns}{seq·Ns}.

(18) If seq is convergent and for every n holds seq(n) ≤ 0, then lim seq ≤ 0.

(19) If for every n holds seq(n) ∈ Y, then rng seq ⊆ Y.

Let us consider h, seq. Let us assume that rng seq ⊆ dom h. The functor h · seq yields a sequence of real numbers and is defined by:

h · seq = (h qua a function) · seq.

The following propositions are true:

(20) If rng seq ⊆ dom h, then h · seq = (h qua a function) · seq.

(21) If rng seq ⊆ dom h, then (h · seq)(n) = h(seq(n)).

(22) If rng seq ⊆ dom h, then (h · seq)^{}n = h · (seq^{}n).

(23) Suppose rng seq ⊆ dom h_{1} ∩ dom h_{2}. Then (h_{1} + h_{2}) · seq = h_{1} · seq + h_{2} · seq and (h_{1} − h_{2}) · seq = h_{1} · seq − h_{2} · seq and (h_{1} ⋄ h_{2}) · seq = (h_{1} · seq) ⋄ (h_{2} · seq).

(24) If rng seq ⊆ dom h, then (r ⋄ h) · seq = r ⋄ (h · seq).

(25) If rng seq ⊆ dom h, then |h · seq| = |h| · seq and −h · seq = (−h) · seq.

(26) If rng seq ⊆ dom \frac{1}{h}, then h · seq is non-zero.

(27) If rng seq ⊆ dom \frac{1}{h}, then \frac{1}{h} · seq = (h · seq)^{−1}.

(28) If rng seq ⊆ dom h, then (h · seq) · Ns = h · (seq · Ns).

(29) If rng seq_{1} ⊆ dom h and seq_{2} is a subsequence of seq_{1}, then h · seq_{2} is a subsequence of h · seq_{1}.

(30) If h is total, then (h · seq)(n) = h(seq(n)).

(31) If h is total, then h · (seq^{}n) = (h · seq)^{}n.

(32) If h_{1} is total and h_{2} is total, then (h_{1} + h_{2}) · seq = h_{1} · seq + h_{2} · seq and (h_{1} − h_{2}) · seq = h_{1} · seq − h_{2} · seq and (h_{1} ⋄ h_{2}) · seq = (h_{1} · seq) ⋄ (h_{2} · seq).

(33) If h is total, then (r ⋄ h) · seq = r ⋄ (h · seq).

(34) If rng seq ⊆ dom(h X), then h X · seq = h · seq.

(35) If rng seq ⊆ dom(h X) but rng seq ⊆ dom(h Y) or X ⊆ Y, then h X · seq = h Y · seq.

(36) If rng seq ⊆ dom(h X), then |h X · seq| = |h| X · seq.

(37) If rng seq ⊆ dom(h X) and h ^{−1} \{0\} = ∅, then \frac{1}{h} X · seq = (h X · seq)^{−1}.

(38) If rng seq ⊆ dom h, then h^{∘} rng seq = rng(h · seq).

(39) If rng seq ⊆ dom(h_{2} · h_{1}), then h_{2} · (h_{1} · seq) = (h_{2} · h_{1}) · seq.

(40) If h is one-to-one, then (h X)^{−1} = h^{−1} (h^{∘} X).

(41) If rng h is bounded and sup(rng h) = inf(rng h), then h is a constant on dom h.

(42) If Y ⊆ dom h and h^{∘} Y is bounded and sup(h^{∘} Y) = inf(h^{∘} Y), then h

is a constant on Y. We now define four new predicates. Let us consider h, Y. We say that h is

increasing on Y if and only if:

for all r_{1}, r_{2} such that r_{1} ∈ Y ∩ dom h and r_{2} ∈ Y ∩ dom h and r_{1} < r_{2} holds h(r_{1}) < h(r_{2}). We say that h is decreasing on Y if and only if:

for all r_{1}, r_{2} such that r_{1} ∈ Y ∩ dom h and r_{2} ∈ Y ∩ dom h and r_{1} < r_{2} holds h(r_{2}) < h(r_{1}). We say that h is non-decreasing on Y if and only if:

for all r_{1}, r_{2} such that r_{1} ∈ Y ∩ dom h and r_{2} ∈ Y ∩ dom h and r_{1} < r_{2} holds h(r_{1}) ≤ h(r_{2}). We say that h is non-increasing on Y if and only if:

for all r_{1}, r_{2} such that r_{1} ∈ Y ∩ dom h and r_{2} ∈ Y ∩ dom h and r_{1} < r_{2} holds h(r_{2}) ≤ h(r_{1}).

Let us consider h, Y. We say that h is monotone on Y if and only if:

h is non-decreasing on Y or h is non-increasing on Y.

Next we state a number of propositions:

(43) h is increasing on Y if and only if for all r_{1}, r_{2} such that r_{1} ∈ Y ∩dom h and r_{2} ∈ Y ∩ dom h and r_{1} < r_{2} holds h(r_{1}) < h(r_{2}).

