FORMALIZED MATHEMATICS Volume 3, Number 1, 1992 Université Catholique de Louvain

Continuity of Mappings over the Union of Subspaces

Zbigniew Karno Warsaw University Białystok

Summary. Let X and Y be topological spaces and let X_{1} and X_{2} be subspaces of X. Let f : X_{1} ∪ X_{2} → Y be a mapping defined on the union of X_{1} and X_{2} such that the restriction mappings f_{|X_{1}} and f_{|X_{2}} are continuous. It is well known that if X_{1} and X_{2} are both open (closed) subspaces of X, then f is continuous (see e.g. [6, p.106]).

The aim is to show, using Mizar System, the following theorem (see Section 5): If X_{1} and X_{2} are weakly separated, then f is continuous (compare also [15, p.358] for related results). This theorem generalizes the preceding one because if X_{1} and X_{2} are both open (closed), then these subspaces are weakly separated (see [5]). However, the following problem remains open.

Problem 1. Characterize the class of pairs of subspaces X_{1} and X_{2} of a topological space X such that (∗) for any topological space Y and for any mapping f : X_{1} ∪ X_{2} → Y, f is continuous if the restrictions f_{|X_{1}} and f_{|X_{2}} are continuous.

In some special case we have the following characterization: X_{1} and X_{2} are separated iff X_{1} misses X_{2} and the condition (∗) is fulfilled. In connection with this fact we hope that the following specification of the preceding problem has an affirmative answer.

Problem 2. Suppose the condition (∗) is fulfilled. Must X_{1} and X_{2} be weakly separated ?

Note that in the last section the concept of the union of two mappings is introduced and studied. In particular, all results presented above are reformulated using this notion. In the remaining sections we introduce concepts needed for the formulation and the proof of theorems on properties of continuous mappings, restriction mappings and modifications of the topology.

MML Identifier: TMAP 1.

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

1. Set-Theoretic Preliminaries

In the sequel A, B will denote non-empty sets. Next we state several propositions:

(1) For every function f from A into B and for every subset A_{0} of A and for every subset B_{0} of B holds f ^{∘} A_{0} ⊆ B_{0} if and only if A_{0} ⊆ f ^{−1} B_{0}.

(2) For every function f from A into B and for every non-empty subset A_{0} of A and for every function f_{0} from A_{0} into B such that for every element c of A such that c ∈ A_{0} holds f(c) = f_{0}(c) holds f A_{0} = f_{0}.

(3) For every function f from A into B and for every non-empty subset A_{0} of A and for every element c of A such that c ∈ A_{0} holds f(c) = (f A_{0})(c).

(4) For every function f from A into B and for every non-empty subset A_{0} of A and for every subset C of A such that C ⊆ A_{0} holds f^{∘}C = (f A_{0})^{∘}C.

(5) For every function f from A into B and for every non-empty subset A_{0} of A and for every subset D of B such that f ^{−1} D ⊆ A_{0} holds f ^{−1} D = (f A_{0})^{−1} D.

Let A, B be non-empty sets, and let A_{1}, A_{2} be non-empty subsets of A, and let f_{1} be a function from A_{1} into B, and let f_{2} be a function from A_{2} into B. Let us assume that f_{1} (A_{1} ∩ A_{2}) = f_{2} (A_{1} ∩ A_{2}). The functor f_{1} ∪ f_{2} yielding a function from A_{1} ∪ A_{2} into B is defined by: (Def.1) (f_{1} ∪ f_{2}) A_{1} = f_{1} and (f_{1} ∪ f_{2}) A_{2} = f_{2}.

The following proposition is true

(6) Let A, B be non-empty sets. Then for all non-empty subsets A_{1}, A_{2} of A such that A_{1} misses A_{2} and for every function f_{1} from A_{1} into B and for every function f_{2} from A_{2} into B holds f_{1} (A_{1} ∩ A_{2}) = f_{2} (A_{1} ∩ A_{2}) and (f_{1} ∪ f_{2}) A_{1} = f_{1} and (f_{1} ∪ f_{2}) A_{2} = f_{2}.

We follow the rules: A, B are non-empty sets and A_{1}, A_{2}, A_{3} are non-empty subsets of A. We now state four propositions:

(7) For every function g from A_{1} ∪A_{2} into B and for every function g_{1} from A_{1} into B and for every function g_{2} from A_{2} into B such that g A_{1} = g_{1} and g A_{2} = g_{2} holds g = g_{1} ∪ g_{2}.

(8) For every function f_{1} from A_{1} into B and for every function f_{2} from A_{2} into B such that f_{1} (A_{1} ∩ A_{2}) = f_{2} (A_{1} ∩ A_{2}) holds f_{1} ∪ f_{2} = f_{2} ∪ f_{1}.

(9) Let A_{12}, A_{23} be non-empty subsets of A. Suppose A_{12} = A_{1} ∪ A_{2} and A_{23} = A_{2} ∪ A_{3}. Let f_{1} be a function from A_{1} into B. Let f_{2} be a function from A_{2} into B. Let f_{3} be a function from A_{3} into B. Suppose f_{1} (A_{1} ∩ A_{2}) = f_{2} (A_{1} ∩ A_{2}) and f_{2} (A_{2} ∩ A_{3}) = f_{3} (A_{2} ∩ A_{3}) and f_{1} (A_{1} ∩ A_{3}) = f_{3} (A_{1} ∩ A_{3}). Then for every function f_{12} from A_{12} into B and for every function f_{23} from A_{23} into B such that f_{12} = f_{1} ∪ f_{2} and f_{23} = f_{2} ∪ f_{3} holds f_{12} ∪ f_{3} = f_{1} ∪ f_{23}.

(10) For every function f_{1} from A_{1} into B and for every function f_{2} from A_{2} into B such that f_{1} (A_{1} ∩ A_{2}) = f_{2} (A_{1} ∩ A_{2}) holds A_{1} is a subset of A_{2} if and only if f_{1} ∪ f_{2} = f_{2} but A_{2} is a subset of A_{1} if and only if f_{1} ∪ f_{2} = f_{1}.

2. Selected Properties of Subspaces of Topological Spaces

In the sequel X is a topological space. Next we state four propositions:

(11) For every subspace X_{0} of X holds the topological structure of X_{0} is a strict subspace of X.

(12) For all topological spaces X_{1}, X_{2} such that X_{1} = the topological structure of X_{2} holds X_{1} is a subspace of X if and only if X_{2} is a subspace of X.

(13) For all topological spaces X_{1}, X_{2} such that X_{2} = the topological structure of X_{1} holds X_{1} is a closed subspace of X if and only if X_{2} is a closed subspace of X.

(14) For all topological spaces X_{1}, X_{2} such that X_{2} = the topological structure of X_{1} holds X_{1} is an open subspace of X if and only if X_{2} is an open subspace of X.

In the sequel X_{1}, X_{2} will denote subspaces of X. Next we state several propositions:

(15) If X_{1} is a subspace of X_{2}, then for every point x_{1} of X_{1} there exists a point x_{2} of X_{2} such that x_{2} = x_{1}.

(16) For every point x of X_{1} ∪ X_{2} holds there exists a point x_{1} of X_{1} such that x_{1} = x or there exists a point x_{2} of X_{2} such that x_{2} = x.

(17) If X_{1} meets X_{2}, then for every point x of X_{1} ∩ X_{2} holds there exists a point x_{1} of X_{1} such that x_{1} = x and there exists a point x_{2} of X_{2} such that x_{2} = x.

(18) For every point x of X_{1} ∪ X_{2} and for every subset F_{1} of X_{1} and for every subset F_{2} of X_{2} such that F_{1} is closed and x ∈ F_{1} and F_{2} is closed and x ∈ F_{2} there exists a subset H of X_{1} ∪ X_{2} such that H is closed and x ∈ H and H ⊆ F_{1} ∪ F_{2}.

(19) For every point x of X_{1} ∪ X_{2} and for every subset U_{1} of X_{1} and for every subset U_{2} of X_{2} such that U_{1} is open and x ∈ U_{1} and U_{2} is open and x ∈ U_{2} there exists a subset V of X_{1} ∪ X_{2} such that V is open and x ∈ V and V ⊆ U_{1} ∪ U_{2}.

(20) For every point x of X_{1} ∪ X_{2} and for every point x_{1} of X_{1} and for every point x_{2} of X_{2} such that x_{1} = x and x_{2} = x and for every neighbourhood A_{1} of x_{1} and for every neighbourhood A_{2} of x_{2} there exists a subset V of X_{1} ∪ X_{2} such that V is open and x ∈ V and V ⊆ A_{1} ∪ A_{2}.

(21) For every point x of X_{1} ∪ X_{2} and for every point x_{1} of X_{1} and for every point x_{2} of X_{2} such that x_{1} = x and x_{2} = x and for every neighbourhood A_{1} of x_{1} and for every neighbourhood A_{2} of x_{2} there exists a neighbourhood A of x such that A ⊆ A_{1} ∪ A_{2}.

In the sequel X_{0}, X_{1}, X_{2}, Y_{1}, Y_{2} will be subspaces of X. One can prove the following propositions:

(22) If X_{0} is a subspace of X_{1}, then X_{0} meets X_{1} and X_{1} meets X_{0}.

(23) If X_{0} is a subspace of X_{1} but X_{0} meets X_{2} or X_{2} meets X_{0}, then X_{1} meets X_{2} and X_{2} meets X_{1}.

(24) If X_{0} is a subspace of X_{1} but X_{1} misses X_{2} or X_{2} misses X_{1}, then X_{0} misses X_{2} and X_{2} misses X_{0}.

(25) X_{0} ∪ X_{0} = the topological structure of X_{0}.

(26) X_{0} ∩ X_{0} = the topological structure of X_{0}.

(27) If Y_{1} is a subspace of X_{1} and Y_{2} is a subspace of X_{2}, then Y_{1} ∪ Y_{2} is a subspace of X_{1} ∪ X_{2}.

(28) If Y_{1} meets Y_{2} and Y_{1} is a subspace of X_{1} and Y_{2} is a subspace of X_{2}, then Y_{1} ∩ Y_{2} is a subspace of X_{1} ∩ X_{2}.

(29) If X_{1} is a subspace of X_{0} and X_{2} is a subspace of X_{0}, then X_{1} ∪ X_{2} is a subspace of X_{0}.

(30) If X_{1} meets X_{2} and X_{1} is a subspace of X_{0} and X_{2} is a subspace of X_{0}, then X_{1} ∩ X_{2} is a subspace of X_{0}.

(31) (i) If X_{1} misses X_{0} or X_{0} misses X_{1} but X_{2} meets X_{0} or X_{0} meets X_{2},

then (X_{1} ∪ X_{2}) ∩ X_{0} = X_{2} ∩ X_{0} and X_{0} ∩ (X_{1} ∪ X_{2}) = X_{0} ∩ X_{2}, (ii) if X_{1} meets X_{0} or X_{0} meets X_{1} but X_{2} misses X_{0} or X_{0} misses X_{2},

then (X_{1} ∪ X_{2}) ∩ X_{0} = X_{1} ∩ X_{0} and X_{0} ∩ (X_{1} ∪ X_{2}) = X_{0} ∩ X_{1}.

(32) If X_{1} meets X_{2}, then if X_{1} is a subspace of X_{0}, then X_{1} ∩ X_{2} is a subspace of X_{0} ∩ X_{2} but if X_{2} is a subspace of X_{0}, then X_{1} ∩ X_{2} is a subspace of X_{1} ∩ X_{0}.

(33) If X_{1} is a subspace of X_{0} but X_{0} misses X_{2} or X_{2} misses X_{0}, then X_{0}∩(X_{1}∪X_{2}) = the topological structure of X_{1} and X_{0}∩(X_{2}∪X_{1}) = the topological structure of X_{1}.

(34) If X_{1} meets X_{2}, then if X_{1} is a subspace of X_{0}, then X_{0} ∩ X_{2} meets X_{1} and X_{2} ∩ X_{0} meets X_{1} but if X_{2} is a subspace of X_{0}, then X_{1} ∩ X_{0} meets X_{2} and X_{0} ∩ X_{1} meets X_{2}.

(35) If X_{1} is a subspace of Y_{1} and X_{2} is a subspace of Y_{2} but Y_{1} misses Y_{2} or Y_{1} ∩ Y_{2} misses X_{1} ∪ X_{2}, then Y_{1} misses X_{2} and Y_{2} misses X_{1}.

(36) Suppose X_{1} is not a subspace of X_{2} and X_{2} is not a subspace of X_{1} and X_{1} ∪ X_{2} is a subspace of Y_{1} ∪ Y_{2} and Y_{1} ∩ (X_{1} ∪ X_{2}) is a subspace of X_{1} and Y_{2} ∩ (X_{1} ∪ X_{2}) is a subspace of X_{2}. Then Y_{1} meets X_{1} ∪ X_{2} and Y_{2} meets X_{1} ∪ X_{2}.

(37) Suppose that (i) X_{1} meets X_{2}, (ii) X_{1} is not a subspace of X_{2}, (iii) X_{2} is not a subspace of X_{1}, (iv) the topological structure of X = Y_{1} ∪ Y_{2} ∪ X_{0},

(v) Y_{1} ∩ (X_{1} ∪ X_{2}) is a subspace of X_{1}, (vi) Y_{2} ∩ (X_{1} ∪ X_{2}) is a subspace of X_{2}, (vii) X_{0} ∩ (X_{1} ∪ X_{2}) is a subspace of X_{1} ∩ X_{2}.

Then Y_{1} meets X_{1} ∪ X_{2} and Y_{2} meets X_{1} ∪ X_{2}.

(38) Suppose that (i) X_{1} meets X_{2}, (ii) X_{1} is not a subspace of X_{2}, (iii) X_{2} is not a subspace of X_{1}, (iv) X_{1} ∪ X_{2} is not a subspace of Y_{1} ∪ Y_{2},

(v) the topological structure of X = Y_{1} ∪ Y_{2} ∪ X_{0}, (vi) Y_{1} ∩ (X_{1} ∪ X_{2}) is a subspace of X_{1}, (vii) Y_{2} ∩ (X_{1} ∪ X_{2}) is a subspace of X_{2}, (viii) X_{0} ∩ (X_{1} ∪ X_{2}) is a subspace of X_{1} ∩ X_{2}.

Then Y_{1} ∪ Y_{2} meets X_{1} ∪ X_{2} and X_{0} meets X_{1} ∪ X_{2}.

(39) X_{1} ∪ X_{2} meets X_{0} if and only if X_{1} meets X_{0} or X_{2} meets X_{0} but X_{0} meets X_{1} ∪ X_{2} if and only if X_{0} meets X_{1} or X_{0} meets X_{2}.

(40) X_{1} ∪ X_{2} misses X_{0} if and only if X_{1} misses X_{0} and X_{2} misses X_{0} but X_{0} misses X_{1} ∪ X_{2} if and only if X_{0} misses X_{1} and X_{0} misses X_{2}.

(41) If X_{1} meets X_{2}, then if X_{1} ∩ X_{2} meets X_{0}, then X_{1} meets X_{0} and X_{2} meets X_{0} but if X_{0} meets X_{1} ∩ X_{2}, then X_{0} meets X_{1} and X_{0} meets X_{2}.

(42) If X_{1} meets X_{2}, then if X_{1} misses X_{0} or X_{2} misses X_{0}, then X_{1} ∩ X_{2} misses X_{0} but if X_{0} misses X_{1} or X_{0} misses X_{2}, then X_{0} misses X_{1} ∩ X_{2}.

(43) For every closed subspace X_{0} of X such that X_{0} meets X_{1} holds X_{0}∩X_{1} is a closed subspace of X_{1}.

(44) For every open subspace X_{0} of X such that X_{0} meets X_{1} holds X_{0} ∩X_{1} is an open subspace of X_{1}.

(45) For every closed subspace X_{0} of X such that X_{1} is a subspace of X_{0} and X_{0} misses X_{2} holds X_{1} is a closed subspace of X_{1} ∪ X_{2} and X_{1} is a closed subspace of X_{2} ∪ X_{1}.

(46) For every open subspace X_{0} of X such that X_{1} is a subspace of X_{0} and X_{0} misses X_{2} holds X_{1} is an open subspace of X_{1} ∪ X_{2} and X_{1} is an open subspace of X_{2} ∪ X_{1}.

3. Continuity of Mappings

We now define two new constructions. Let X, Y be topological spaces. A mapping from X into Y is a function from the carrier of X into the carrier of Y.

We say that f is continuous at x if and only if: (Def.2) for every neighbourhood G of f(x) there exists a neighbourhood H of

x such that f ^{∘} H ⊆ G. In the sequel X, Y denote topological spaces and f denotes a mapping from

X into Y. One can prove the following propositions:

(47) For every point x of X holds f is continuous at x if and only if for every neighbourhood G of f(x) holds f ^{−1} G is a neighbourhood of x.

(48) For every point x of X holds f is continuous at x if and only if for every subset G of Y such that G is open and f(x) ∈ G there exists a subset H of X such that H is open and x ∈ H and f ^{∘} H ⊆ G.

(49) f is continuous if and only if for every point x of X holds f is continuous at x.

(50) For all topological spaces X, Y, Z such that the carrier of Y = the carrier of Z and the topology of Z ⊆ the topology of Y and for every mapping f from X into Y and for every mapping g from X into Z such that f = g and for every point x of X such that f is continuous at x holds g is continuous at x.

(51) Let X, Y, Z be topological spaces. Then if the carrier of X = the carrier of Y and the topology of Y ⊆ the topology of X, then for every mapping f from X into Z and for every mapping g from Y into Z such that f = g and for every point x of X and for every point y of Y such that x = y holds if g is continuous at y, then f is continuous at x.

Let X, Y, Z be topological spaces, and let f be a mapping from X into Y, and let g be a mapping from Y into Z. Then g · f is a mapping from X into Z.

We follow a convention: X, Y, Z are topological spaces, f is a mapping from X into Y, and g is a mapping from Y into Z. The following propositions are true:

(52) For every point x of X and for every point y of Y such that y = f(x) holds if f is continuous at x and g is continuous at y, then g·f is continuous at x.

(53) For every point y of Y such that f is continuous and g is continuous at y and for every point x of X such that x ∈ f ^{−1} {y} holds g · f is continuous at x.

(54) For every point x of X such that f is continuous at x and g is continuous

holds g · f is continuous at x. Let X, Y be topological spaces. We introduce continuous mapping from X

into Y as a synonym of continuous map from X into Y.

The following propositions are true:

(55) f is a continuous mapping from X into Y if and only if for every point x of X holds f is continuous at x.

(56) For all topological spaces X, Y, Z such that the carrier of Y = the carrier of Z and the topology of Z ⊆ the topology of Y every continuous mapping from X into Y is a continuous mapping from X into Z.

(57) For all topological spaces X, Y, Z such that the carrier of X = the carrier of Y and the topology of Y ⊆ the topology of X every continuous mapping from Y into Z is a continuous mapping from X into Z.

Let X, Y be topological spaces, and let X_{0} be a subspace of X, and let f be a mapping from X into Y. The functor f X_{0} yielding a mapping from X_{0} into Y is defined by:

(Def.3) f X_{0} = f the carrier of X_{0}.

In the sequel X, Y will denote topological spaces, X_{0} will denote a subspace of X, and f will denote a mapping from X into Y. The following propositions are true:

(58) For every point x of X such that x ∈ the carrier of X_{0} holds f(x) = (f X_{0})(x).

(59) For every mapping f_{0} from X_{0} into Y such that for every point x of X such that x ∈ the carrier of X_{0} holds f(x) = f_{0}(x) holds f X_{0} = f_{0}.

(60) If the topological structure of X_{0} = the topological structure of X, then f = f X_{0}.

(61) For every subset A of X such that A ⊆ the carrier of X_{0} holds f ^{∘} A = (f X_{0})^{∘} A.

(62) For every subset B of Y such that f ^{−1} B ⊆ the carrier of X_{0} holds f ^{−1} B = (f X_{0})^{−1} B.

(63) For every mapping g from X_{0} into Y there exists a mapping h from X into Y such that h X_{0} = g.

In the sequel f is a mapping from X into Y and X_{0} is a subspace of X. Next we state several propositions:

(64) For every point x of X and for every point x_{0} of X_{0} such that x = x_{0} holds if f is continuous at x, then f X_{0} is continuous at x_{0}.

(65) For every subset A of X and for every point x of X and for every point x_{0} of X_{0} such that A ⊆ the carrier of X_{0} and A is a neighbourhood of x and x = x_{0} holds f is continuous at x if and only if f X_{0} is continuous at x_{0}.

(66) For every subset A of X and for every point x of X and for every point x_{0} of X_{0} such that A is open and x ∈ A and A ⊆ the carrier of X_{0} and x = x_{0} holds f is continuous at x if and only if f X_{0} is continuous at x_{0}.

(67) For every open subspace X_{0} of X and for every point x of X and for every point x_{0} of X_{0} such that x = x_{0} holds f is continuous at x if and only if f X_{0} is continuous at x_{0}.

(68) For every continuous mapping f from X into Y and for every subspace X_{0} of X holds f X_{0} is a continuous mapping from X_{0} into Y.

(69) For all topological spaces X, Y, Z and for every subspace X_{0} of X and for every mapping f from X into Y and for every mapping g from Y into Z holds (g · f) X_{0} = g · (f X_{0}).

(70) For all topological spaces X, Y, Z and for every subspace X_{0} of X and for every mapping g from Y into Z and for every mapping f from X into Y such that g is continuous and f X_{0} is continuous holds (g · f) X_{0} is continuous.

(71) For all topological spaces X, Y, Z and for every subspace X_{0} of X and for every continuous mapping g from Y into Z and for every mapping f from X into Y such that f X_{0} is a continuous mapping from X_{0} into Y holds (g · f) X_{0} is a continuous mapping from X_{0} into Z.

Let X, Y be topological spaces, and let X_{0}, X_{1} be subspaces of X, and let g be a mapping from X_{0} into Y. Let us assume that X_{1} is a subspace of X_{0}. The functor g X_{1} yielding a mapping from X_{1} into Y is defined as follows:

(Def.4) g X_{1} = g the carrier of X_{1}.

For simplicity we follow a convention: X, Y denote topological spaces, X_{0}, X_{1} denote subspaces of X, f denotes a mapping from X into Y, and g denotes a mapping from X_{0} into Y. The following propositions are true:

(72) If X_{1} is a subspace of X_{0}, then for every point x_{0} of X_{0} such that x_{0} ∈ the carrier of X_{1} holds g(x_{0}) = (g X_{1})(x_{0}).

(73) If X_{1} is a subspace of X_{0}, then for every mapping g_{1} from X_{1} into Y such that for every point x_{0} of X_{0} such that x_{0} ∈ the carrier of X_{1} holds g(x_{0}) = g_{1}(x_{0}) holds g X_{1} = g_{1}.

(74) g = g X_{0}.

(75) If X_{1} is a subspace of X_{0}, then for every subset A of X_{0} such that A ⊆ the carrier of X_{1} holds g^{∘} A = (g X_{1})^{∘} A.

(76) If X_{1} is a subspace of X_{0}, then for every subset B of Y such that g^{−1} B ⊆ the carrier of X_{1} holds g^{−1} B = (g X_{1})^{−1} B.

(77) For every mapping g from X_{0} into Y such that g = f X_{0} holds if X_{1} is a subspace of X_{0}, then g X_{1} = f X_{1}.

(78) If X_{1} is a subspace of X_{0}, then f X_{0} X_{1} = f X_{1}.

(79) For all subspaces X_{0}, X_{1}, X_{2} of X such that X_{1} is a subspace of X_{0} and X_{2} is a subspace of X_{1} and for every mapping g from X_{0} into Y holds g X_{1} X_{2} = g X_{2}.

(80) For every mapping f from X into Y and for every mapping f_{0} from X_{1} into Y and for every mapping g from X_{0} into Y such that X_{0} = X and f = g holds g X_{1} = f_{0} if and only if f X_{1} = f_{0}.

We follow the rules: X_{0}, X_{1}, X_{2} are subspaces of X, f is a mapping from X into Y, and g is a mapping from X_{0} into Y. One can prove the following propositions:

(81) For every point x_{0} of X_{0} and for every point x_{1} of X_{1} such that x_{0} = x_{1} holds if X_{1} is a subspace of X_{0} and g is continuous at x_{0}, then g X_{1} is continuous at x_{1}.

(82) If X_{1} is a subspace of X_{0}, then for every point x_{0} of X_{0} and for every point x_{1} of X_{1} such that x_{0} = x_{1} holds if f X_{0} is continuous at x_{0}, then f X_{1} is continuous at x_{1}.

(83) If X_{1} is a subspace of X_{0}, then for every subset A of X_{0} and for every point x_{0} of X_{0} and for every point x_{1} of X_{1} such that A ⊆ the carrier of X_{1} and A is a neighbourhood of x_{0} and x_{0} = x_{1} holds g is continuous at x_{0} if and only if g X_{1} is continuous at x_{1}.

(84) If X_{1} is a subspace of X_{0}, then for every subset A of X_{0} and for every point x_{0} of X_{0} and for every point x_{1} of X_{1} such that A is open and x_{0} ∈ A and A ⊆ the carrier of X_{1} and x_{0} = x_{1} holds g is continuous at x_{0} if and only if g X_{1} is continuous at x_{1}.

(85) If X_{1} is a subspace of X_{0}, then for every subset A of X and for every point x_{0} of X_{0} and for every point x_{1} of X_{1} such that A is open and x_{0} ∈ A and A ⊆ the carrier of X_{1} and x_{0} = x_{1} holds g is continuous at x_{0} if and only if g X_{1} is continuous at x_{1}.

(86) If X_{1} is an open subspace of X_{0}, then for every point x_{0} of X_{0} and for every point x_{1} of X_{1} such that x_{0} = x_{1} holds g is continuous at x_{0} if and only if g X_{1} is continuous at x_{1}.

(87) If X_{1} is an open subspace of X and X_{1} is a subspace of X_{0}, then for every point x_{0} of X_{0} and for every point x_{1} of X_{1} such that x_{0} = x_{1} holds g is continuous at x_{0} if and only if g X_{1} is continuous at x_{1}.

(88) If the topological structure of X_{1} = X_{0}, then for every point x_{0} of X_{0} and for every point x_{1} of X_{1} such that x_{0} = x_{1} holds if g X_{1} is continuous at x_{1}, then g is continuous at x_{0}.

(89) For every continuous mapping g from X_{0} into Y such that X_{1} is a subspace of X_{0} holds g X_{1} is a continuous mapping from X_{1} into Y.

(90) If X_{1} is a subspace of X_{0} and X_{2} is a subspace of X_{1}, then for every mapping g from X_{0} into Y such that g X_{1} is a continuous mapping from X_{1} into Y holds g X_{2} is a continuous mapping from X_{2} into Y.

Let X be a topological space. The functor id_{X} yielding a mapping from X into X is defined as follows: (Def.5) id_{X} = id_{(the carrier of X)}.

One can prove the following four propositions:

(91) For every point x of X holds id_{X}(x) = x.

(92) For every mapping f from X into X such that for every point x of X holds f(x) = x holds f = id_{X}.

(93) For every mapping f from X into Y holds f · id_{X} = f and id_{Y} · f = f.

(94) id_{X} is a continuous mapping from X into X.

We now define two new functors. Let X be a topological space, and let X_{0} be a subspace of X. The functor {X_{0} \atop ↪} yielding a mapping from X_{0} into X is defined by: (Def.6) {X_{0} \atop ↪} = id_{X} X_{0}.

We introduce the functor X_{0} ↪ X as a synonym of {X_{0} \atop ↪}.

Next we state four propositions:

(95) For every subspace X_{0} of X and for every point x of X such that x ∈ the carrier of X_{0} holds ({X_{0} \atop ↪})(x) = x.

(96) For every subspace X_{0} of X and for every mapping f_{0} from X_{0} into X such that for every point x of X such that x ∈ the carrier of X_{0} holds x = f_{0}(x) holds {X_{0} \atop ↪} = f_{0}.

(97) For every subspace X_{0} of X and for every mapping f from X into Y holds f X_{0} = f · ({X_{0} \atop ↪}).

(98) For every subspace X_{0} of X holds {X_{0} \atop ↪} is a continuous mapping from X_{0} into X.

4. A Modification of the Topology of Topological Spaces

In the sequel X will denote a topological space and H, G will denote subsets of X. Let us consider X, and let A be a subset of X. The A-extension of the topology of X yielding a family of subsets of X is defined as follows: (Def.7) the A-extension of the topology of X = {H ∪ G ∩ A : H ∈ the topology

of X ∧ G ∈ the topology of X}. We now state several propositions:

(99) For every subset A of X holds the topology of X ⊆ the A-extension of the topology of X.

(100) For every subset A of X holds {G ∩ A : G ∈ the topology of X} ⊆ the A-extension of the topology of X, where G ranges over subsets of X.

(101) For every subset A of X and for all subsets C, D of X such that C ∈ the topology of X and D ∈ {G ∩ A : G ∈ the topology of X}, where G ranges over subsets of X holds C ∪ D ∈ the A-extension of the topology of X and C ∩ D ∈ the A-extension of the topology of X.

(102) For every subset A of X holds A ∈ the A-extension of the topology of X.

(103) For every subset A of X holds A ∈ the topology of X if and only if the

topology of X = the A-extension of the topology of X. Let X be a topological space, and let A be a subset of X. The X modified

w.r.t. A yields a strict topological space and is defined by: (Def.8) the X modified w.r.t. A = ⟨the carrier of X, the A-extension of the

topology of X⟩. In the sequel A will be a subset of X. The following three propositions are

true:

(104) The carrier of the X modified w.r.t. A = the carrier of X and the topology of the X modified w.r.t. A = the A-extension of the topology of X.

(105) For every subset B of the X modified w.r.t. A such that B = A holds B is open.

(106) A is open if and only if the topological structure of X = the X modified w.r.t. A.

Let X be a topological space, and let A be a subset of X. The functor modid_{X,A} yields a mapping from X into the X modified w.r.t. A and is defined as follows: (Def.9) modid_{X,A} = id_{(the carrier of X)}.

We now state several propositions:

(107) If A is open, then modid_{X,A} = id_{X}.

(108) For every point x of X such that x ∉ A holds modid_{X,A} is continuous at x.

(109) For every subspace X_{0} of X such that (the carrier of X_{0}) ∩ A = ∅ and for every point x_{0} of X_{0} holds modid_{X,A} X_{0} is continuous at x_{0}.

(110) For every subspace X_{0} of X such that the carrier of X_{0} = A and for every point x_{0} of X_{0} holds modid_{X,A} X_{0} is continuous at x_{0}.

(111) For every subspace X_{0} of X such that (the carrier of X_{0}) ∩ A = ∅ holds modid_{X,A} X_{0} is a continuous mapping from X_{0} into the X modified w.r.t. A.

(112) For every subspace X_{0} of X such that the carrier of X_{0} = A holds modid_{X,A} X_{0} is a continuous mapping from X_{0} into the X modified w.r.t. A.

(113) For every subset A of X holds A is open if and only if modid_{X,A} is a

continuous mapping from X into the X modified w.r.t. A. Let X be a topological space, and let X_{0} be a subspace of X. The X modified

w.r.t. X_{0} yielding a strict topological space is defined as follows: (Def.10) for every subset A of X such that A = the carrier of X_{0} holds the X

modified w.r.t. X_{0} = the X modified w.r.t. A.

In the sequel X_{0} will denote a subspace of X. The following three propositions are true:

(114) The carrier of the X modified w.r.t. X_{0} = the carrier of X and for every subset A of X such that A = the carrier of X_{0} holds the topology of the X modified w.r.t. X_{0} = the A-extension of the topology of X.

(115) For every subspace Y_{0} of the X modified w.r.t. X_{0} such that the carrier of Y_{0} = the carrier of X_{0} holds Y_{0} is an open subspace of the X modified w.r.t. X_{0}.

(116) X_{0} is an open subspace of X if and only if the topological structure of

X = the X modified w.r.t. X_{0}. Let X be a topological space, and let X_{0} be a subspace of X. The functor

modid_{X,X_{0}} yielding a mapping from X into the X modified w.r.t. X_{0} is defined as follows:

(Def.11) for every subset A of X such that A = the carrier of X_{0} holds modid_{X,X_{0}} = modid_{X,A}.

We now state several propositions:

(117) If X_{0} is an open subspace of X, then modid_{X,X_{0}} = id_{X}.

(118) For all subspaces X_{0}, X_{1} of X such that X_{0} misses X_{1} and for every point x_{1} of X_{1} holds modid_{X,X_{0}} X_{1} is continuous at x_{1}.

(119) For every subspace X_{0} of X and for every point x_{0} of X_{0} holds

modid_{X,X_{0}} X_{0}

is continuous at x_{0}.

(120) For all subspaces X_{0}, X_{1} of X such that X_{0} misses X_{1} holds modid_{X,X_{0}} X_{1} is a continuous mapping from X_{1} into the X modified w.r.t. X_{0}.

(121) For every subspace X_{0} of X holds modid_{X,X_{0}} X_{0} is a continuous mapping from X_{0} into the X modified w.r.t. X_{0}.

(122) For every subspace X_{0} of X holds X_{0} is an open subspace of X if and only if modid_{X,X_{0}} is a continuous mapping from X into the X modified w.r.t. X_{0}.