(44) h is decreasing on Y if and only if for all r_{1}, r_{2} such that r_{1} ∈ Y ∩dom h and r_{2} ∈ Y ∩ dom h and r_{1} < r_{2} holds h(r_{2}) < h(r_{1}).

(45) h is non-decreasing on Y if and only if for all r_{1}, r_{2} such that r_{1} ∈ Y ∩ dom h and r_{2} ∈ Y ∩ dom h and r_{1} < r_{2} holds h(r_{1}) ≤ h(r_{2}).

(46) h is non-increasing on Y if and only if for all r_{1}, r_{2} such that r_{1} ∈ Y ∩ dom h and r_{2} ∈ Y ∩ dom h and r_{1} < r_{2} holds h(r_{2}) ≤ h(r_{1}).

(47) h is monotone on Y if and only if h is non-decreasing on Y or h is non-increasing on Y.

(48) h is non-decreasing on Y if and only if for all r_{1}, r_{2} such that r_{1} ∈ Y ∩ dom h and r_{2} ∈ Y ∩ dom h and r_{1} ≤ r_{2} holds h(r_{1}) ≤ h(r_{2}).

(49) h is non-increasing on Y if and only if for all r_{1}, r_{2} such that r_{1} ∈ Y ∩ dom h and r_{2} ∈ Y ∩ dom h and r_{1} ≤ r_{2} holds h(r_{2}) ≤ h(r_{1}).

(50) h is increasing on X if and only if h X is increasing on X.

(51) h is decreasing on X if and only if h X is decreasing on X.

(52) h is non-decreasing on X if and only if h X is non-decreasing on X.

(53) h is non-increasing on X if and only if h X is non-increasing on X.

(54) If Y ∩ dom h = ∅, then h is increasing on Y and h is decreasing on Y and h is non-decreasing on Y and h is non-increasing on Y and h is monotone on Y.

(55) If h is increasing on Y, then h is non-decreasing on Y.

(56) If h is decreasing on Y, then h is non-increasing on Y.

(57) If h is a constant on Y, then h is non-decreasing on Y.

(58) If h is a constant on Y, then h is non-increasing on Y.

(59) If h is non-decreasing on Y and h is non-increasing on X, then h is a constant on Y ∩ X.

(60) If X ⊆ Y and h is increasing on Y, then h is increasing on X.

(61) If X ⊆ Y and h is decreasing on Y, then h is decreasing on X.

(62) If X ⊆ Y and h is non-decreasing on Y, then h is non-decreasing on X.

(63) If X ⊆ Y and h is non-increasing on Y, then h is non-increasing on X.

(64) If h is increasing on Y and 0 < r, then r ⋄ h is increasing on Y but if r = 0, then r ⋄ h is a constant on Y but if h is increasing on Y and r < 0, then r ⋄ h is decreasing on Y.

(65) If h is decreasing on Y and 0 < r, then r ⋄ h is decreasing on Y but if h is decreasing on Y and r < 0, then r ⋄ h is increasing on Y.

(66) If h is non-decreasing on Y and 0 ≤ r, then r ⋄ h is non-decreasing on Y but if h is non-decreasing on Y and r ≤ 0, then r ⋄ h is non-increasing on Y.

(67) If h is non-increasing on Y and 0 ≤ r, then r ⋄ h is non-increasing on Y but if h is non-increasing on Y and r ≤ 0, then r ⋄ h is non-decreasing on Y.

(68) If r ∈ (X ∩Y)∩dom(h_{1} +h_{2}), then r ∈ X ∩dom h_{1} and r ∈ Y ∩dom h_{2}.

(69) (i) If h_{1} is increasing on X and h_{2} is increasing on Y, then h_{1} + h_{2} is

increasing on X ∩ Y, (ii) if h_{1} is decreasing on X and h_{2} is decreasing on Y, then h_{1} + h_{2} is

decreasing on X ∩ Y, (iii) if h_{1} is non-decreasing on X and h_{2} is non-decreasing on Y, then h_{1}+h_{2}

is non-decreasing on X ∩ Y, (iv) if h_{1} is non-increasing on X and h_{2} is non-increasing on Y, then h_{1}+h_{2}

is non-increasing on X ∩ Y.

(70) If h_{1} is increasing on X and h_{2} is a constant on Y, then h_{1} + h_{2} is increasing on X ∩ Y but if h_{1} is decreasing on X and h_{2} is a constant on Y, then h_{1} + h_{2} is decreasing on X ∩ Y.

(71) If h_{1} is increasing on X and h_{2} is non-decreasing on Y, then h_{1} + h_{2} is increasing on X ∩ Y.

(72) If h_{1} is non-increasing on X and h_{2} is a constant on Y, then h_{1} + h_{2} is non-increasing on X ∩ Y.

(73) If h_{1} is decreasing on X and h_{2} is non-increasing on Y, then h_{1} + h_{2} is decreasing on X ∩ Y.

(74) If h_{1} is non-decreasing on X and h_{2} is a constant on Y, then h_{1} + h_{2} is non-decreasing on X ∩ Y.

(75) h is increasing on {x}.

(76) h is decreasing on {x}.

(77) h is non-decreasing on {x}.