5. Continuity of Mappings over the Union of Subspaces

In the sequel X, Y denote topological spaces. We now state three propositions:

(123) For all subspaces X_{1}, X_{2} of X and for every mapping g from X_{1} ∪ X_{2} into Y and for every point x_{1} of X_{1} and for every point x_{2} of X_{2} and for every point x of X_{1} ∪ X_{2} such that x = x_{1} and x = x_{2} holds g is continuous at x if and only if g X_{1} is continuous at x_{1} and g X_{2} is continuous at x_{2}.

(124) Let f be a mapping from X into Y. Then for all subspaces X_{1}, X_{2} of X and for every point x of X_{1} ∪ X_{2} and for every point x_{1} of X_{1} and for every point x_{2} of X_{2} such that x = x_{1} and x = x_{2} holds f (X_{1} ∪ X_{2}) is continuous at x if and only if f X_{1} is continuous at x_{1} and f X_{2} is continuous at x_{2}.

(125) Let f be a mapping from X into Y. Then for all subspaces X_{1}, X_{2} of X such that X = X_{1} ∪ X_{2} and for every point x of X and for every point x_{1} of X_{1} and for every point x_{2} of X_{2} such that x = x_{1} and x = x_{2} holds f is continuous at x if and only if f X_{1} is continuous at x_{1} and f X_{2} is continuous at x_{2}.

In the sequel X_{1}, X_{2} will denote subspaces of X. One can prove the following propositions:

(126) If X_{1} and X_{2} are weakly separated, then for every mapping g from X_{1} ∪ X_{2} into Y holds g is a continuous mapping from X_{1} ∪ X_{2} into Y if and only if g X_{1} is a continuous mapping from X_{1} into Y and g X_{2} is a continuous mapping from X_{2} into Y.

(127) For all closed subspaces X_{1}, X_{2} of X and for every mapping g from X_{1} ∪ X_{2} into Y holds g is a continuous mapping from X_{1} ∪ X_{2} into Y if and only if g X_{1} is a continuous mapping from X_{1} into Y and g X_{2} is a continuous mapping from X_{2} into Y.

(128) For all open subspaces X_{1}, X_{2} of X and for every mapping g from X_{1} ∪ X_{2} into Y holds g is a continuous mapping from X_{1} ∪ X_{2} into Y if and only if g X_{1} is a continuous mapping from X_{1} into Y and g X_{2} is a continuous mapping from X_{2} into Y.

(129) If X_{1} and X_{2} are weakly separated, then for every mapping f from X into Y holds f (X_{1} ∪ X_{2}) is a continuous mapping from X_{1} ∪ X_{2} into Y if and only if f X_{1} is a continuous mapping from X_{1} into Y and f X_{2} is a continuous mapping from X_{2} into Y.

(130) For every mapping f from X into Y and for all closed subspaces X_{1}, X_{2} of X holds f (X_{1} ∪X_{2}) is a continuous mapping from X_{1} ∪X_{2} into Y if and only if f X_{1} is a continuous mapping from X_{1} into Y and f X_{2} is a continuous mapping from X_{2} into Y.

(131) For every mapping f from X into Y and for all open subspaces X_{1}, X_{2} of X holds f (X_{1} ∪ X_{2}) is a continuous mapping from X_{1} ∪ X_{2} into Y if and only if f X_{1} is a continuous mapping from X_{1} into Y and f X_{2} is a continuous mapping from X_{2} into Y.

(132) For every mapping f from X into Y and for all subspaces X_{1}, X_{2} of X such that X = X_{1} ∪ X_{2} and X_{1} and X_{2} are weakly separated holds f is a continuous mapping from X into Y if and only if f X_{1} is a continuous mapping from X_{1} into Y and f X_{2} is a continuous mapping from X_{2} into Y.

(133) For every mapping f from X into Y and for all closed subspaces X_{1}, X_{2} of X such that X = X_{1} ∪ X_{2} holds f is a continuous mapping from X into Y if and only if f X_{1} is a continuous mapping from X_{1} into Y and f X_{2} is a continuous mapping from X_{2} into Y.

(134) For every mapping f from X into Y and for all open subspaces X_{1}, X_{2} of X such that X = X_{1} ∪ X_{2} holds f is a continuous mapping from X into Y if and only if f X_{1} is a continuous mapping from X_{1} into Y and f X_{2} is a continuous mapping from X_{2} into Y.

(135) X_{1} and X_{2} are separated if and only if X_{1} misses X_{2} and for every topological space Y and for every mapping g from X_{1} ∪ X_{2} into Y such that g X_{1} is a continuous mapping from X_{1} into Y and g X_{2} is a continuous mapping from X_{2} into Y holds g is a continuous mapping from X_{1} ∪ X_{2} into Y.

(136) X_{1} and X_{2} are separated if and only if X_{1} misses X_{2} and for every topological space Y and for every mapping f from X into Y such that f X_{1} is a continuous mapping from X_{1} into Y and f X_{2} is a continuous mapping from X_{2} into Y holds f (X_{1} ∪ X_{2}) is a continuous mapping from X_{1} ∪ X_{2} into Y.

(137) For all subspaces X_{1}, X_{2} of X such that X = X_{1} ∪ X_{2} holds X_{1} and X_{2} are separated if and only if X_{1} misses X_{2} and for every topological space Y and for every mapping f from X into Y such that f X_{1} is a continuous mapping from X_{1} into Y and f X_{2} is a continuous mapping from X_{2} into Y holds f is a continuous mapping from X into Y.

6. The Union of Continuous Mappings

Let X, Y be topological spaces, and let X_{1}, X_{2} be subspaces of X, and let f_{1} be a mapping from X_{1} into Y, and let f_{2} be a mapping from X_{2} into Y. Let us assume that X_{1} misses X_{2} or f_{1} (X_{1} ∩ X_{2}) = f_{2} (X_{1} ∩ X_{2}). The functor f_{1} ∪ f_{2} yielding a mapping from X_{1} ∪ X_{2} into Y is defined as follows:

(Def.12) (f_{1} ∪ f_{2}) X_{1} = f_{1} and (f_{1} ∪ f_{2}) X_{2} = f_{2}.

In the sequel X, Y will denote topological spaces. We now state a number of propositions:

(138) For all subspaces X_{1}, X_{2} of X and for every mapping g from X_{1} ∪ X_{2} into Y holds g = g X_{1} ∪ g X_{2}.

(139) For all subspaces X_{1}, X_{2} of X such that X = X_{1} ∪ X_{2} and for every mapping g from X into Y holds g = g X_{1} ∪ g X_{2}.

(140) For all subspaces X_{1}, X_{2} of X such that X_{1} meets X_{2} and for every mapping f_{1} from X_{1} into Y and for every mapping f_{2} from X_{2} into Y holds (f_{1}∪f_{2}) X_{1} = f_{1} and (f_{1}∪f_{2}) X_{2} = f_{2} if and only if f_{1} (X_{1}∩X_{2}) = f_{2} (X_{1} ∩ X_{2}).

(141) For all subspaces X_{1}, X_{2} of X and for every mapping f_{1} from X_{1} into Y and for every mapping f_{2} from X_{2} into Y such that f_{1} (X_{1} ∩ X_{2}) = f_{2} (X_{1} ∩ X_{2}) holds X_{1} is a subspace of X_{2} if and only if f_{1} ∪ f_{2} = f_{2} but X_{2} is a subspace of X_{1} if and only if f_{1} ∪ f_{2} = f_{1}.

(142) For all subspaces X_{1}, X_{2} of X and for every mapping f_{1} from X_{1} into Y and for every mapping f_{2} from X_{2} into Y such that X_{1} misses X_{2} or f_{1} (X_{1} ∩ X_{2}) = f_{2} (X_{1} ∩ X_{2}) holds f_{1} ∪ f_{2} = f_{2} ∪ f_{1}.

(143) Let X_{1}, X_{2}, X_{3} be subspaces of X. Let f_{1} be a mapping from X_{1} into Y. Let f_{2} be a mapping from X_{2} into Y. Let f_{3} be a mapping from X_{3} into Y. Suppose X_{1} misses X_{2} or f_{1} (X_{1} ∩ X_{2}) = f_{2} (X_{1} ∩ X_{2}) but X_{1} misses X_{3} or f_{1} (X_{1} ∩ X_{3}) = f_{3} (X_{1} ∩ X_{3}) but X_{2} misses X_{3} or f_{2} (X_{2} ∩ X_{3}) = f_{3} (X_{2} ∩ X_{3}). Then (f_{1} ∪ f_{2}) ∪ f_{3} = f_{1} ∪ (f_{2} ∪ f_{3}).

(144) For all subspaces X_{1}, X_{2} of X such that X_{1} meets X_{2} and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y such that f_{1} (X_{1} ∩ X_{2}) = f_{2} (X_{1} ∩ X_{2}) holds if X_{1} and X_{2} are weakly separated, then f_{1} ∪ f_{2} is a continuous mapping from X_{1} ∪ X_{2} into Y.

(145) For all subspaces X_{1}, X_{2} of X such that X_{1} misses X_{2} and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y such that X_{1} and X_{2} are weakly separated holds f_{1} ∪f_{2} is a continuous mapping from X_{1} ∪ X_{2} into Y.

(146) For all closed subspaces X_{1}, X_{2} of X such that X_{1} meets X_{2} and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y such that f_{1} (X_{1} ∩ X_{2}) = f_{2} (X_{1} ∩ X_{2}) holds f_{1} ∪ f_{2} is a continuous mapping from X_{1} ∪ X_{2} into Y.

(147) For all open subspaces X_{1}, X_{2} of X such that X_{1} meets X_{2} and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y such that f_{1} (X_{1} ∩ X_{2}) = f_{2} (X_{1} ∩ X_{2}) holds f_{1} ∪ f_{2} is a continuous mapping from X_{1} ∪ X_{2} into Y.

(148) For all closed subspaces X_{1}, X_{2} of X such that X_{1} misses X_{2} and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y holds f_{1} ∪ f_{2} is a continuous mapping from X_{1} ∪ X_{2} into Y.

(149) For all open subspaces X_{1}, X_{2} of X such that X_{1} misses X_{2} and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y holds f_{1} ∪ f_{2} is a continuous mapping from X_{1} ∪ X_{2} into Y.

(150) For all subspaces X_{1}, X_{2} of X holds X_{1} and X_{2} are separated if and only if X_{1} misses X_{2} and for every topological space Y and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y holds f_{1} ∪ f_{2} is a continuous mapping from X_{1} ∪ X_{2} into Y.

(151) For all subspaces X_{1}, X_{2} of X such that X = X_{1} ∪ X_{2} and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y such that (f_{1} ∪ f_{2}) X_{1} = f_{1} and (f_{1} ∪ f_{2}) X_{2} = f_{2} holds if X_{1} and X_{2} are weakly separated, then f_{1} ∪ f_{2} is a continuous mapping from X into Y.

(152) For all closed subspaces X_{1}, X_{2} of X and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y such that X = X_{1} ∪ X_{2} and (f_{1} ∪ f_{2}) X_{1} = f_{1} and (f_{1} ∪ f_{2}) X_{2} = f_{2} holds f_{1} ∪ f_{2} is a continuous mapping from X into Y.

(153) For all open subspaces X_{1}, X_{2} of X and for every continuous mapping f_{1} from X_{1} into Y and for every continuous mapping f_{2} from X_{2} into Y such that X = X_{1} ∪ X_{2} and (f_{1} ∪ f_{2}) X_{1} = f_{1} and (f_{1} ∪ f_{2}) X_{2} = f_{2} holds f_{1} ∪ f_{2} is a continuous mapping from X into Y.

Acknowledgments

I would like to express my thanks to Professors A. Trybulec and Cz. Byliński for their active interest in the publication of this article and for elucidating to me new advanced Mizar constructions.

References

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

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

16 zbigniew karno

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

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

[5] Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Formalized Mathematics, 2(5):665–674, 1991.

[6] Kazimierz Kuratowski. Topology. Volume I, PWN - Polish Scientific Publishers, Academic Press, Warsaw, New York and London, 1966.

[7] Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239–244, 1990.

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

[9] Beata Padlewska. Locally connected spaces. Formalized Mathematics, 2(1):93–96, 1991.

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

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

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

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

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

[15] Eduard Čech. Topological Spaces. Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1966.

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

Received May 22, 1992

Functional Sequence from a Domain to a Domain

Beata Perkowska Warsaw University Białystok

Summary. Definitions of functional sequences and basic operations on functional sequences from a domain to a domain, point and uniform convergent, limit of functional sequence from a domain to the set of real numbers and facts about properties of the limit of functional sequences are proved.

MML Identifier: SEQFUNC.

The articles [11], [1], [2], [3], [13], [5], [6], [9], [8], [4], [12], [7], and [10] provide the notation and terminology for this paper. For simplicity we adopt the following rules: D, D_{1}, D_{2} denote non-empty sets, n, k denote natural numbers, p, r denote real numbers, and f denotes a function. Let us consider D_{1}, D_{2}. A function is called a sequence of partial functions from D_{1} into D_{2} if: (Def.1) dom it = and rng it ⊆ D_{1} ˙→D_{2}.

In the sequel F, F_{1}, F_{2} are sequences of partial functions from D_{1} into D_{2}. Let us consider D_{1}, D_{2}, F, n. Then F (n) is a partial function from D_{1} to D_{2}.

In the sequel G, H, H_{1}, H_{2}, J are sequences of partial functions from D into . One can prove the following two propositions:

(1) f is a sequence of partial functions from D_{1} into D_{2} if and only if dom f = and for every n holds f(n) is a partial function from D_{1} to D_{2}.

(2) For all F_{1}, F_{2} such that for every n holds F_{1}(n) = F_{2}(n) holds F_{1} = F_{2}. The scheme ExFuncSeq deals with a non-empty set A, a non-empty set B,

and a unary functor F yielding a partial function from A to B and states that:

there exists a sequence G of partial functions from A into B such that for every n holds G(n) = F(n) for all values of the parameters.

We now define several new functors. Let us consider D, H, r. The functor r H yields a sequence of partial functions from D into and is defined as follows: (Def.2) for every n holds (r H)(n) = r H(n). Let us consider D, H. The functor H^{−1} yielding a sequence of partial functions from D into is defined by: (Def.3) for every n holds H^{−1}(n) = \frac{1}{H(n)}. The functor −H yields a sequence of partial functions from D into and is defined by: (Def.4) for every n holds (−H)(n) = −H(n). The functor |H| yields a sequence of partial functions from D into and is defined as follows: (Def.5) for every n holds |H|(n) = |H(n)|. Let us consider D, G, H. The functor G + H yields a sequence of partial functions from D into and is defined by: (Def.6) for every n holds (G + H)(n) = G(n) + H(n). The functor G − H yielding a sequence of partial functions from D into is defined as follows: (Def.7) G − H = G + −H. The functor G H yields a sequence of partial functions from D into and is defined as follows: (Def.8) for every n holds (G H)(n) = G(n) H(n). Let us consider D, H, G. The functor \frac{G}{H} yielding a sequence of partial functions from D into is defined as follows: (Def.9) \frac{G}{H} = G H^{−1}.

Next we state a number of propositions:

(3) H_{1} = \frac{G}{H} if and only if for every n holds H_{1}(n) = \frac{G(n)}{H(n)}.

(4) H_{1} = G − H if and only if for every n holds H_{1}(n) = G(n) − H(n).

(5) G + H = H + G and (G + H) + J = G + (H + J).

(6) G H = H G and (G H) J = G (H J).

(7) (G + H) J = G J + H J and J (G + H) = J G + J H.

(8) −H = (−1) H.

(9) (G − H) J = G J − H J and J G − J H = J (G − H).

(10) r (G + H) = r G + r H and r (G − H) = r G − r H.

(11) (r · p) H = r (p H).

(12) 1 H = H.

(13) −−H = H.

(14) G^{−1} H^{−1} = (G H)^{−1}.

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

(16) |H|^{−1} = |H^{−1}|.

(17) |G H| = |G| |H|.

(18) |\frac{G}{H} | = \frac{|G|}{|H|}.

(19) |r H| = |r| |H|.

In the sequel x is an element of D, X, Y are sets, and f is a partial function from D to . We now define three new constructions. Let us consider D_{1}, D_{2}, F, X. We say that X is common for elements of F if and only if:

(Def.10) X ≠ ∅ and for every n holds X ⊆ dom F (n).

Let us consider D, H, x. The functor H#x yielding a sequence of real numbers is defined as follows:

(Def.11) for every n holds (H#x)(n) = H(n)(x).

Let us consider D, H, X. We say that H is point-convergent on X if and only if:

(Def.12) X is common for elements of H and there exists f such that X = dom f and for every x such that x ∈ X and for every p such that p > 0 there exists k such that for every n such that n ≥ k holds |H(n)(x) − f(x)| < p.

Next we state two propositions:

(20) H is point-convergent on X if and only if X is common for elements of H and there exists f such that X = dom f and for every x such that x ∈ X holds H#x is convergent and lim(H#x) = f(x).

(21) H is point-convergent on X if and only if X is common for elements of

H and for every x such that x ∈ X holds H#x is convergent. We now define two new constructions. Let us consider D, H, X. We say

that H is uniform-convergent on X if and only if: (Def.13) X is common for elements of H and there exists f such that X = dom f

and for every p such that p > 0 there exists k such that for all n, x such that n ≥ k and x ∈ X holds |H(n)(x) − f(x)| < p.

Let us assume that H is point-convergent on X. The functor lim_{X}H yielding a partial function from D to is defined as follows:

(Def.14) dom lim_{X}H = X and for every x such that x ∈ dom lim_{X}H holds (lim_{X}H)(x) = lim(H#x).

We now state a number of propositions:

(22) If H is point-convergent on X, then f = lim_{X}H if and only if dom f = X and for every x such that x ∈ X and for every p such that p > 0 there exists k such that for every n such that n ≥ k holds |H(n)(x) − f(x)| < p.

(23) If H is uniform-convergent on X, then H is point-convergent on X.

(24) If Y ⊆ X and Y ≠ ∅ and X is common for elements of H, then Y is common for elements of H.

(25) If Y ⊆ X and Y ≠ ∅ and H is point-convergent on X, then H is point-convergent on Y and lim_{X}H Y = lim_{Y} H.

(26) If Y ⊆ X and Y ≠ ∅ and H is uniform-convergent on X, then H is uniform-convergent on Y.

(27) If X is common for elements of H, then for every x such that x ∈ X holds {x} is common for elements of H.

(28) If H is point-convergent on X, then for every x such that x ∈ X holds {x} is common for elements of H.

(29) Suppose {x} is common for elements of H_{1} and {x} is common for elements of H_{2}. Then H_{1}#x+H_{2}#x = (H_{1}+H_{2})#x and H_{1}#x−H_{2}#x = (H_{1} − H_{2})#x and (H_{1}#x) (H_{2}#x) = (H_{1} H_{2})#x.

(30) If {x} is common for elements of H, then |H|#x = |H#x| and (−H)#x = −H#x.

(31) If {x} is common for elements of H, then (r H)#x = r (H#x).

(32) Suppose X is common for elements of H_{1} and X is common for elements of H_{2}. Then for every x such that x ∈ X holds H_{1}#x + H_{2}#x = (H_{1} + H_{2})#x and H_{1}#x − H_{2}#x = (H_{1} − H_{2})#x and (H_{1}#x) (H_{2}#x) = (H_{1} H_{2})#x.

(33) If X is common for elements of H, then for every x such that x ∈ X holds |H|#x = |H#x| and (−H)#x = −H#x.

(34) If X is common for elements of H, then for every x such that x ∈ X holds (r H)#x = r (H#x).

(35) Suppose H_{1} is point-convergent on X and H_{2} is point-convergent on X. Then for every x such that x ∈ X holds H_{1}#x + H_{2}#x = (H_{1} + H_{2})#x and H_{1}#x − H_{2}#x = (H_{1} − H_{2})#x and (H_{1}#x) (H_{2}#x) = (H_{1} H_{2})#x.

(36) If H is point-convergent on X, then for every x such that x ∈ X holds |H|#x = |H#x| and (−H)#x = −H#x.

(37) If H is point-convergent on X, then for every x such that x ∈ X holds (r H)#x = r (H#x).

(38) If X is common for elements of H_{1} and X is common for elements of H_{2}, then X is common for elements of H_{1} + H_{2} and X is common for elements of H_{1} − H_{2} and X is common for elements of H_{1} H_{2}.

(39) If X is common for elements of H, then X is common for elements of |H| and X is common for elements of −H.

(40) If X is common for elements of H, then X is common for elements of r H.

(41) Suppose H_{1} is point-convergent on X and H_{2} is point-convergent on X.

Then

(i) H_{1} + H_{2} is point-convergent on X, (ii) lim_{X}(H_{1} + H_{2}) = lim_{X}H_{1} + lim_{X}H_{2}, (iii) H_{1} − H_{2} is point-convergent on X, (iv) lim_{X}(H_{1} − H_{2}) = lim_{X}H_{1} − lim_{X}H_{2},

(v) H_{1} H_{2} is point-convergent on X, (vi) lim_{X}(H_{1} H_{2}) = lim_{X}H_{1} lim_{X}H_{2}.

(42) If H is point-convergent on X, then |H| is point-convergent on X and lim_{X}|H| = |lim_{X}H| and −H is point-convergent on X and lim_{X}(−H) =

functional sequence from a domain to a domain 21

−lim_{X}H.

(43) If H is point-convergent on X, then r H is point-convergent on X and lim_{X}(r H) = r lim_{X}H.

(44) H is uniform-convergent on X if and only if X is common for elements of H and H is point-convergent on X and for every r such that 0 < r there exists k such that for all n, x such that n ≥ k and x ∈ X holds |H(n)(x) − (lim_{X}H)(x)| < r.

In the sequel H will be a sequence of partial functions from into . Let us consider n, k. Then max(n, k) is a natural number.

We now state the proposition

(45) If H is uniform-convergent on X and for every n holds H(n) is continuous on X, then lim_{X}H is continuous on X.

References

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

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

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

[4] 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. Real function continuity. Formalized Mathematics, 1(4):787–791, 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.

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

Received May 22, 1992 22

Reper Algebras

Michał Muzalewski Warsaw University Białystok

Summary. We shall describe n-dimensional spaces with the reper operation [10, pages 72–79]. An inspiration to such approach comes from the monograph [12] and so-called Leibniz program. Let us recall that the Leibniz program is a program of algebraization of geometry using purely geometric notions. Leibniz formulated his program in opposition to algebraization method developed by Descartes. The Euclidean geometry in Szmielew’s approach [12] is a theory of structures ⟨S; ∥, ⊕, O⟩, where ⟨S; ∥, ⊕, O⟩ is Desarguean midpoint plane and O ⊆ S × S × S is the relation of equi-orthogonal basis. Points o, p, q are in relation O if they form an isosceles triangle with the right angle in vertex a. If we fix vertices a, p, then there exist exactly two points q, q′ such that O(apq), O(apq′). Moreover q ⊕ q′ = a. In accordance with the Leibniz program we replace the relation of equi-orthogonal basis by a binary operation ∗ : S ×S → S, called the reper operation. A standard model for the Euclidean geometry in the above sense is the oriented plane over the field of real numbers with the reper operations ∗ defined by the condition: a ∗ b = q iff the point q is the result of rotating of p about right angle around the center a.

MML Identifier: MIDSP 3.

The terminology and notation used here are introduced in the following articles: [13], [5], [6], [3], [7], [2], [4], [1], [8], [11], and [9].

1. Substitutions in tuples

For simplicity we adopt the following rules: n, i, j, k, l are natural numbers, D is a non-empty set, c, d are elements of D, and p, q, r are finite sequences of elements of D. The following propositions are true:

(1) If len p = j + 1 + k, then there exist q, r, c such that len q = j and len r = k and p = q^{}⟨c⟩^{}r.

(2) If i ∈ Seg n, then there exist j, k such that n = j + 1 + k and i = j + 1.

(3) Suppose p = q ^{}⟨c⟩^{}r and i = len q + 1. Then for every l such that 1 ≤ l and l ≤ len q holds p(l) = q(l) and p(i) = c and for every l such that i + 1 ≤ l and l ≤ len p holds p(l) = r(l − i).

(4) l ≤ j or l = j + 1 or j + 2 ≤ l.

(5) If l ∈ Seg n \ {i} and i = j + 1, then 1 ≤ l and l ≤ j or i + 1 ≤ l and l ≤ n.

Let us consider n, i, D, d, and let p be an element of D^{n+1}. Let us assume that i ∈ Seg(n + 1). The functor p(i/d) yielding an element of D^{n+1} is defined as follows: (Def.1) p(i/d)(i) = d and for every l such that l ∈ Seg len p\{i} holds p(i/d)(l) =

p(l).

2. Reper Algebra Structure and its Properties

Let us consider n. We consider structures of reper algebra over n which are extension of a midpoint algebra structure and are systems

⟨a carrier, a midpoint operation, a reper⟩, where the carrier is a non-empty set, the midpoint operation is a binary operation on the carrier, and the reper is a function from (the carrier)^{n} into the carrier. Let us observe that there exists a structure of reper algebra over n + 2 which is midpoint algebra-like.

We adopt the following rules: R_{1} will denote a midpoint algebra-like structure of reper algebra over n + 2 and a, b, d, p_{1}, p′_{1} will denote points of R_{1}. We now define two new modes. Let us consider i, D. A tuple of i and D is an element of D^{i}.

Let us consider n, R_{1}, i. A tuple of i and R_{1} is a tuple of i and the carrier of R_{1}.

In the sequel p, q will denote tuples of n + 1 and R_{1}. Let us consider n, R_{1}, a. Then ⟨a⟩ is a tuple of 1 and R_{1}. Let us consider n, R_{1}, i, j, and let p be a tuple of i and R_{1}, and let q be a tuple of j and R_{1}. Then p^{}q is a tuple of i + j and R_{1}.

We now state the proposition

(6) ⟨a⟩^{}p is a tuple of n + 2 and R_{1}.

We now define two new functors. Let us consider n, R_{1}, a, p. The functor ∗(a, p) yielding a point of R_{1} is defined as follows: (Def.2) ∗(a, p) = (the reper of R_{1})(⟨a⟩^{}p). Let us consider n, i, R_{1}, d, p. The functor p_{i ˙→d} yields a tuple of n + 1 and R_{1} and is defined as follows: (Def.3) for every D and for every element p′ of D^{n+1} and for every element

d′ of D such that D = the carrier of R_{1} and p′ = p and d′ = d holds p_{i ˙→d} = p′(i/d′).

We now state the proposition

(7) If i ∈ Seg(n + 1), then p_{i ˙→d}(i) = d and for every l such that l ∈ Seg len p \ {i} holds p_{i ˙→d}(l) = p(l).

Let us consider n. A natural number is said to be a natural number of n if: (Def.4) 1 ≤ it and it ≤ n + 1.

In the sequel m is a natural number of n. We now state several propositions:

(8) i is a natural number of n if and only if i ∈ Seg(n + 1).

(9) 1 ≤ i + 1.

(10) If i ≤ n, then i + 1 is a natural number of n.

(11) If for every m holds p(m) = q(m), then p = q.

(12) For every natural number l of n such that l = i holds p_{i ˙→d}(l) = d and

for all natural numbers l, i of n such that l ≠ i holds p_{i ˙→d}(l) = p(l). We now define three new predicates. Let us consider n, D, and let p be an

element of D^{n+1}, and let us consider m. Then p(m) is an element of D. Let us consider n, R_{1}. We say that R_{1} is invariance if and only if: (Def.5) for all a, b, p, q such that for every m holds a ⊕ q(m) = b ⊕ p(m) holds

a ⊕ ∗(b, q) = b ⊕ ∗(a, p). Let us consider n, i, R_{1}. We say that R_{1} has property of zero in i if and only if: (Def.6) for all a, p holds ∗(a, p_{i ˙→a}) = a. We say that R_{1} is semi additive in i if and only if: (Def.7) for all a, p_{1}, p such that p(i) = p_{1} holds ∗(a, p_{i ˙→a⊕p_{1}}) = a ⊕ ∗(a, p).

The following proposition is true

(13) If R_{1} is semi additive in m, then for all a, d, p, q such that q = p_{m ˙→d}

holds ∗(a, p_{m ˙→a⊕d}) = a ⊕ ∗(a, q). We now define two new predicates. Let us consider n, i, R_{1}. We say that R_{1}

is additive in i if and only if: (Def.8) for all a, p_{1}, p′_{1}, p such that p(i) = p_{1} holds ∗(a, p_{i ˙→p_{1}⊕p′_{1}}) = ∗(a, p) ⊕

∗(a, p_{i ˙→p′_{1}}). We say that R_{1} is alternative in i if and only if: (Def.9) for all a, p, p_{1} such that p(i) = p_{1} holds ∗(a, p_{i+1 ˙→p_{1}}) = a.

In the sequel W is an atlas of R_{1} and v is a vector of W. Let us consider n, R_{1}, W, i. A tuple of i and W is a tuple of i and the carrier of the algebra of W.

In the sequel x, y are tuples of n + 1 and W. Let us consider n, R_{1}, W, x, i, v. The functor x_{i ˙→v} yields a tuple of n + 1 and W and is defined by:

(Def.10) for every D and for every element x′ of D^{n+1} and for every element v′ of D such that D = the carrier of the algebra of W and x′ = x and v′ = v holds x_{i ˙→v} = x′(i/v′).

Next we state three propositions:

(14) If i ∈ Seg(n + 1), then x_{i ˙→v}(i) = v and for every l such that l ∈ Seg len x \ {i} holds x_{i ˙→v}(l) = x(l).

(15) For every natural number l of n such that l = i holds x_{i ˙→v}(l) = v and for all natural numbers l, i of n such that l ≠ i holds x_{i ˙→v}(l) = x(l).

(16) If for every m holds x(m) = y(m), then x = y.

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

there exists a finite sequence z of elements of B such that len z = A + 1 and for every natural number j of A holds z(j) = F(j) for all values of the parameters.

We now define two new functors. Let us consider n, R_{1}, W, a, x. The functor (a, x).W yielding a tuple of n + 1 and R_{1} is defined as follows:

(Def.11) ((a, x).W)(m) = (a, x(m)).W. Let us consider n, R_{1}, W, a, p. The functor W (a, p) yielding a tuple of n + 1 and W is defined by:

(Def.12) W (a, p)(m) = W (a, p(m)).

The following three propositions are true:

(17) W (a, p) = x if and only if (a, x).W = p.

(18) W (a, (a, x).W) = x.

(19) (a, W (a, p)).W = p.

Let us consider n, R_{1}, W, a, x. The functor Φ(a, x) yields a vector of W and is defined by:

(Def.13) Φ(a, x) = W (a, ∗(a, (a, x).W)).

One can prove the following propositions:

(20) If W (a, p) = x and W (a, b) = v, then ∗(a, p) = b if and only if Φ(a, x) = v.

(21) R_{1} is invariance if and only if for all a, b, x holds Φ(a, x) = Φ(b, x).

(22) 1 ∈ Seg(n + 1).

(23) 1 is an element of Seg(n + 1).

(24) 1 is a natural number of n.

3. Reper Algebra and its Atlas

Let us consider n. A midpoint algebra-like structure of reper algebra over n + 2 is called a reper algebra of n if:

(Def.14) it is invariance.

For simplicity we adopt the following convention: R_{1} will be a reper algebra of n, a, b will be points of R_{1}, p will be a tuple of n + 1 and R_{1}, W will be an atlas of R_{1}, v will be a vector of W, and x will be a tuple of n + 1 and W. Next we state the proposition

(25) Φ(a, x) = Φ(b, x).

reper algebras 27

Let us consider n, R_{1}, W, x. The functor Φ(x) yields a vector of W and is defined by:

(Def.15) for every a holds Φ(x) = Φ(a, x).

We now state a number of propositions:

(26) If W (a, p) = x and W (a, b) = v and Φ(x) = v, then ∗(a, p) = b.

(27) If (a, x).W = p and (a, v).W = b and ∗(a, p) = b, then Φ(x) = v.

(28) If W (a, p) = x and W (a, b) = v, then W (a, p_{m ˙→b}) = x_{m ˙→v}.

(29) If (a, x).W = p and (a, v).W = b, then (a, x_{m ˙→v}).W = p_{m ˙→b}.

(30) R_{1} has property of zero in m if and only if for every x holds Φ((x_{m ˙→0_{W}} )) = 0_{W}.

(31) R_{1} is semi additive in m if and only if for every x holds Φ((x_{m ˙→2x(m)})) = 2 Φ(x).

(32) If R_{1} has property of zero in m and R_{1} is additive in m, then R_{1} is semi additive in m.

(33) If R_{1} has property of zero in m, then R_{1} is additive in m if and only if for all x, v holds Φ((x_{m ˙→x(m)+v})) = Φ(x) + Φ((x_{m ˙→v})).

(34) If W (a, p) = x and m ≤ n, then W (a, p_{m+1 ˙→p(m)}) = x_{m+1 ˙→x(m)}.

(35) If (a, x).W = p and m ≤ n, then (a, x_{m+1 ˙→x(m)}).W = p_{m+1 ˙→p(m)}.

(36) If m ≤ n, then R_{1} is alternative in m if and only if for every x holds Φ((x_{m+1 ˙→x(m)})) = 0_{W}.

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

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

[9] Michał Muzalewski. Atlas of Midpoint Algebra. Formalized Mathematics, 2(4):487–491, 1991.

[10] Michał Muzalewski. Foundations of Metric-Affine Geometry. Dział Wydawnictw Filii UW w Białymstoku, Filia UW w Białymstoku, 1990.