(78) h is non-increasing on {x}.

(79) id_{R} is increasing on R.

(80) If h is increasing on X, then −h is decreasing on X.

(81) If h is non-decreasing on X, then −h is non-increasing on X.

(82) If h is increasing on [p, g] or h is decreasing on [p, g], then h [p, g] is one-to-one.

(83) If h is increasing on [p, g], then (h [p, g])^{−1} is increasing on h^{∘} [p, g].

(84) If h is decreasing on [p, g], then (h [p, g])^{−1} is decreasing on h^{∘} [p, g].

References

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

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

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

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

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

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

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

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

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

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

786 Jarosław Kotowicz

[11] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathemat-

ics, 1(1):9–11, 1990. Received June 18, 1990

Real Function Continuity ^{1}

Konrad Raczkowski Warsaw University Białystok Paweł Sadowski Warsaw University Białystok

Summary. The continuity of real functions is discussed. There is a function defined on some domain in real numbers which is continuous in a single point and on a subset of domain of the function. Main properties of real continuous functions are proved. Among them there is the Weierstraß Theorem. Algebraic features for real continuous functions are shown. Lipschitzian functions are introduced. The Lipschitz condition entails continuity.

MML Identifier: FCONT 1.

The papers [11], [2], [9], [8], [4], [3], [12], [1], [5], [6], [7], and [10] provide the terminology and notation for this paper. For simplicity we adopt the following rules: n is a natural number, X, X_{1}, Z, Z_{1} are sets, s, g, r, p, x_{0}, x_{1}, x_{2} are real numbers, s_{1} is a sequence of real numbers, Y is a subset of , and f, f_{1}, f_{2} are partial functions from to . Let us consider f, x_{0}. We say that f is continuous in x_{0} if and only if:

x_{0} ∈ dom f and for every s_{1} such that rng s_{1} ⊆ dom f and s_{1} is convergent and lim s_{1} = x_{0} holds f · s_{1} is convergent and f(x_{0}) = lim(f · s_{1}).

Next we state a number of propositions:

(1) For all f, x_{0} holds f is continuous in x_{0} if and only if x_{0} ∈ dom f and for every s_{1} such that rng s_{1} ⊆ dom f and s_{1} is convergent and lim s_{1} = x_{0} holds f · s_{1} is convergent and f(x_{0}) = lim(f · s_{1}).

(2) f is continuous in x_{0} if and only if x_{0} ∈ dom f and for every s_{1} such that rng s_{1} ⊆ dom f and s_{1} is convergent and lim s_{1} = x_{0} and for every n holds s_{1}(n) ≠ x_{0} holds f · s_{1} is convergent and f(x_{0}) = lim(f · s_{1}).

(3) f is continuous in x_{0} if and only if x_{0} ∈ dom f and for every r such that 0 < r there exists s such that 0 < s and for every x_{1} such that x_{1} ∈ dom f and |x_{1} − x_{0}| < s holds |f(x_{1}) − f(x_{0})| < r.

1Supported by RPBP.III-24.C8.

(4) For all f, x_{0} holds f is continuous in x_{0} if and only if x_{0} ∈ dom f and for every neighbourhood N_{1} of f(x_{0}) there exists a neighbourhood N of x_{0} such that for every x_{1} such that x_{1} ∈ dom f and x_{1} ∈ N holds f(x_{1}) ∈ N_{1}.

(5) For all f, x_{0} holds f is continuous in x_{0} if and only if x_{0} ∈ dom f and for every neighbourhood N_{1} of f(x_{0}) there exists a neighbourhood N of x_{0} such that f ^{∘} N ⊆ N_{1}.

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

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

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

(9) If f is continuous in x_{0}, then |f| is continuous in x_{0} and −f is continuous in x_{0}.

(10) If f is continuous in x_{0} and f(x_{0}) ≠ 0, then \frac{1}{f} is continuous in x_{0}.

(11) If f_{1} is continuous in x_{0} and f_{1}(x_{0}) ≠ 0 and f_{2} is continuous in x_{0}, then \frac{f_{2}}{f_{1}} is continuous in x_{0}.

(12) If f_{1} is continuous in x_{0} and f_{2} is continuous in f_{1}(x_{0}), then f_{2} · f_{1} is

continuous in x_{0}. Let us consider f, X. We say that f is continuous on X if and only if: X ⊆ dom f and for every x_{0} such that x_{0} ∈ X holds f X is continuous in

x_{0}.

One can prove the following propositions:

(13) For all f, X holds f is continuous on X if and only if X ⊆ dom f and for every x_{0} such that x_{0} ∈ X holds f X is continuous in x_{0}.

(14) For all X, f holds f is continuous on X if and only if X ⊆ dom f and for every s_{1} such that rng s_{1} ⊆ X and s_{1} is convergent and lim s_{1} ∈ X holds f · s_{1} is convergent and f(lim s_{1}) = lim(f · s_{1}).

(15) f is continuous on X if and only if X ⊆ dom f and for all x_{0}, r such that x_{0} ∈ X and 0 < r there exists s such that 0 < s and for every x_{1} such that x_{1} ∈ X and |x_{1} − x_{0}| < s holds |f(x_{1}) − f(x_{0})| < r.

(16) f is continuous on X if and only if f X is continuous on X.

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

(18) If x_{0} ∈ dom f, then f is continuous on {x_{0}}.

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

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

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

(22) If f is continuous on X, then |f| is continuous on X and −f is continuous on X.

(23) If f is continuous on X and f ^{−1} \{0\} = ∅, then \frac{1}{f} is continuous on X.

(24) If f is continuous on X and (f X)^{−1} \{0\} = ∅, then \frac{1}{f} is continuous on X.

(25) If f_{1} is continuous on X and f_{1} ^{−1} {0} = ∅ and f_{2} is continuous on X, then \frac{f_{2}}{f_{1}} is continuous on X.

(26) If f_{1} is continuous on X and f_{2} is continuous on f_{1} ^{∘} X, then f_{2} · f_{1} is continuous on X.

(27) If f_{1} is continuous on X and f_{2} is continuous on X_{1}, then f_{2} · f_{1} is continuous on X ∩ f_{1} ^{−1} X_{1}.

(28) If f is total and for all x_{1}, x_{2} holds f(x_{1} + x_{2}) = f(x_{1}) + f(x_{2}) and there exists x_{0} such that f is continuous in x_{0}, then f is continuous on .

(29) For every f such that dom f is compact and f is continuous on dom f holds rng f is compact.

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

(31) For every f such that dom f ≠ ∅ and dom f is compact and f is continuous on dom f there exist x_{1}, x_{2} such that x_{1} ∈ dom f and x_{2} ∈ dom f and f(x_{1}) = sup(rng f) and f(x_{2}) = inf(rng f).

(32) For all f, Y such that Y ≠ ∅ and Y ⊆ dom f and Y is compact and f is continuous on Y there exist x_{1}, x_{2} such that x_{1} ∈ Y and x_{2} ∈ Y and f(x_{1}) = sup(f ^{∘} Y) and f(x_{2}) = inf(f ^{∘} Y).

Let us consider f, X. We say that f is Lipschitzian on X if and only if: X ⊆ dom f and there exists r such that 0 < r and for all x_{1}, x_{2} such that

x_{1} ∈ X and x_{2} ∈ X holds |f(x_{1}) − f(x_{2})| ≤ r · |x_{1} − x_{2}|.

One can prove the following propositions:

(33) For every f holds f is Lipschitzian on X if and only if X ⊆ dom f and there exists r such that 0 < r and for all x_{1}, x_{2} such that x_{1} ∈ X and x_{2} ∈ X holds |f(x_{1}) − f(x_{2})| ≤ r · |x_{1} − x_{2}|.

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

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

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

(37) If f_{1} is Lipschitzian on X and f_{2} is Lipschitzian on X_{1} and f_{1} is bounded on Z and f_{2} is bounded on Z_{1}, then f_{1} ⋄ f_{2} is Lipschitzian on ((X ∩ Z) ∩ X_{1}) ∩ Z_{1}.

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

(39) If f is Lipschitzian on X, then −f is Lipschitzian on X and |f| is Lipschitzian on X.

(40) If X ⊆ dom f and f is a constant on X, then f is Lipschitzian on X.

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

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

(43) For every f such that there exists r such that rng f = {r} holds f is continuous on dom f.

(44) If X ⊆ dom f and f is a constant on X, then f is continuous on X.

(45) For every f such that for every x_{0} such that x_{0} ∈ dom f holds f(x_{0}) = x_{0} holds f is continuous on dom f.

(46) If f = id_{dom f}, then f is continuous on dom f.

(47) If Y ⊆ dom f and f Y = id_{Y}, then f is continuous on Y.

(48) If X ⊆ dom f and for every x_{0} such that x_{0} ∈ X holds f(x_{0}) = r·x_{0} +p, then f is continuous on X.

(49) If for every x_{0} such that x_{0} ∈ dom f holds f(x_{0}) = x_{0}^{2}, then f is continuous on dom f.

(50) If X ⊆ dom f and for every x_{0} such that x_{0} ∈ X holds f(x_{0}) = x_{0}^{2}, then f is continuous on X.

(51) If for every x_{0} such that x_{0} ∈ dom f holds f(x_{0}) = |x_{0}|, then f is continuous on dom f.

(52) If X ⊆ dom f and for every x_{0} such that x_{0} ∈ X holds f(x_{0}) = |x_{0}|, then f is continuous on X.

(53) If X ⊆ dom f and f is monotone on X and there exist p, g such that p ≤ g and f ^{∘} X = [p, g], then f is continuous on X.

(54) If p ≤ g and [p, g] ⊆ dom f but f is increasing on [p, g] or f is decreasing on [p, g], then (f [p, g])^{−1} is continuous on f ^{∘} [p, g].

References

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

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

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

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

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

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

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

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

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

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

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

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

792

Real Function Uniform Continuity ^{1}

Jarosław Kotowicz Warsaw University Białystok Konrad Raczkowski Warsaw University Białystok

Summary. The uniform continuity for real functions is introduced. More theorems concerning continuous functions are given. (See [10]) The Darboux Theorem is exposed. Algebraic features for uniformly continuous functions are presented. Various facts, e.g., a continuous function on a compact set is uniformly continuous are proved.