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

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

28 michał muzalewski

[13] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized

Mathematics, 1(1):17–23, 1990.

Received May 28, 1992

Isomorphisms of Cyclic Groups. Some Properties of Cyclic Groups

Dariusz Surowik Warsaw University Białystok

Summary. Some theorems and properties of cyclic groups have been proved with special regard to isomorphisms of these groups. Among other things it has been proved that an arbitrary cyclic group is isomorphic with groups of integers with addition or group of integers with addition modulo m. Moreover, it has been proved that two arbitrary cyclic groups of the same order are isomorphic and that the class of cyclic groups is closed in consideration of homomorphism images. Some other properties of groups of this type have been proved too.

MML Identifier: GR CY 2.

The terminology and notation used in this paper have been introduced in the following articles: [19], [6], [11], [7], [12], [2], [18], [1], [10], [4], [14], [17], [21], [13], [31], [25], [29], [23], [3], [27], [26], [24], [30], [15], [16], [5], [28], [22], [20], [9], and [8]. For simplicity we adopt the following rules: F, G will be groups, G_{1} will be a subgroup of G, G_{2} will be a cyclic group, H will be a subgroup of G_{2}, f will be a homomorphism from G to G_{2}, a, b will be elements of G, g will be an element of G_{2}, a_{1} will be an element of G_{1}, k, m, n, p, s will be natural numbers, and i, i_{1}, i_{2} will be integers. The following propositions are true:

(1) For all n, m such that 0 < m holds n mod m = n − m · (n ÷ m).

(2) If i_{2} > 0, then i_{1} mod i_{2} ≥ 0.

(3) If i_{2} > 0, then i_{1} mod i_{2} < i_{2}.

(4) i_{1} = (i_{1} ÷ i_{2}) · i_{2} + (i_{1} mod i_{2}).

(5) For all m, n such that m > 0 or n > 0 there exist i, i_{1} such that i · m + i_{1} · n = gcd(m, n).

(6) If ord(a) > 1 and a = b^{k}, then k ≠ 0.

(7) If G is finite, then ord(G) > 0.

(8) a ∈ gr({a}).

(9) If a = a_{1}, then gr({a}) = gr({a_{1}}).

(10) gr({a}) is a cyclic group.

(11) For every strict group G and for every element b of G holds for every element a of G there exists i such that a = b^{i} if and only if G = gr({b}).

(12) For every strict group G and for every element b of G such that G is finite holds for every element a of G there exists p such that a = b^{p} if and only if G = gr({b}).

(13) For every strict group G and for every element a of G such that G is finite and G = gr({a}) and for every strict subgroup G_{1} of G there exists p such that G_{1} = gr({a^{p}}).

(14) If G is finite and G = gr({a}) and ord(G) = n and n = p · s, then ord(a^{p}) = s.

(15) If s | k, then a^{k} ∈ gr({a^{s}}).

(16) If G is finite and ord(gr({a^{s}})) = ord(gr({a^{k}})) and a^{k} ∈ gr({a^{s}}), then gr({a^{s}}) = gr({a^{k}}).

(17) If G is finite and ord(G) = n and G = gr({a}) and ord(G_{1}) = p and G_{1} = gr({a^{k}}), then n | k · p.

(18) For every strict group G and for every element a of G such that G is finite and G = gr({a}) and ord(G) = n holds G = gr({a^{k}}) if and only if gcd(k, n) = 1.

(19) If G_{2} = gr({g}) and g ∈ H, then the half group structure of G_{2} = the half group structure of H.

(20) If G_{2} = gr({g}), then G_{2} is finite if and only if there exist i, i_{1} such that i ≠ i_{1} and g^{i} = g^{i_{1}}.

Let us consider n satisfying the condition: n > 0. Let h be an element of _{n}^{+}. The functor ^{@}h yielding a natural number is defined as follows:

(Def.1) ^{@}h = h.

The following propositions are true:

(21) For every strict cyclic group G_{2} such that G_{2} is finite and ord(G_{2}) = n holds _{n}^{+} and G_{2} are isomorphic.

(22) For every strict cyclic group G_{2} such that G_{2} is infinite holds ^{+} and G_{2} are isomorphic.

(23) For all strict cyclic groups G_{2}, H_{1} such that H_{1} is finite and G_{2} is finite and ord(H_{1}) = ord(G_{2}) holds H_{1} and G_{2} are isomorphic.

(24) For all strict groups F, G such that F is finite and G is finite and ord(F) = p and ord(G) = p and p is prime holds F and G are isomorphic.

(25) For all strict groups F, G such that F is finite and G is finite and ord(F) = 2 and ord(G) = 2 holds F and G are isomorphic.

(26) For every strict group G such that G is finite and ord(G) = 2 and for every strict subgroup H of G holds H = {1}_{G} or H = G.

isomorphisms of cyclic groups. … 31

(27) For every strict group G such that G is finite and ord(G) = 2 holds G is a cyclic group.

(28) For every strict group G such that G is finite and G is a cyclic group and ord(G) = n and for every p such that p | n there exists a strict subgroup G_{1} of G such that ord(G_{1}) = p and for every strict subgroup G_{3} of G such that ord(G_{3}) = p holds G_{3} = G_{1}.

Let us note that every group which is cyclic is also Abelian.

We now state two propositions:

(29) If G_{2} = gr({g}), then for all G, f such that g ∈ Im f holds f is an epimorphism.

(30) For every strict cyclic group G_{2} such that G_{2} is finite and ord(G_{2}) = n and there exists k such that n = 2·k there exists an element g_{1} of G_{2} such that ord(g_{1}) = 2 and for every element g_{2} of G_{2} such that ord(g_{2}) = 2 holds g_{1} = g_{2}.

Let us consider G. Then Z(G) is a strict normal subgroup of G.

One can prove the following propositions:

(31) For every strict cyclic group G_{2} such that G_{2} is finite and ord(G_{2}) = n and there exists k such that n = 2 · k there exists a subgroup H of G_{2} such that ord(H) = 2 and H is a cyclic group.

(32) For every strict group G and for every homomorphism g from G to F such that G is a cyclic group holds Im g is a cyclic group.

(33) For all strict groups G, F such that G and F are isomorphic but G is a cyclic group or F is a cyclic group holds G is a cyclic group and F is a cyclic group.

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Some Isomorphisms Between Functor Categories

Andrzej Trybulec Warsaw University Białystok

Summary. We define some well known isomorphisms between functor categories: between A^{^{}˙ (o,m)} and A, between C^{[:A,B:]} and (C^{B})^{A}, and between [:B, C:]^{A} and [:B^{A}, C^{A}:]. Compare [12] and [11]. Unfortunately in this paper ”functor” is used in two different meanings, as a lingual function and as a functor between categories.

MML Identifier: ISOCAT 2.

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

1. Preliminaries

The scheme ChoiceD concerns a non-empty set A, a non-empty set B, and a binary predicate P, and states that:

there exists a function h from A into B such that for every element a of A holds P[a, h(a)] provided the parameters meet the following requirement:

• for every element a of A there exists an element b of B such that P[a, b].

Let A, B, C be non-empty sets, and let f be a function from A into C^{B}. Then uncurry f is a function from [: A, B:] into C.

We now state several propositions:

(1) For all non-empty sets A, B, C and for every function f from A into C^{B} holds curry uncurry f = f.

(2) For all non-empty sets A, B, C and for every function f from A into C^{B} and for every element a of A and for every element b of B holds (uncurry f)(⟨⟨a, b⟩⟩) = f(a)(b).

(3) For an arbitrary x and for every non-empty set A and for all functions f, g from {x} into A such that f(x) = g(x) holds f = g.

(4) For all non-empty sets A, B and for every element x of A and for every function f from A into B holds f(x) ∈ rng f.

(5) For all non-empty sets A, B, C and for all functions f, g from A into [: B, C:] such that π_{1}(B×C)·f = π_{1}(B×C)·g and π_{2}(B×C)·f = π_{2}(B×C)·g holds f = g.

We adopt the following rules: A, B, C will be categories and F, F_{1}, F_{2} will be functors from A to B. The following two propositions are true:

(6) For every morphism f of A holds id_{cod f} · f = f.

(7) For every morphism f of A holds f · id_{dom f} = f. In the sequel o, m will be arbitrary. The following two propositions are true:

(8) o is an object of B^{A} if and only if o is a functor from A to B.

(9) For every morphism f of B^{A} there exist functors F_{1}, F_{2} from A to B and there exists a natural transformation t from F_{1} to F_{2} such that F_{1} is naturally transformable to F_{2} and dom f = F_{1} and cod f = F_{2} and f = ⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t⟩⟩.

2. The isomorphism between A^{_{}˙ (o,m)} and A

Let us consider A, B, and let a be an object of A. The functor a ↦ B yields a functor from B^{A} to B and is defined by: (Def.1) for all functors F_{1}, F_{2} from A to B and for every natural transformation

t from F_{1} to F_{2} such that F_{1} is naturally transformable to F_{2} holds (a ↦ B)(⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t⟩⟩) = t(a).

One can prove the following two propositions:

(10) The objects of ˙ (o, m) = {o} and the morphisms of ˙ (o, m) = {m}.

(11) A^{_{}˙ (o,m)} ≅ A.

3. The isomorphism between C^{[:A,B:]} and (C^{B})^{A}

Next we state four propositions:

(12) For every functor F from [: A, B:] to C and for every object a of A and for every object b of B holds F (a, −)(b) = F (⟨⟨a, b⟩⟩).

(13) For all objects a_{1}, a_{2} of A and for all objects b_{1}, b_{2} of B holds hom(a_{1}, a_{2}) ≠ ∅ and hom(b_{1}, b_{2}) ≠ ∅ if and only if hom(⟨⟨a_{1}, b_{1}⟩⟩, ⟨⟨a_{2}, b_{2}⟩⟩) ≠ ∅.

(14) Let a_{1}, a_{2} be objects of A. Then for all objects b_{1}, b_{2} of B such that hom(⟨⟨a_{1}, b_{1}⟩⟩, ⟨⟨a_{2}, b_{2}⟩⟩) ≠ ∅ and for every morphism f of A and for every morphism g of B holds ⟨⟨f, g⟩⟩ is a morphism from ⟨⟨a_{1}, b_{1}⟩⟩ to ⟨⟨a_{2}, b_{2}⟩⟩ if and only if f is a morphism from a_{1} to a_{2} and g is a morphism from b_{1} to b_{2}.

(15) For all functors F_{1}, F_{2} from [: A, B:] to C such that F_{1} is naturally transformable to F_{2} and for every natural transformation t from F_{1} to F_{2} and for every object a of A holds F_{1}(a, −) is naturally transformable to F_{2}(a, −) and (curry t)(a) is a natural transformation from F_{1}(a, −) to F_{2}(a, −).

Let us consider A, B, C, and let F be a functor from [: A, B:] to C, and let f be a morphism of A. The functor curry(F, f) yields a function from the morphisms of B into the morphisms of C and is defined by: (Def.2) curry(F, f) = (curry F)(f).

The following two propositions are true:

(16) For all objects a_{1}, a_{2} of A and for all objects b_{1}, b_{2} of B and for every morphism f of A and for every morphism g of B such that f ∈ hom(a_{1}, a_{2}) and g ∈ hom(b_{1}, b_{2}) holds ⟨⟨f, g⟩⟩ ∈ hom(⟨⟨a_{1}, b_{1}⟩⟩, ⟨⟨a_{2}, b_{2}⟩⟩).

(17) For every functor F from [: A, B:] to C and for all objects a, b of A such that hom(a, b) ≠ ∅ and for every morphism f from a to b holds F (a, −) is naturally transformable to F (b, −) and curry(F, f) · the id-map of B is a natural transformation from F (a, −) to F (b, −).

Let us consider A, B, C, and let F be a functor from [: A, B:] to C, and let f be a morphism of A. The functor F (f, −) yielding a natural transformation from F (dom f, −) to F (cod f, −) is defined by: (Def.3) F (f, −) = curry(F, f) · the id-map of B.

We now state four propositions:

(18) For every functor F from [: A, B:] to C and for every morphism g of A holds F (dom g, −) is naturally transformable to F (cod g, −).

(19) For every functor F from [: A, B:] to C and for every morphism f of A and for every object b of B holds F (f, −)(b) = F (⟨⟨f, id_{b}⟩⟩).

(20) For every functor F from [: A, B:] to C and for every object a of A holds id_{F (a,−)} = F (id_{a}, −).

(21) For every functor F from [: A, B:] to C and for all morphisms g, f of A such that dom g = cod f and for every natural transformation t from F (dom f, −) to F (dom g, −) such that t = F (f, −) holds F (g · f, −) = F (g, −)^{∘} t.

Let us consider A, B, C, and let F be a functor from [: A, B:] to C. The functor export(F) yielding a functor from A to C^{B} is defined as follows: (Def.4) for every morphism f of A holds (export(F))(f) = ⟨⟨⟨⟨F (dom f, −),

F (cod f, −)⟩⟩, F (f, −)⟩⟩.

We now state several propositions:

(22) For every functor F from [: A, B:] to C and for every morphism f of A holds (export(F))(f) = ⟨⟨⟨⟨F (dom f, −), F (cod f, −)⟩⟩, F (f, −)⟩⟩.

(23) For all functors F_{1}, F_{2} from A to B such that F_{1} is transformable to F_{2} and for every transformation t from F_{1} to F_{2} and for every object a of A holds t(a) ∈ hom(F_{1}(a), F_{2}(a)).

(24) For every functor F from [: A, B:] to C and for every object a of A holds (export(F))(a) = F (a, −).

(25) For every functor F from [: A, B:] to C and for every object a of A holds (export(F))(a) is a functor from B to C.

(26) For all functors F_{1}, F_{2} from [: A, B:] to C such that export(F_{1}) = export(F_{2}) holds F_{1} = F_{2}.

(27) Let F_{1}, F_{2} be functors from [: A, B:] to C. Suppose F_{1} is naturally transformable to F_{2}. Let t be a natural transformation from F_{1} to F_{2}. Then export(F_{1}) is naturally transformable to export(F_{2}) and there exists a natural transformation G from export(F_{1}) to export(F_{2}) such that for every function s from [: the objects of A, the objects of B:] into the morphisms of C such that t = s and for every object a of A holds G(a) = ⟨⟨⟨⟨(export(F_{1}))(a), (export(F_{2}))(a)⟩⟩, (curry s)(a)⟩⟩.

Let us consider A, B, C, and let F_{1}, F_{2} be functors from [: A, B:] to C satisfying the condition: F_{1} is naturally transformable to F_{2}. Let t be a natural transformation from F_{1} to F_{2}. The functor export(t) yielding a natural transformation from export(F_{1}) to export(F_{2}) is defined as follows:

(Def.5) for every function s from [: the objects of A, the objects of B:] into the morphisms of C such that t = s and for every object a of A holds (export(t))(a) = ⟨⟨⟨⟨(export(F_{1}))(a), (export(F_{2}))(a)⟩⟩, (curry s)(a)⟩⟩.

We now state several propositions:

(28) For every functor F from [: A, B:] to C holds id_{export(F)} = export(id_{F}).

(29) For all functors F_{1}, F_{2}, F_{3} from [: A, B:] to C such that F_{1} is naturally transformable to F_{2} and F_{2} is naturally transformable to F_{3} and for every natural transformation t_{1} from F_{1} to F_{2} and for every natural transformation t_{2} from F_{2} to F_{3} holds export(t_{2} ^{∘} t_{1}) = export(t_{2})^{∘} export(t_{1}).

(30) For all functors F_{1}, F_{2} from [: A, B:] to C such that F_{1} is naturally transformable to F_{2} and for all natural transformations t_{1}, t_{2} from F_{1} to F_{2} such that export(t_{1}) = export(t_{2}) holds t_{1} = t_{2}.

(31) For every functor G from A to C^{B} there exists a functor F from [: A, B:] to C such that G = export(F).

(32) For all functors F_{1}, F_{2} from [: A, B:] to C such that export(F_{1}) is naturally transformable to export(F_{2}) and for every natural transformation t from export(F_{1}) to export(F_{2}) holds F_{1} is naturally transformable to F_{2} and there exists a natural transformation u from F_{1} to F_{2} such that t = export(u).

Let us consider A, B, C. The functor export_{A,B,C} yields a functor from C^{[: A, B:]} to (C^{B})^{A} and is defined by: (Def.6) for all functors F_{1}, F_{2} from [: A, B:] to C such that F_{1} is naturally trans-

formable to F_{2} and for every natural transformation t from F_{1} to F_{2} holds export_{A,B,C}(⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t⟩⟩) = ⟨⟨⟨⟨export(F_{1}), export(F_{2})⟩⟩, export(t)⟩⟩.

Next we state two propositions:

(33) export_{A,B,C} is an isomorphism.