MML Identifier: FCONT 2.

The notation and terminology used in this paper have been introduced in the following articles: [12], [13], [3], [1], [9], [8], [4], [2], [5], [6], [7], [11], and [10]. For simplicity we adopt the following convention: X, X_{1}, Z, Z_{1} are sets, s, g, r, p, x_{1}, x_{2} are real numbers, Y is a subset of , and f, f_{1}, f_{2} are partial functions from to . Let us consider f, X. We say that f is uniformly continuous on X if and only if:

X ⊆ dom f and for every r such that 0 < r there exists s such that 0 < s and for all x_{1}, x_{2} such that x_{1} ∈ X and x_{2} ∈ X and |x_{1} − x_{2}| < s holds |f(x_{1}) − f(x_{2})| < r.

We now state a number of propositions:

(1) Given f, X. Then f is uniformly continuous on X if and only if X ⊆ dom f and for every r such that 0 < r there exists s such that 0 < s and for all x_{1}, x_{2} such that x_{1} ∈ X and x_{2} ∈ X and |x_{1} − x_{2}| < s holds |f(x_{1}) − f(x_{2})| < r.

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

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

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

1Supported by RPBP.III-24.C8.

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

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

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

(8) If f_{1} is uniformly continuous on X and f_{2} is uniformly continuous on X_{1} and f_{1} is bounded on Z and f_{2} is bounded on Z_{1}, then f_{1} ⋄f_{2} is uniformly continuous on ((X ∩ Z) ∩ X_{1}) ∩ Z_{1}.

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

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

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

(12) For every f such that dom f is compact and f is continuous on dom f holds f is uniformly continuous on dom f.

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

(14) For all f, Y such that Y ≠ ∅ and Y ⊆ dom f and Y is compact and f is uniformly continuous on Y there exist x_{1}, x_{2} such that x_{1} ∈ Y and x_{2} ∈ Y and f(x_{1}) = sup(f ^{∘} Y) and f(x_{2}) = inf(f ^{∘} Y).

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

(16) If p ≤ g and f is continuous on [p, g], then for every r such that r ∈ [f(p), f(g)] ∪ [f(g), f(p)] there exists s such that s ∈ [p, g] and r = f(s).

(17) If p ≤ g and f is continuous on [p, g], then for every r such that r ∈ [inf(f^{∘}[p, g]), sup(f^{∘}[p, g])] there exists s such that s ∈ [p, g] and r = f(s).

(18) If f is one-to-one and p ≤ g and f is continuous on [p, g], then f is increasing on [p, g] or f is decreasing on [p, g].

(19) Suppose f is one-to-one and p ≤ g and f is continuous on [p, g]. Then inf(f ^{∘} [p, g]) = f(p) and sup(f ^{∘} [p, g]) = f(g) or inf(f ^{∘} [p, g]) = f(g) and sup(f ^{∘} [p, g]) = f(p).

(20) If p ≤ g and f is continuous on [p, g], then f^{∘}[p, g] = [inf(f^{∘}[p, g]), sup(f^{∘} [p, g])].

(21) If f is one-to-one and p ≤ g and f is continuous on [p, g], then f^{−1} is continuous on [inf(f ^{∘} [p, g]), sup(f ^{∘} [p, g])].

References

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

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

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

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

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

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

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

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

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

[10] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.

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

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

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

796

Real Function Differentiability ^{1}

Konrad Raczkowski Warsaw University Białystok Paweł Sadowski Warsaw University Białystok

Summary. For a real valued function defined on its domain in real numbers the differentiability in a single point and on a subset of the domain is presented. The main elements of differential calculus are developed. The algebraic properties of differential real functions are shown.

MML Identifier: FDIFF 1.

The terminology and notation used here have been introduced in the following articles: [11], [2], [8], [3], [4], [1], [5], [6], [7], [10], and [9]. For simplicity we follow the rules: x, x_{0}, r, p will be real numbers, n will be a natural number, Y will be a subset of , Z will be a real open subset, X will be a set, s_{1} will be a sequence of real numbers, and f, f_{1}, f_{2} will be partial functions from to . We now state the proposition

(1) For every r holds r ∈ Y if and only if r ∈ if and only if Y = .

A sequence of real numbers is called a real sequence convergent to 0 if:

it is non-zero and it is convergent and lim it = 0.

The following proposition is true

(2) For every s_{1} holds s_{1} is a real sequence convergent to 0 if and only if s_{1} is non-zero and s_{1} is convergent and lim s_{1} = 0.

A sequence of real numbers is called a constant real sequence if:

it is constant.

We now state the proposition

(3) For every s_{1} holds s_{1} is a constant real sequence if and only if s_{1} is constant.

In the sequel h will be a real sequence convergent to 0 and c will be a constant real sequence. A partial function from to is called a rest if:

1Supported by RPBP.III-24.C8.

it is total and for every h holds h^{−1}⋄(it·h) is convergent and lim(h^{−1}⋄(it·h)) = 0.