(34) C^{[: A, B:]} ≅ (C^{B})^{A}.

4. The isomorphism between [:B, C:]^{A} and [:B^{A}, C^{A}:]

We now state the proposition

(35) For all functors F_{1}, F_{2} from A to B and for every functor G from B to C such that F_{1} is naturally transformable to F_{2} and for every natural transformation t from F_{1} to F_{2} holds G · t = G · t qua a function .

We now define two new functors. Let us consider A, B. Then π_{1}(A × B) is a functor from [: A, B:] to A. Then π_{2}(A × B) is a functor from [: A, B:] to B. Let us consider A, B, C, and let F be a functor from A to B, and let G be a functor from A to C. Then ⟨F, G⟩ is a functor from A to [: B, C:]. Let F be a functor from A to [: B, C:]. The functor π_{1}· F yielding a functor from A to B is defined as follows: (Def.7) π_{1}· F = π_{1}(B × C) · F. The functor π_{2}· F yielding a functor from A to C is defined by: (Def.8) π_{2}· F = π_{2}(B × C) · F.

The following two propositions are true:

(36) For every functor F from A to B and for every functor G from A to C holds π_{1}· ⟨F, G⟩ = F and π_{2}· ⟨F, G⟩ = G.

(37) For all functors F, G from A to [: B, C:] such that π_{1}· F = π_{1}· G and

π_{2}· F = π_{2}· G holds F = G. We now define two new functors. Let us consider A, B, C, and let F_{1}, F_{2}

be functors from A to [: B, C:], and let t be a natural transformation from F_{1} to F_{2}. The functor π_{1}· t yielding a natural transformation from π_{1}· F_{1} to π_{1}· F_{2} is defined as follows: (Def.9) π_{1}· t = π_{1}(B × C) · t. The functor π_{2}·t yielding a natural transformation from π_{2}·F_{1} to π_{2}·F_{2} is defined as follows:

(Def.10) π_{2}· t = π_{2}(B × C) · t.

We now state several propositions:

(38) For all functors F, G from A to [: B, C:] such that F is naturally transformable to G holds π_{1}· F is naturally transformable to π_{1}· G and π_{2}· F is naturally transformable to π_{2}· G.

(39) For all functors F_{1}, F_{2}, G_{1}, G_{2} from A to [: B, C:] such that F_{1} is naturally transformable to F_{2} and G_{1} is naturally transformable to G_{2} and for every natural transformation s from F_{1} to F_{2} and for every natural transformation t from G_{1} to G_{2} such that π_{1}· s = π_{1}· t and π_{2}· s = π_{2}· t holds s = t.

(40) For every functor F from A to [: B, C:] holds id_{π_{1}·F} = π_{1} · (id_{F}) and id_{π_{2}·F} = π_{2}· (id_{F}).

(41) For all functors F, G, H from A to [: B, C:] such that F is naturally transformable to G and G is naturally transformable to H and for every natural transformation s from F to G and for every natural transformation t from G to H holds π_{1}·(t^{∘} s) = π_{1}·t^{∘} π_{1}·s and π_{2}·(t^{∘} s) = π_{2}·t^{∘} π_{2}·s.

(42) For every functor F from A to B and for every functor G from A to C and for all objects a, b of A such that hom(a, b) ≠ ∅ and for every morphism f from a to b holds ⟨F, G⟩(f) = ⟨⟨F (f), G(f)⟩⟩.

(43) For every functor F from A to B and for every functor G from A to C and for every object a of A holds ⟨F, G⟩(a) = ⟨⟨F (a), G(a)⟩⟩.

(44) For all functors F_{1}, G_{1} from A to B and for all functors F_{2}, G_{2} from A to C such that F_{1} is transformable to G_{1} and F_{2} is transformable to G_{2} holds ⟨F_{1}, F_{2}⟩ is transformable to ⟨G_{1}, G_{2}⟩.

Let us consider A, B, C, and let F_{1}, G_{1} be functors from A to B, and let F_{2}, G_{2} be functors from A to C satisfying the condition: F_{1} is transformable to G_{1} and F_{2} is transformable to G_{2}. Let t_{1} be a transformation from F_{1} to G_{1}, and let t_{2} be a transformation from F_{2} to G_{2}. The functor ⟨t_{1}, t_{2}⟩ yielding a transformation from ⟨F_{1}, F_{2}⟩ to ⟨G_{1}, G_{2}⟩ is defined as follows:

(Def.11) ⟨t_{1}, t_{2}⟩ = ⟨t_{1}, t_{2}⟩.

One can prove the following propositions:

(45) For all functors F_{1}, G_{1} from A to B and for all functors F_{2}, G_{2} from A to C such that F_{1} is transformable to G_{1} and F_{2} is transformable to G_{2} and for every transformation t_{1} from F_{1} to G_{1} and for every transformation t_{2} from F_{2} to G_{2} and for every object a of A holds ⟨t_{1}, t_{2}⟩(a) = ⟨⟨t_{1}(a), t_{2}(a)⟩⟩.

(46) For all functors F_{1}, G_{1} from A to B and for all functors F_{2}, G_{2} from A to C such that F_{1} is naturally transformable to G_{1} and F_{2} is naturally transformable to G_{2} holds ⟨F_{1}, F_{2}⟩ is naturally transformable to ⟨G_{1}, G_{2}⟩.

Let us consider A, B, C, and let F_{1}, G_{1} be functors from A to B, and let F_{2}, G_{2} be functors from A to C satisfying the conditions: F_{1} is naturally transformable to G_{1} and F_{2} is naturally transformable to G_{2}. Let t_{1} be a natural transformation from F_{1} to G_{1}, and let t_{2} be a natural transformation from F_{2} to G_{2}. The functor ⟨t_{1}, t_{2}⟩ yielding a natural transformation from ⟨F_{1}, F_{2}⟩ to ⟨G_{1}, G_{2}⟩ is defined as follows:

some isomorphisms between functor categories 39

(Def.12) ⟨t_{1}, t_{2}⟩ = ⟨t_{1}, t_{2}⟩.

Next we state the proposition

(47) For all functors F_{1}, G_{1} from A to B and for all functors F_{2}, G_{2} from A to C such that F_{1} is naturally transformable to G_{1} and F_{2} is naturally transformable to G_{2} and for every natural transformation t_{1} from F_{1} to G_{1} and for every natural transformation t_{2} from F_{2} to G_{2} holds π_{1}·⟨t_{1}, t_{2}⟩ = t_{1} and π_{2}· ⟨t_{1}, t_{2}⟩ = t_{2}.

Let us consider A, B, C. The functor distribute_{A,B,C} yielding a functor from [: B, C:]^{A} to [: B^{A}, C^{A}:] is defined by:

(Def.13) for all functors F_{1}, F_{2} from A to [: B, C:] such that F_{1} is naturally transformable to F_{2} and for every natural transformation t from F_{1} to F_{2} holds distribute_{A,B,C}(⟨⟨⟨⟨F_{1}, F_{2}⟩⟩, t⟩⟩) = ⟨⟨⟨⟨⟨⟨π_{1}· F_{1}, π_{1}· F_{2}⟩⟩, π_{1}· t⟩⟩, ⟨⟨⟨⟨π_{2}· F_{1}, π_{2}· F_{2}⟩⟩, π_{2}· t⟩⟩⟩⟩.

One can prove the following two propositions:

(48) distribute_{A,B,C} is an isomorphism.

(49) [: B, C:]^{A} ≅ [: B^{A}, C^{A}:].

References

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

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

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

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

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

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

[7] 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.

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

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

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

[11] Saunders Mac Lane. Categories for the Working Mathematician. Volume 5 of Graduate Texts in Mathematics, Springer Verlag, New York, Heidelberg, Berlin, 1971.

[12] Zbigniew Semadeni and Antoni Wiweger. Wstȩp do teorii kategorii i funktorów. Volume 45 of Biblioteka Matematyczna, PWN, Warszawa, 1978.

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

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

[15] Andrzej Trybulec. Isomorphisms of categories. Formalized Mathematics, 2(5):629–634, 1991.

[16] Andrzej Trybulec. Natural transformations. Discrete categories. Formalized Mathematics, 2(4):467–474, 1991.

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

40 andrzej trybulec

[18] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics,

1(1):97–105, 1990.

Received June 5, 1992

The Lattice of Domains of a Topological Space ^{1}

Toshihiko Watanabe Shinshu University Nagano

Summary. Let T be a topological space and let A be a subset of T. Recall that A is said to be a closed domain of T if A = \overline{Int A} and A is said to be an open domain of T if A = Int \overline{A} (see e.g. [8], [15]). Some simple generalization of these notions is the following one. A is said to be a domain of T provided Int \overline{A} ⊆ A ⊆ \overline{Int A} (see [15] and compare [7]). In this paper certain connections between these concepts are introduced and studied.

Our main results are concerned with the following well–known theorems (see e.g. [9], [2]). For a given topological space all its closed domains form a Boolean lattice, and similarly all its open domains form a Boolean lattice, too. It is proved that all domains of a given topological space form a complemented lattice. Moreover, it is shown that both the lattice of open domains and the lattice of closed domains are sublattices of the lattice of all domains. In the beginning some useful theorems about subsets of topological spaces are proved and certain properties of domains, closed domains and open domains are discussed.

MML Identifier: TDLAT 1.

The terminology and notation used in this paper are introduced in the following articles: [14], [11], [4], [5], [16], [3], [13], [10], [15], [1], [12], and [6].

1. Preliminary Theorems on Subset of Topological Spaces

In the sequel T is a topological space. We now state a number of propositions: (1) For all subsets A, B of T holds A ∪ B = Ω_{T} if and only if A^{c} ⊆ B.

(2) For all subsets A, B of T holds A ∩ B = ∅_{T} if and only if B ⊆ A^{c}.

1This paper was done under the supervision of Z. Karno while the author was visiting the Institute of Mathematics of Warsaw University in Białystok.

(3) For every subset A of T holds \overline{Int \overline{A}} ⊆ \overline{A}.

(4) For every subset A of T holds Int A ⊆ Int \overline{Int A}.

(5) For every subset A of T holds Int \overline{A} = Int \overline{Int \overline{A}}.

(6) For all subsets A, B of T such that A is closed or B is closed holds \overline{Int A} ∪ \overline{Int B} = \overline{Int(A ∪ B)}.

(7) For all subsets A, B of T such that A is open or B is open holds Int \overline{A} ∩ Int \overline{B} = Int \overline{A ∩ B}.

(8) For every subset A of T holds Int(A ∩ \overline{A^{c}}) = ∅_{T}.

(9) For every subset A of T holds \overline{A ∪ Int(A^{c})} = Ω_{T}.

(10) For all subsets A, B of T holds Int \overline{A ∪ (Int \overline{B} ∪ B)}∪(A∪(Int \overline{B} ∪B)) = Int \overline{A ∪ B} ∪ (A ∪ B).

(11) For all subsets A, C of T holds Int \overline{Int \overline{A} ∪ A ∪ C} ∪ (Int \overline{A} ∪ A ∪ C) = Int \overline{A ∪ C} ∪ (A ∪ C).

(12) For all subsets A, B of T holds \overline{Int(A ∩ (\overline{Int B} ∩ B))}∩(A∩(\overline{Int B}∩B)) = \overline{Int(A ∩ B)} ∩ (A ∩ B).

(13) For all subsets A, C of T holds \overline{Int(\overline{Int A} ∩ A ∩ C)} ∩ (\overline{Int A} ∩ A ∩ C) = \overline{Int(A ∩ C)} ∩ (A ∩ C).

2. Properties of Domains of Topological Spaces

In the sequel T will be a topological space. Next we state a number of propositions:

(14) ∅_{T} is a domain.

(15) Ω_{T} is a domain.

(16) For every subset A of T such that A is a domain holds A^{c} is a domain.

(17) For all subsets A, B of T such that A is a domain and B is a domain holds Int \overline{A ∪ B}∪(A∪B) is a domain and \overline{Int(A ∩ B)}∩(A∩B) is a domain.

(18) ∅_{T} is a closed domain.

(19) Ω_{T} is a closed domain.

(20) ∅_{T} is an open domain.

(21) Ω_{T} is an open domain.

(22) For every subset A of T holds \overline{Int A} is a closed domain.

(23) For every subset A of T holds Int \overline{A} is an open domain.

(24) For every subset A of T such that A is a domain holds \overline{A} is a closed domain.

(25) For every subset A of T such that A is a domain holds Int A is an open domain.

(26) For every subset A of T such that A is a domain holds \overline{A^{c}} is a closed domain.

(27) For every subset A of T such that A is a domain holds Int(A^{c}) is an open domain.

(28) For all subsets A, B, C of T such that A is a closed domain and B is a closed domain and C is a closed domain holds \overline{Int(A ∩ \overline{Int(B ∩ C)})} = \overline{Int(\overline{Int(A ∩ B)} ∩ C)}.

(29) For all subsets A, B, C of T such that A is an open domain and B is an open domain and C is an open domain holds Int \overline{A ∪ Int \overline{B ∪ C}} = Int \overline{Int \overline{A ∪ B} ∪ C}.

3. The Lattice of Domains

We now define five new functors. Let T be a topological space. The domains of T yields a non-empty family of subsets of the carrier of T and is defined as follows: (Def.1) the domains of T = {A : A is a domain}, where A ranges over subsets

of T. The domains union of T yielding a binary operation on the domains of T is defined by: (Def.2) for all elements A, B of the domains of T holds (the domains union of

T)(A, B) = Int \overline{A ∪ B} ∪ (A ∪ B). We introduce the functor D-Union(T) as a synonym of the domains union of T. The domains meet of T yields a binary operation on the domains of T and is defined as follows: (Def.3) for all elements A, B of the domains of T holds (the domains meet of

T)(A, B) = \overline{Int(A ∩ B)} ∩ (A ∩ B). We introduce the functor D-Meet(T) as a synonym of the domains meet of T.

One can prove the following proposition

(30) For every topological space T holds ⟨the domains of T , D-Union(T),

D-Meet(T)⟩ is a complemented lattice. Let T be a topological space. The lattice of domains of T yields a comple-

mented lattice and is defined by: (Def.4) the lattice of domains of T = ⟨the domains of T , the domains union of

T , the domains meet of T ⟩.

4. The Lattice of Closed Domains

Let T be a topological space. The closed domains of T yielding a non-empty family of subsets of the carrier of T is defined as follows:

(Def.5) the closed domains of T = {A : A is a closed domain}, where A ranges over subsets of T.

Next we state the proposition

(31) For every topological space T holds the closed domains of T ⊆ the

domains of T. We now define two new functors. Let T be a topological space. The closed

domains union of T yielding a binary operation on the closed domains of T is defined by: (Def.6) for all elements A, B of the closed domains of T holds (the closed

domains union of T)(A, B) = A ∪ B. We introduce the functor CLD-Union(T) as a synonym of the closed domains union of T.

Next we state the proposition

(32) For all elements A, B of the closed domains of T holds

(CLD-Union(T))(A, B) = (D-Union(T))(A, B). We now define two new functors. Let T be a topological space. The closed

domains meet of T yielding a binary operation on the closed domains of T is defined as follows: (Def.7) for all elements A, B of the closed domains of T holds (the closed

domains meet of T)(A, B) = \overline{Int(A ∩ B)}. We introduce the functor CLD-Meet(T) as a synonym of the closed domains meet of T.

One can prove the following two propositions:

(33) For all elements A, B of the closed domains of T holds (CLD-Meet(T))(A, B) = (D-Meet(T))(A, B).

(34) For every topological space T holds ⟨the closed domains of T , CLD-Union(T), CLD-Meet(T)⟩ is a Boolean lattice.

Let T be a topological space. The lattice of closed domains of T yielding a Boolean lattice is defined as follows: (Def.8) the lattice of closed domains of T = ⟨the closed domains of T , the closed

domains union of T , the closed domains meet of T ⟩.

5. The Lattice of Open Domains

Let T be a topological space. The open domains of T yields a non-empty family of subsets of the carrier of T and is defined by: (Def.9) the open domains of T = {A : A is an open domain}, where A ranges

over subsets of T.

Next we state the proposition (35) For every topological space T holds the open domains of T ⊆ the do-

mains of T. We now define two new functors. Let T be a topological space. The open

domains union of T yielding a binary operation on the open domains of T is defined by:

(Def.10) for all elements A, B of the open domains of T holds (the open domains

union of T)(A, B) = Int \overline{A ∪ B}. We introduce the functor OPD-Union(T) as a synonym of the open domains union of T.

One can prove the following proposition

(36) For all elements A, B of the open domains of T holds (OPD-Union(T))(A,

B) = (D-Union(T))(A, B). We now define two new functors. Let T be a topological space. The open

domains meet of T yielding a binary operation on the open domains of T is defined by:

(Def.11) for all elements A, B of the open domains of T holds (the open domains

meet of T)(A, B) = A ∩ B. We introduce the functor OPD-Meet(T) as a synonym of the open domains meet of T.

We now state two propositions:

(37) For all elements A, B of the open domains of T holds (OPD-Meet(T))(A, B) = (D-Meet(T))(A, B).

(38) For every topological space T holds ⟨the open domains of T , OPD-Union(T), OPD-Meet(T)⟩ is a Boolean lattice.

Let T be a topological space. The lattice of open domains of T yielding a Boolean lattice is defined by:

(Def.12) the lattice of open domains of T = ⟨the open domains of T , the open domains union of T , the open domains meet of T ⟩.

6. Connections between Lattices of Domains

In the sequel T will be a topological space. The following propositions are true:

(39) CLD-Union(T) = D-Union(T) [: the closed domains of T , the closed domains of T :].

(40) CLD-Meet(T) = D-Meet(T) [: the closed domains of T , the closed domains of T :].

(41) The lattice of closed domains of T is a sublattice of the lattice of domains of T.

(42) OPD-Union(T) = D-Union(T) [: the open domains of T , the open domains of T :].

46 toshihiko watanabe

(43) OPD-Meet(T) = D-Meet(T) [: the open domains of T , the open domains of T :].

(44) The lattice of open domains of T is a sublattice of the lattice of domains of T.

Acknowledgments

The author wishes to express his thanks to Professors A. Trybulec and Z. Karno for their active interest in the publication of this article and for useful suggestions and many valuable comments.

References

[1] Grzegorz Bancerek. Filters - part II. Quotient lattices modulo filters and direct product of two lattices. Formalized Mathematics, 2(3):433–438, 1991.

[2] Garrett Birkhoff. Lattice Theory. Providence, Rhode Island, New York, 1967.

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

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

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

[6] Marek Chmur. The lattice of natural numbers and the sublattice of it. The set of prime numbers. Formalized Mathematics, 2(4):453–459, 1991.

[7] Yoshinori Isomichi. New concepts in the theory of topological space – supercondensed set, subcondensed set, and condensed set. Pacific Journal of Mathematics, 38(3):657– 668, 1971.

[8] Kazimierz Kuratowski. Topology. Volume I, PWN - Polish Scientific Publishers, Academic Press, Warsaw, New York and London, 1966.

[9] Kazimierz Kuratowski and Andrzej Mostowski. Set Theory (with an introduction to descriptive set theory). Volume 86 of Studies in Logic and The Foundations of Mathematics, PWN - Polish Scientific Publishers and North-Holland Publishing Company, Warsaw-Amsterdam, 1976.

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

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

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

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

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

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

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

Submodules

Michał Muzalewski Warsaw University Białystok

Summary. This article contains the notions of trivial and nontrivial leftmodules and rings, cyclic submodules and inclusion of submodules. A few basic theorems related to these notions are proved.

MML Identifier: LMOD 6.

The notation and terminology used here are introduced in the following papers: [15], [16], [3], [4], [2], [1], [5], [6], [7], [14], [9], [13], [12], [10], [11], and [8].

1. Preliminaries

For simplicity we adopt the following rules: x is arbitrary, K denotes an associative ring, r denotes a scalar of K, V, M, M_{1}, M_{2}, N denote left modules over K, a denotes a vector of V, m, m_{1}, m_{2} denote vectors of M, n, n_{1}, n_{2} denote vectors of N, A denotes a subset of V, l denotes a linear combination of A, and W, W_{1}, W_{2}, W_{3} denote submodules of V. Next we state four propositions:

(1) If M_{1} = the left module structure of M_{2}, then x ∈ M_{1} if and only if x ∈ M_{2}.

(2) For every vector v of the left module structure of V such that a = v holds r · a = r · v.

(3) The left module structure of V is a strict submodule of V.

(4) V is a submodule of Ω_{V}.

2. Trivial and non-trivial modules and rings

We now define two new predicates. Let us consider K, V. We say that V is non-trivial if and only if: (Def.1) there exists a vector a of V such that a ≠ Θ_{V}. Let us consider K. We say that K is non-trivial if and only if: (Def.2) 0_{K} ≠ 1_{K}.

We now state three propositions:

(5) If K is trivial, then for every r holds r = 0_{K} and for every a holds a = Θ_{V}.

(6) If K is trivial, then V is trivial.

(7) V is trivial if and only if the left module structure of V = 0_{V}.

3. Submodules and subsets

We now define two new functors. Let us consider K, V, and let W be a strict submodule of V. The functor ë(W) yields an element of Sub(V) and is defined by: (Def.3) ë(W) = W. The functor ç(V) yields a non-empty subset of V and is defined as follows: (Def.4) ç(V) = the carrier of V.

The following two propositions are true:

(8) For all sets X, Y, A such that X ⊆ Y and A is a subset of X holds A is a subset of Y.

(9) Every subset of W is a subset of V.

Let us consider K, V, W, and let A be a subset of W. The functor ï(A) yields a subset of V and is defined by: (Def.5) ï(A) = A. Let A be a non-empty subset of W. Then ï(A) is a non-empty subset of V.

The following propositions are true:

(10) x ∈ ç(V) if and only if x ∈ V.

(11) x ∈ ï(ç(W)) if and only if x ∈ W.

(12) A ⊆ ç(Lin(A)).

(13) If A ≠ ∅ and A is linearly closed, then ∑ l ∈ A.

(14) If Θ_{V} ∈ A and A is linearly closed, then ∑ l ∈ A.

(15) If Θ_{V} ∈ A and A is linearly closed, then A = ç(Lin(A)).

4. Cyclic submodules

Let us consider K, V, a. Then {a} is a non-empty subset of V. The functor ∏^{∗} a yielding a strict submodule of V is defined by: (Def.6) ∏^{∗} a = Lin({a}).

5. Inclusion of left R-modules

Let us consider K, M, N. The predicate M ⊆ N is defined as follows: (Def.7) M is a submodule of N.

We now state a number of propositions:

(16) If M ⊆ N, then if x ∈ M, then x ∈ N but if x is a vector of M, then x is a vector of N.

(17) Suppose M ⊆ N. Then (i) Θ_{M} = Θ_{N}, (ii) if m_{1} = n_{1} and m_{2} = n_{2}, then m_{1} + m_{2} = n_{1} + n_{2}, (iii) if m = n, then r · m = r · n, (iv) if m = n, then −n = −m,

(v) if m_{1} = n_{1} and m_{2} = n_{2}, then m_{1} − m_{2} = n_{1} − n_{2}, (vi) Θ_{N} ∈ M, (vii) Θ_{M} ∈ N, (viii) if n_{1} ∈ M and n_{2} ∈ M, then n_{1} + n_{2} ∈ M, (ix) if n ∈ M, then r · n ∈ M,

(x) if n ∈ M, then −n ∈ M, (xi) if n_{1} ∈ M and n_{2} ∈ M, then n_{1} − n_{2} ∈ M.

(18) Suppose M_{1} ⊆ N and M_{2} ⊆ N. Then (i) Θ_{M_{1}} = Θ_{M_{2}}, (ii) Θ_{M_{1}} ∈ M_{2}, (iii) if the carrier of M_{1} ⊆ the carrier of M_{2}, then M_{1} ⊆ M_{2}, (iv) if for every n such that n ∈ M_{1} holds n ∈ M_{2}, then M_{1} ⊆ M_{2},

(v) if the carrier of M_{1} = the carrier of M_{2} and M_{1} is strict and M_{2} is strict, then M_{1} = M_{2},

(vi) 0_{M_{1}} ⊆ M_{2}.

(19) W_{1} + W_{2} ⊆ V and W_{1} ∩ W_{2} ⊆ V.

(20) N ⊆ N.

(21) For all strict left modules V, M over K such that V ⊆ M and M ⊆ V holds V = M.

(22) If V ⊆ M and M ⊆ N, then V ⊆ N.

(23) If M ⊆ N, then 0_{M} ⊆ N.

(24) If M ⊆ N, then 0_{N} ⊆ M.

(25) If M ⊆ N, then M ⊆ Ω_{N}.

50 michał muzalewski

(26) W_{1} ⊆ W_{1} + W_{2} and W_{2} ⊆ W_{1} + W_{2}.

(27) W_{1} ∩ W_{2} ⊆ W_{1} and W_{1} ∩ W_{2} ⊆ W_{2}.

(28) If W_{1} ⊆ W_{2}, then W_{1} ∩ W_{3} ⊆ W_{2} ∩ W_{3}.

(29) If W_{1} ⊆ W_{3}, then W_{1} ∩ W_{2} ⊆ W_{3}.

(30) If W_{1} ⊆ W_{2} and W_{1} ⊆ W_{3}, then W_{1} ⊆ W_{2} ∩ W_{3}.

(31) W_{1} ∩ W_{2} ⊆ W_{1} + W_{2}.

(32) W_{1} ∩ W_{2} + W_{2} ∩ W_{3} ⊆ W_{2} ∩ (W_{1} + W_{3}).

(33) If W_{1} ⊆ W_{2}, then W_{2} ∩ (W_{1} + W_{3}) = W_{1} ∩ W_{2} + W_{2} ∩ W_{3}.

(34) W_{2} + W_{1} ∩ W_{3} ⊆ (W_{1} + W_{2}) ∩ (W_{2} + W_{3}).

(35) If W_{1} ⊆ W_{2}, then W_{2} + W_{1} ∩ W_{3} = (W_{1} + W_{2}) ∩ (W_{2} + W_{3}).

(36) If W_{1} ⊆ W_{2}, then W_{1} ⊆ W_{2} + W_{3}.

(37) If W_{1} ⊆ W_{3} and W_{2} ⊆ W_{3}, then W_{1} + W_{2} ⊆ W_{3}.

(38) For all subsets A, B of V such that A ⊆ B holds Lin(A) ⊆ Lin(B).

(39) For all subsets A, B of V holds Lin(A ∩ B) ⊆ Lin(A) ∩ Lin(B).

(40) If M_{1} ⊆ M_{2}, then ç(M_{1}) ⊆ ç(M_{2}).

(41) W_{1} ⊆ W_{2} if and only if for every a such that a ∈ W_{1} holds a ∈ W_{2}.

(42) W_{1} ⊆ W_{2} if and only if ç(W_{1}) ⊆ ç(W_{2}).

(43) W_{1} ⊆ W_{2} if and only if ï(ç(W_{1})) ⊆ ï(ç(W_{2})).

(44) 0_{W} ⊆ V and 0_{V} ⊆ W and 0_{W_{1}} ⊆ W_{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. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

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

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

[7] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[8] Michał Muzalewski. Free modules. Formalized Mathematics, 2(4):587–589, 1991.

[9] Michał Muzalewski and Wojciech Skaba. Finite sums of vectors in left module over associative ring. Formalized Mathematics, 2(2):279–282, 1991.

[10] Michał Muzalewski and Wojciech Skaba. Linear combinations in left module over associative ring. Formalized Mathematics, 2(2):295–300, 1991.

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

submodules 51

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

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

52

Oriented Metric-Affine Plane - Part II

Jarosław Zajkowski Warsaw University Białystok

Summary. Axiomatic description of properties of the oriented orthogonality relation. Next we construct (with the help of the oriented orthogonality relation) vector space and give the definitions of left-, right-, and semi-transitives.

MML Identifier: DIRORT.

The articles [1], [6], [7], [5], [3], [2], [4], and [8] provide the notation and terminology for this paper. In the sequel V will be a real linear space, A_{1} will be an affine structure, and x, y will be vectors of V. One can prove the following propositions:

(1) Suppose x, y span the space. Then

(i) for all elements u, u_{1}, v, v_{1}, w, w_{1}, w_{2} of the carrier of CESpace(V, x, y) holds u, u⊤^{>}v, w and u, v⊤^{>}w, w but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}v_{1}, u_{1}, then u = v or u_{1} = v_{1} but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}u_{1}, w, then u, v⊤^{>}v_{1}, w or u, v⊤^{>}w, v_{1} but if u, v⊤^{>}u_{1}, v_{1}, then v, u⊤^{>}v_{1}, u_{1} but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}v_{1}, w, then u, v⊤^{>}u_{1}, w but if u, u_{1}⊤^{>}v, v_{1}, then v, v_{1}⊤^{>}u, u_{1} or v, v_{1}⊤^{>}u_{1}, u,

(ii) for every elements u, v, w of the carrier of CESpace(V, x, y) there exists an element u_{1} of the carrier of CESpace(V, x, y) such that w ≠ u_{1} and w, u_{1}⊤^{>}u, v,

(iii) for every elements u, v, w of the carrier of CESpace(V, x, y) there exists an element u_{1} of the carrier of CESpace(V, x, y) such that w ≠ u_{1} and u, v⊤^{>}w, u_{1}.

(2) Suppose x, y span the space. Then

(i) for all elements u, u_{1}, v, v_{1}, w, w_{1}, w_{2} of the carrier of CMSpace(V, x, y) holds u, u⊤^{>}v, w and u, v⊤^{>}w, w but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}v_{1}, u_{1}, then u = v or u_{1} = v_{1} but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}u_{1}, w, then u, v⊤^{>}v_{1}, w or u, v⊤^{>}w, v_{1} but if u, v⊤^{>}u_{1}, v_{1}, then v, u⊤^{>}v_{1}, u_{1} but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}v_{1}, w, then u, v⊤^{>}u_{1}, w but if u, u_{1}⊤^{>}v, v_{1}, then v, v_{1}⊤^{>}u, u_{1} or v, v_{1}⊤^{>}u_{1}, u,

(ii) for every elements u, v, w of the carrier of CMSpace(V, x, y) there exists an element u_{1} of the carrier of CMSpace(V, x, y) such that w ≠ u_{1} and w, u_{1}⊤^{>}u, v,

(iii) for every elements u, v, w of the carrier of CMSpace(V, x, y) there exists an element u_{1} of the carrier of CMSpace(V, x, y) such that w ≠ u_{1} and u, v⊤^{>}w, u_{1}.

We now define two new constructions. An affine structure is oriented orthogonality if it satisfies the conditions (Def.1). (Def.1) (i) For all elements u, u_{1}, v, v_{1}, w, w_{1}, w_{2} of the carrier of it holds

u, u⊤^{>}v, w and u, v⊤^{>}w, w but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}v_{1}, u_{1}, then u = v or u_{1} = v_{1} but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}u_{1}, w, then u, v⊤^{>}v_{1}, w or u, v⊤^{>}w, v_{1} but if u, v⊤^{>}u_{1}, v_{1}, then v, u⊤^{>}v_{1}, u_{1} but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}v_{1}, w, then u, v⊤^{>}u_{1}, w but if u, u_{1}⊤^{>}v, v_{1}, then v, v_{1}⊤^{>}u, u_{1} or v, v_{1}⊤^{>}u_{1}, u,

(ii) for every elements u, v, w of the carrier of it there exists an element

u_{1} of the carrier of it such that w ≠ u_{1} and w, u_{1}⊤^{>}u, v, (iii) for every elements u, v, w of the carrier of it there exists an element

u_{1} of the carrier of it such that w ≠ u_{1} and u, v⊤^{>}w, u_{1}. An oriented orthogonality space is an oriented orthogonality affine structure.

Next we state three propositions:

(3) The following conditions are equivalent:

(i) for all elements u, u_{1}, v, v_{1}, w, w_{1}, w_{2} of the carrier of A_{1} holds u, u⊤^{>}v, w and u, v⊤^{>}w, w but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}v_{1}, u_{1}, then u = v or u_{1} = v_{1} but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}u_{1}, w, then u, v⊤^{>}v_{1}, w or u, v⊤^{>}w, v_{1} but if u, v⊤^{>}u_{1}, v_{1}, then v, u⊤^{>}v_{1}, u_{1} but if u, v⊤^{>}u_{1}, v_{1} and u, v⊤^{>}v_{1}, w, then u, v⊤^{>}u_{1}, w but if u, u_{1}⊤^{>}v, v_{1}, then v, v_{1}⊤^{>}u, u_{1} or v, v_{1}⊤^{>}u_{1}, u and for every elements u, v, w of the carrier of A_{1} there exists an element u_{1} of the carrier of A_{1} such that w ≠ u_{1} and w, u_{1}⊤^{>}u, v and for every elements u, v, w of the carrier of A_{1} there exists an element u_{1} of the carrier of A_{1} such that w ≠ u_{1} and u, v⊤^{>}w, u_{1},

(ii) A_{1} is an oriented orthogonality space.

(4) If x, y span the space, then CMSpace(V, x, y) is an oriented orthogonality space.

(5) If x, y span the space, then CESpace(V, x, y) is an oriented orthogonality space.

We follow a convention: A_{1} will denote an oriented orthogonality space and u, u_{1}, u_{2}, v, v_{1}, v_{2}, w, w_{1} will denote elements of the carrier of A_{1}. We now state three propositions:

(6) For every elements u, v, w of the carrier of A_{1} there exists an element u_{1} of the carrier of A_{1} such that u_{1}, w⊤^{>}u, v and u_{1} ≠ w.