One can prove the following proposition

(4) For every f holds f is a rest if and only if f is total and for every h holds h^{−1} ⋄ (f · h) is convergent and lim(h^{−1} ⋄ (f · h)) = 0.

A partial function from to is called a linear function if: it is total and there exists r such that for every p holds it(p) = r · p. The following proposition is true

(5) For every f holds f is a linear function if and only if f is total and there exists r such that for every p holds f(p) = r · p.

We follow the rules: R, R_{1}, R_{2} are rests and L, L_{1}, L_{2} are linear functions. We now state several propositions:

(6) For all L_{1}, L_{2} holds L_{1} + L_{2} is a linear function and L_{1} − L_{2} is a linear function.

(7) For all r, L holds r ⋄ L is a linear function.

(8) For all R_{1}, R_{2} holds R_{1} + R_{2} is a rest and R_{1} − R_{2} is a rest and R_{1} ⋄ R_{2} is a rest.

(9) For all r, R holds r ⋄ R is a rest.

(10) L_{1} ⋄ L_{2} is a rest.

(11) R ⋄ L is a rest and L ⋄ R is a rest.

Let us consider f, x_{0}. We say that f is differentiable in x_{0} if and only if: there exists a neighbourhood N of x_{0} such that N ⊆ dom f and there exist

L, R such that for every x such that x ∈ N holds f(x) − f(x_{0}) = L(x − x_{0}) + R(x − x_{0}).

The following proposition is true

(12) For all f, x_{0} holds f is differentiable in x_{0} if and only if there exists a neighbourhood N of x_{0} such that N ⊆ dom f and there exist L, R such that for every x such that x ∈ N holds f(x)−f(x_{0}) = L(x−x_{0})+R(x−x_{0}).

Let us consider f, x_{0}. Let us assume that f is differentiable in x_{0}. The functor f′(x_{0}) yields a real number and is defined as follows:

there exists a neighbourhood N of x_{0} such that N ⊆ dom f and there exist L, R such that f′(x_{0}) = L(1) and for every x such that x ∈ N holds f(x)−f(x_{0}) = L(x − x_{0}) + R(x − x_{0}).

The following proposition is true

(13) Given r, f, x_{0}. Suppose f is differentiable in x_{0}. Then r = f′(x_{0}) if and only if there exists a neighbourhood N of x_{0} such that N ⊆ dom f and there exist L, R such that r = L(1) and for every x such that x ∈ N holds f(x) − f(x_{0}) = L(x − x_{0}) + R(x − x_{0}).

Let us consider f, X. We say that f is differentiable on X if and only if: X ⊆ dom f and for every x such that x ∈ X holds f X is differentiable in

x.

The following four propositions are true:

(14) For all f, X holds f is differentiable on X if and only if X ⊆ dom f and for every x such that x ∈ X holds f X is differentiable in x.

(15) If f is differentiable on X, then X is a subset of .

(16) f is differentiable on Z if and only if Z ⊆ dom f and for every x such that x ∈ Z holds f is differentiable in x.

(17) If f is differentiable on Y, then Y is open.

Let us consider f, X. Let us assume that f is differentiable on X. The functor f′_{X} yielding a partial function from to is defined by:

dom(f′_{X}) = X and for every x such that x ∈ X holds (f′_{X})(x) = f′(x). One can prove the following two propositions:

(18) For all f, X and for every partial function F from to such that f is differentiable on X holds F = f′_{X} if and only if dom F = X and for every x such that x ∈ X holds F (x) = f′(x).

(19) For all f, Z such that Z ⊆ dom f and there exists r such that rng f = {r} holds f is differentiable on Z and for every x such that x ∈ Z holds (f′_{Z})(x) = 0.

Let us consider h, n. Then h^{}n is a real sequence convergent to 0. Let us consider c, n. Then c^{}n is a constant real sequence. Next we state a number of propositions:

(20) Given f, x_{0}. Let N be a neighbourhood of x_{0}. Suppose f is differentiable in x_{0} and N ⊆ dom f. Then for all h, c such that rng c = {x_{0}} and rng(h + c) ⊆ N holds h^{−1} ⋄ (f · (h + c) − f · c) is convergent and f′(x_{0}) = lim(h^{−1} ⋄ (f · (h + c) − f · c)).

(21) For all f_{1}, f_{2}, x_{0} such that f_{1} is differentiable in x_{0} and f_{2} is differentiable in x_{0} holds f_{1} + f_{2} is differentiable in x_{0} and (f_{1} + f_{2})′(x_{0}) = f′_{1}(x_{0}) + f′_{2}(x_{0}).

(22) For all f_{1}, f_{2}, x_{0} such that f_{1} is differentiable in x_{0} and f_{2} is differentiable in x_{0} holds f_{1} − f_{2} is differentiable in x_{0} and (f_{1} − f_{2})′(x_{0}) = f′_{1}(x_{0}) − f′_{2}(x_{0}).

(23) For all r, f, x_{0} such that f is differentiable in x_{0} holds r ⋄ f is differentiable in x_{0} and (r ⋄ f)′(x_{0}) = r · (f′(x_{0})).

(24) For all f_{1}, f_{2}, x_{0} such that f_{1} is differentiable in x_{0} and f_{2} is differentiable in x_{0} holds f_{1} ⋄ f_{2} is differentiable in x_{0} and (f_{1} ⋄ f_{2})′(x_{0}) = f_{2}(x_{0}) · (f′_{1}(x_{0})) + f_{1}(x_{0}) · (f′_{2}(x_{0})).

(25) For all f, Z such that Z ⊆ dom f and f Z = id_{Z} holds f is differentiable on Z and for every x such that x ∈ Z holds (f′_{Z})(x) = 1.

(26) For all f_{1}, f_{2}, Z such that Z ⊆ dom(f_{1} + f_{2}) and f_{1} is differentiable on Z and f_{2} is differentiable on Z holds f_{1} + f_{2} is differentiable on Z and for every x such that x ∈ Z holds ((f_{1} + f_{2})′_{Z})(x) = f′_{1}(x) + f′_{2}(x).

(27) For all f_{1}, f_{2}, Z such that Z ⊆ dom(f_{1} − f_{2}) and f_{1} is differentiable on Z and f_{2} is differentiable on Z holds f_{1} − f_{2} is differentiable on Z and for every x such that x ∈ Z holds ((f_{1} − f_{2})′_{Z})(x) = f′_{1}(x) − f′_{2}(x).

(28) For all r, f, Z such that Z ⊆ dom(r ⋄ f) and f is differentiable on Z holds r ⋄ f is differentiable on Z and for every x such that x ∈ Z holds ((r ⋄ f)′_{Z})(x) = r · (f′(x)).

(29) Given f_{1}, f_{2}, Z. Then if Z ⊆ dom(f_{1} ⋄ f_{2}) and f_{1} is differentiable on Z and f_{2} is differentiable on Z, then f_{1}⋄f_{2} is differentiable on Z and for every x such that x ∈ Z holds ((f_{1} ⋄ f_{2})′_{Z})(x) = f_{2}(x) · (f′_{1}(x)) + f_{1}(x) · (f′_{2}(x)).

(30) If Z ⊆ dom f and f is a constant on Z, then f is differentiable on Z and for every x such that x ∈ Z holds (f′_{Z})(x) = 0.

(31) If Z ⊆ dom f and for every x such that x ∈ Z holds f(x) = r·x+p, then f is differentiable on Z and for every x such that x ∈ Z holds (f′_{Z})(x) = r.

(32) If f is differentiable in x_{0}, then f is continuous in x_{0}.

(33) If f is differentiable on X, then f is continuous on X.

(34) If f is differentiable on X and Z ⊆ X, then f is differentiable on Z.

(35) If f is differentiable in x_{0}, then there exists R such that R(0) = 0 and R is continuous in 0.

References

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

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

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

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

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

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

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

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

[9] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.

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

Real Function Differentiability 801

[11] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathemat-

ics, 1(1):9–11, 1990. Received June 18, 1990 802

Average Value Theorems for Real Functions of One Variable ^{1}

Jarosław Kotowicz Warsaw University

Białystok Konrad Raczkowski Warsaw University Białystok Paweł Sadowski Warsaw University Białystok

Summary. Three basic theorems in differential calculus of one variable functions are presented: Rolle Theorem, Lagrange Theorem and Cauchy Theorem. There are also direct conclusions.

MML Identifier: ROLLE.

The terminology and notation used here have been introduced in the following papers: [2], [1], [3], [4], [5], [8], [6], and [7]. We adopt the following rules: g, r, s, p, t, x, x_{0}, x_{1} will denote real numbers and f, f_{1}, f_{2} will denote partial functions from to . We now state a number of propositions:

(1) For all p, g such that p < g for every f such that f is continuous on [p, g] and f(p) = f(g) and f is differentiable on ]p, g[ there exists x_{0} such that x_{0} ∈ ]p, g[ and f′(x_{0}) = 0.

(2) Given x, t. Suppose 0 < t. Then for every f such that f is continuous on [x, x + t] and f(x) = f(x + t) and f is differentiable on ]x, x + t[ there exists s such that 0 < s and s < 1 and f′(x + s · t) = 0.

(3) For all p, g such that p < g for every f such that f is continuous on [p, g] and f is differentiable on ]p, g[ there exists x_{0} such that x_{0} ∈ ]p, g[ and f′(x_{0}) = \frac{f(g)−f(p)}{g−p}.

(4) Given x, t. Suppose 0 < t. Then for every f such that f is continuous on [x, x + t] and f is differentiable on ]x, x + t[ there exists s such that 0 < s and s < 1 and f(x + t) = f(x) + t · (f′(x + s · t)).

(5) Given p, g. Suppose p < g. Given f_{1}, f_{2}. Suppose f_{1} is continuous on [p, g] and f_{1} is differentiable on ]p, g[ and f_{2} is continuous on [p, g] and f_{2} is differentiable on ]p, g[. Then there exists x_{0} such that x_{0} ∈ ]p, g[ and (f_{1}(g) − f_{1}(p)) · (f′_{2}(x_{0})) = (f_{2}(g) − f_{2}(p)) · (f′_{1}(x_{0})).

1Supported by RPBP.III-24.C8.

(6) Given x, t. Suppose 0 < t. Given f_{1}, f_{2}. Suppose f_{1} is continuous on [x, x + t] and f_{1} is differentiable on ]x, x + t[ and f_{2} is continuous on [x, x + t] and f_{2} is differentiable on ]x, x + t[ and for every x_{1} such that x_{1} ∈ ]x, x + t[ holds f′_{2}(x_{1}) ≠ 0. Then there exists s such that 0 < s and s < 1 and \frac{f_{1}(x+t)−f_{1}(x)}{f_{2}(x+t)−f_{2}(x)} = \frac{f′_{1}(x+s·t)}{f′_{2}(x+s·t)}.

(7) For all p, g such that p < g for every f such that f is differentiable on ]p, g[ and for every x such that x ∈ ]p, g[ holds f′(x) = 0 holds f is a constant on ]p, g[.

(8) Given p, g. Suppose p < g. Given f_{1}, f_{2}. Suppose f_{1} is differentiable on ]p, g[ and f_{2} is differentiable on ]p, g[ and for every x such that x ∈ ]p, g[ holds f′_{1}(x) = f′_{2}(x). Then f_{1} − f_{2} is a constant on ]p, g[ and there exists r such that for every x such that x ∈ ]p, g[ holds f_{1}(x) = f_{2}(x) + r.

(9) For all p, g such that p < g for every f such that f is differentiable on ]p, g[ and for every x such that x ∈ ]p, g[ holds 0 < f′(x) holds f is increasing on ]p, g[.

(10) For all p, g such that p < g for every f such that f is differentiable on ]p, g[ and for every x such that x ∈ ]p, g[ holds f′(x) < 0 holds f is decreasing on ]p, g[.

(11) For all p, g such that p < g for every f such that f is differentiable on ]p, g[ and for every x such that x ∈ ]p, g[ holds 0 ≤ f′(x) holds f is non-decreasing on ]p, g[.

(12) For all p, g such that p < g for every f such that f is differentiable on ]p, g[ and for every x such that x ∈ ]p, g[ holds f′(x) ≤ 0 holds f is non-increasing on ]p, g[.

References

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

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

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

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

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

[6] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.

[7] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

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

Index of MML Identifiers

AFF 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .625 AFF 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .635 ANPROJ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 ANPROJ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 CAT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .725 COLLSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 CQC LANG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 CQC THE1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 FCONT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 FCONT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793 FDIFF 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 FINSEQOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 MARGREL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 ORDINAL4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 PARTFUN2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 PASCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 PROB 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .745 RCOMP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 REAL LAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 RFUNCT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 RFUNCT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 ROLLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803 RPR 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .755 RVSUM 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 SETWOP 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 TDGROUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 TRANSGEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 TRANSLAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 VALUAT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

Contents Form. Math. 1 (4)

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Classical Configurations in Affine Planes

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

Affine Localizations of Desargues Axiom

By Eugeniusz Kusak et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

Binary Operations Applied to Finite Sequences

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

Semigroup operations on finite subsets

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

The Collinearity Structure

By Wojciech Skaba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

The Sum and Product of Finite Sequences of Real Numbers

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

A Classical First Order Language

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

Classical and Non–classical Pasch Configurations in Ordered Affine Planes

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677

The Lattice of Real Numbers. The Lattice of Real Functions

By Marek Chmur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

A Construction of an Abstract Space of Congruence of Vectors

By Grzegorz Lewandowski et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

A First-Order Predicate Calculus

By Agata Darmochwał . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

Continued on inside back cover

Partial Functions from a Domain to a Domain

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

Partial Functions from a Domain to the Set of Real Numbers

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

Increasing and Continuous Ordinal Sequences

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

Transformations in Affine Spaces

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

Subcategories and Products of Categories

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725

Many-Argument Relations

By Edmund Woronowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

Interpretation and Satisfiability in the First Order Logic

By Edmund Woronowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

Probability

By Andrzej Ne‘dzusiak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

Translations in Affine Planes

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751

Introduction to Probability

By Jan Popiołek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755

A Construction of Analytical Projective Space

By Wojciech Leończuk et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

Projective Spaces - Part I

By Wojciech Leończuk et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

Topological Properties of Subsets in Real Numbers

By Konrad Raczkowski and Paweł Sadowski . . . . . . . . . . . . 777

Properties of Real Functions

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

Real Function Continuity

By Konrad Raczkowski and Paweł Sadowski . . . . . . . . . . . . 787

Real Function Uniform Continuity

By Jarosław Kotowicz and Konrad Raczkowski . . . . . . . 793

Real Function Differentiability

By Konrad Raczkowski and Paweł Sadowski . . . . . . . . . . . . 797

Average Value Theorems for Real Functions of One Variable

By Jarosław Kotowicz et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806

Continued on inside back cover