(7) For all elements u, v, w of the carrier of A_{1} holds u, v⊤^{>}w, w.

(8) For every elements u, v, w of the carrier of A_{1} there exists an element u_{1} of the carrier of A_{1} such that u ≠ u_{1} but v, w⊤^{>}u, u_{1} or v, w⊤^{>}u_{1}, u.

We now define several new constructions. Let A_{1} be an oriented orthogonality space, and let a, b, c, d be elements of the carrier of A_{1}. The predicate a, b ⊥ c, d is defined by: (Def.2) a, b⊤^{>}c, d or a, b⊤^{>}d, c. Let a, b, c, d be elements of the carrier of A_{1}. The predicate a, b \begin{matrix} \\ \\ ∥ \end{matrix} c, d is defined as follows: (Def.3) there exist elements e, f of the carrier of A_{1} such that e ≠ f and

e, f⊤^{>}a, b and e, f⊤^{>}c, d. An oriented orthogonality space is semi transitive if: (Def.4) for all elements u, u_{1}, u_{2}, v, v_{1}, v_{2}, w, w_{1} of the carrier of it such that

u, u_{1}⊤^{>}v, v_{1} and w, w_{1}⊤^{>}v, v_{1} and w, w_{1}⊤^{>}u_{2}, v_{2} holds w = w_{1} or v = v_{1} or u, u_{1}⊤^{>}u_{2}, v_{2}.

An oriented orthogonality space is right transitive if: (Def.5) for all elements u, u_{1}, u_{2}, v, v_{1}, v_{2}, w, w_{1} of the carrier of it such that

u, u_{1}⊤^{>}v, v_{1} and v, v_{1}⊤^{>}w, w_{1} and u_{2}, v_{2}⊤^{>}w, w_{1} holds w = w_{1} or v = v_{1} or u, u_{1}⊤^{>}u_{2}, v_{2}.

An oriented orthogonality space is left transitive if: (Def.6) for all elements u, u_{1}, u_{2}, v, v_{1}, v_{2}, w, w_{1} of the carrier of it such that

u, u_{1}⊤^{>}v, v_{1} and v, v_{1}⊤^{>}w, w_{1} and u, u_{1}⊤^{>}u_{2}, v_{2} holds u = u_{1} or v = v_{1} or u_{2}, v_{2}⊤^{>}w, w_{1}.

An oriented orthogonality space is Euclidean like if: (Def.7) for all elements u, u_{1}, v, v_{1} of the carrier of it such that u, u_{1}⊤^{>}v, v_{1}

holds v, v_{1}⊤^{>}u_{1}, u. An oriented orthogonality space is Minkowskian like if: (Def.8) for all elements u, u_{1}, v, v_{1} of the carrier of it such that u, u_{1}⊤^{>}v, v_{1}

holds v, v_{1}⊤^{>}u, u_{1}.

One can prove the following propositions:

(9) u, u_{1} \begin{matrix} \\ \\ ∥ \end{matrix} w, w and w, w \begin{matrix} \\ \\ ∥ \end{matrix} u, u_{1}.

(10) If u, u_{1} \begin{matrix} \\ \\ ∥ \end{matrix} v, v_{1}, then v, v_{1} \begin{matrix} \\ \\ ∥ \end{matrix} u, u_{1}.

(11) If u, u_{1} \begin{matrix} \\ \\ ∥ \end{matrix} v, v_{1}, then u_{1}, u \begin{matrix} \\ \\ ∥ \end{matrix} v_{1}, v.

(12) A_{1} is left transitive if and only if for all v, v_{1}, w, w_{1}, u_{2}, v_{2} such that

v, v_{1} \begin{matrix} \\ \\ ∥ \end{matrix} u_{2}, v_{2} and v, v_{1}⊤^{>}w, w_{1} and v ≠ v_{1} holds u_{2}, v_{2}⊤^{>}w, w_{1}.

(13) A_{1} is semi transitive if and only if for all u, u_{1}, u_{2}, v, v_{1}, v_{2} such that

u, u_{1}⊤^{>}v, v_{1} and v, v_{1} \begin{matrix} \\ \\ ∥ \end{matrix} u_{2}, v_{2} and v ≠ v_{1} holds u, u_{1}⊤^{>}u_{2}, v_{2}.

(14) If A_{1} is semi transitive, then for all u, u_{1}, v, v_{1}, w, w_{1} such that u, u_{1} \begin{matrix} \\ \\ ∥ \end{matrix}

v, v_{1} and v, v_{1} \begin{matrix} \\ \\ ∥ \end{matrix} w, w_{1} and v ≠ v_{1} holds u, u_{1} \begin{matrix} \\ \\ ∥ \end{matrix} w, w_{1}.

(15) If x, y span the space and A_{1} = CESpace(V, x, y), then A_{1} is Euclidean like, left transitive, right transitive and semi transitive.

56 jarosław zajkowski

One can readily verify that there exists an oriented orthogonality space which is Euclidean like, left transitive, right transitive and semi transitive.

We now state the proposition

(16) If x, y span the space and A_{1} = CMSpace(V, x, y), then A_{1} is Minkowskian like, left transitive, right transitive and semi transitive.

Let us note that there exists an oriented orthogonality space which is Minkowskian like, left transitive, right transitive and semi transitive.

Next we state four propositions:

(17) If A_{1} is left transitive, then A_{1} is right transitive.

(18) If A_{1} is left transitive, then A_{1} is semi transitive.

(19) If A_{1} is semi transitive, then A_{1} is right transitive if and only if for all u, u_{1}, v, v_{1}, u_{2}, v_{2} such that u, u_{1}⊤^{>}u_{2}, v_{2} and v, v_{1}⊤^{>}u_{2}, v_{2} and u_{2} ≠ v_{2}

holds u, u_{1} \begin{matrix} \\ \\ ∥ \end{matrix} v, v_{1}.

(20) If A_{1} is right transitive but A_{1} is Euclidean like or A_{1} is Minkowskian like, then A_{1} is left transitive.

References

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

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical metric affine spaces and planes. Formalized Mathematics, 1(5):891–899, 1990.

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

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. A construction of analytical ordered trapezium spaces. Formalized Mathematics, 2(3):315–322, 1991.

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

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

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

[8] Jarosław Zajkowski. Oriented metric-affine plane - part I. Formalized Mathematics, 2(4):591–597, 1991.

Opposite Rings, Modules and their Morphisms

Michał Muzalewski Warsaw University Białystok

Summary. Let = ⟨S; K, 0, 1, +, ·⟩ be a ring. The structure op = ⟨S; K, 0, 1, +, •⟩ is called anti-ring, if α • β = β · α for elements α, β of K [12, pages 5–7]. It is easily seen that ^{op} is also a ring. If V is a left module over , then V is a right module over ^{op} . If W is a right module over , then W is a left module over ^{op} . Let K, L be rings. A morphism J : K ⟶ L is called anti-homomorphism, if J(α · β) = J(β) · J(α) for elements α, β of K. If J : K ⟶ L is a homomorphism, then J : K ⟶ ^{op}L is an anti-homomorphism. Let K, L be rings, V, W left modules over K, L respectively and J : K ⟶ L an anti-monomorphism. A map f : V ⟶ W is called J - semilinear, if f(x + y) = f(x) + f(y) and f(α · x) = J(α) · f(x) for vectors x, y of V and a scalar α of K.

MML Identifier: MOD 4.

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

1. Opposite functions

In the sequel A, B, C are non-empty sets and f is a function from [: A, B:] into C. Let us consider A, B, C, f. Then f is a function from [: B, A:] into C.

We now state the proposition

(1) For every element x of A and for every element y of B holds f(x, y) = ( f)(y, x).

2. Opposite rings

In the sequel K, L will be field structures. Let us consider K. The functor ^{op}K yielding a strict field structure is defined by: (Def.1) ^{op}K = ⟨the carrier of K, (the multiplication of K), the addition of K,

the reverse-map of K, the unity of K, the zero of K⟩. We now state four propositions:

(2) The group structure of ^{op}K = the group structure of K and for an arbitrary x holds x is a scalar of ^{op}K if and only if x is a scalar of K.

(3) ^{op}(^{op}K) = the field structure of K.

(4) (i) 0_{K} = 0_{^{op}K}, (ii) 1_{K} = 1_{^{op}K}, (iii) for all scalars x, y, z, u of K and for all scalars a, b, c, d of ^{op}K such

that x = a and y = b and z = c and u = d holds x+y = a+b and x·y = b·a and −x = −a and x + y + z = a + b + c and x + (y + z) = a + (b + c) and (x · y) · z = c · (b · a) and x · (y · z) = (c · b) · a and x · (y + z) = (b + c) · a and (y + z) · x = a · (b + c) and x · y + z · u = b · a + d · c.

(5) For every ring K holds ^{op}K is a strict ring. Let K be a ring. Then ^{op}K is a strict ring. One can prove the following proposition

(6) For every associative ring K holds ^{op}K is an associative ring. Let K be an associative ring. Then ^{op}K is a strict associative ring. Next we state the proposition

(7) For every skew field K holds ^{op}K is a skew field. Let K be a skew field. Then ^{op}K is a strict skew field. One can prove the following proposition

(8) For every field K holds ^{op}K is a strict field. Let K be a field. Then ^{op}K is a strict field.

3. Opposite modules

In the sequel V denotes a left module structure over K. Let us consider K, V. The functor ^{op}V yields a strict right module structure over ^{op}K and is defined as follows: (Def.2) for every function o from [: the carrier of V, the carrier of ^{op}K:] into the

carrier of V such that o = (the left multiplication of V) holds^{op}V = ⟨the carrier of V, the addition of V, the reverse-map of V, the zero of V, o⟩.

The following proposition is true

(9) The group structure of ^{op}V = the group structure of V and for an arbitrary x holds x is a vector of V if and only if x is a vector of ^{op}V.

Let us consider K, V, and let o be a function from [: the carrier of K, the carrier of V :] into the carrier of V. The functor ^{op}o yields a function from [: the carrier of ^{op}V, the carrier of ^{op}K:] into the carrier of ^{op}V and is defined by: (Def.3) ^{op}o = o.

One can prove the following two propositions:

(10) The right multiplication of ^{op}V = ^{op}(the left multiplication of V).

(11) ^{op}V = ⟨the carrier of ^{op}V, the addition of ^{op}V, the reverse-map of ^{op}V,

the zero of ^{op}V,^{op}(the left multiplication of V)⟩. In the sequel W denotes a right module structure over K. Let us consider

K, W. The functor ^{op}W yields a strict left module structure over ^{op}K and is defined by: (Def.4) for every function o from [: the carrier of ^{op}K, the carrier of W :] into

the carrier of W such that o = (the right multiplication of W) holds opW = ⟨the carrier of W, the addition of W, the reverse-map of W, the zero of W, o⟩.

We now state the proposition

(12) The group structure of ^{op}W = the group structure of W and for an

arbitrary x holds x is a vector of W if and only if x is a vector of ^{op}W. Let us consider K, W, and let o be a function from [: the carrier of W, the

carrier of K:] into the carrier of W. The functor ^{op}o yielding a function from [: the carrier of ^{op}K, the carrier of ^{op}W :] into the carrier of ^{op}W is defined as follows: (Def.5) ^{op}o = o.

The following propositions are true:

(13) The left multiplication of ^{op}W = ^{op}(the right multiplication of W).

(14) ^{op}W = ⟨the carrier of ^{op}W, the addition of ^{op}W, the reverse-map of ^{op}W, the zero of ^{op}W,^{op}(the right multiplication of W)⟩.

(15) For every function o from [: the carrier of K, the carrier of V :] into the carrier of V holds ^{op}(^{op}o) = o.

(16) For every function o from [: the carrier of K, the carrier of V :] into the carrier of V and for every scalar x of K and for every scalar y of ^{op}K and for every vector v of V and for every vector w of ^{op}V such that x = y and v = w holds (^{op}o)(w, y) = o(x, v).

(17) Let K, L be rings. Then for every V being a left module structure over K and for every W being a right module structure over L and for every scalar x of K and for every scalar y of L and for every vector v of V and for every vector w of W such that L = ^{op}K and W = ^{op}V and x = y and v = w holds w · y = x · v.

(18) For all rings K, L and for every V being a left module structure over K and for every W being a right module structure over L and for all vectors v_{1}, v_{2} of V and for all vectors w_{1}, w_{2} of W such that L = ^{op}K and W = ^{op}V and v_{1} = w_{1} and v_{2} = w_{2} holds w_{1} + w_{2} = v_{1} + v_{2}.

(19) For every function o from [: the carrier of W, the carrier of K:] into the carrier of W holds ^{op}(^{op}o) = o.

(20) For every function o from [: the carrier of W, the carrier of K:] into the carrier of W and for every scalar x of K and for every scalar y of ^{op}K and for every vector v of W and for every vector w of ^{op}W such that x = y and v = w holds (^{op}o)(y, w) = o(v, x).

(21) Let K, L be rings. Then for every V being a left module structure over K and for every W being a right module structure over L and for every scalar x of K and for every scalar y of L and for every vector v of V and for every vector w of W such that K = ^{op}L and V = ^{op}W and x = y and v = w holds w · y = x · v.

(22) For all rings K, L and for every V being a left module structure over K and for every W being a right module structure over L and for all vectors v_{1}, v_{2} of V and for all vectors w_{1}, w_{2} of W such that K = ^{op}L and V = ^{op}W and v_{1} = w_{1} and v_{2} = w_{2} holds w_{1} + w_{2} = v_{1} + v_{2}.

(23) For every K being a strict field structure and for every V being a left module structure over K holds ^{op}(^{op}V) = the left module structure of V.

(24) For every K being a strict field structure and for every W being a right module structure over K holds ^{op}(^{op}W) = the right module structure of W.

(25) For every associative ring K and for every left module V over K holds

opV is a strict right module over ^{op}K. Let K be an associative ring, and let V be a left module over K. Then ^{op}V

is a strict right module over ^{op}K.

One can prove the following proposition

(26) For every associative ring K and for every right module W over K holds

opW is a strict left module over ^{op}K. Let K be an associative ring, and let W be a right module over K. Then

opW is a strict left module over ^{op}K.

4. Morphisms of rings

We now define several new attributes. Let us consider K, L. A map from K into L is antilinear if: (Def.6) for all scalars x, y of K holds it(x + y) = it(x) + it(y) and for all scalars

x, y of K holds it(x · y) = it(y) · it(x) and it(1_{K}) = 1_{L}. A map from K into L is monomorphism if: (Def.7) it is linear and it is one-to-one. A map from K into L is antimonomorphism if: (Def.8) it is antilinear and it is one-to-one. A map from K into L is epimorphism if:

(Def.9) it is linear and rng it = the carrier of L. A map from K into L is antiepimorphism if:

(Def.10) it is antilinear and rng it = the carrier of L.

A map from K into L is isomorphism if: (Def.11) it is monomorphism and rng it = the carrier of L.

A map from K into L is antiisomorphism if: (Def.12) it is antimonomorphism and rng it = the carrier of L.

In the sequel J denotes a map from K into K. We now define four new attributes. Let us consider K. A map from K into K is endomorphism if:

(Def.13) it is linear.

A map from K into K is antiendomorphism if: (Def.14) it is antilinear.

A map from K into K is automorphism if: (Def.15) it is isomorphism.

A map from K into K is antiautomorphism if: (Def.16) it is antiisomorphism.

One can prove the following propositions:

(27) J is automorphism if and only if the following conditions are satisfied: (i) for all scalars x, y of K holds J(x + y) = J(x) + J(y), (ii) for all scalars x, y of K holds J(x · y) = J(x) · J(y), (iii) J(1_{K}) = 1_{K}, (iv) J is one-to-one,

(v) rng J = the carrier of K.

(28) J is antiautomorphism if and only if the following conditions are satis-

fied:

(i) for all scalars x, y of K holds J(x + y) = J(x) + J(y), (ii) for all scalars x, y of K holds J(x · y) = J(y) · J(x), (iii) J(1_{K}) = 1_{K}, (iv) J is one-to-one,

(v) rng J = the carrier of K.

(29) id_{K} is automorphism.

We follow the rules: K, L will denote rings, J will denote a map from K into L, and x, y will denote scalars of K. Next we state three propositions:

(30) If J is linear, then J(0_{K}) = 0_{L} and J(−x) = −J(x) and J(x − y) = J(x) − J(y).

(31) If J is antilinear, then J(0_{K}) = 0_{L} and J(−x) = −J(x) and J(x − y) = J(x) − J(y).

(32) For every associative ring K holds id_{K} is antiautomorphism if and only

if K is a commutative ring. One can prove the following proposition (33) For every skew field K holds id_{K} is antiautomorphism if and only if K

is a field.

5. Opposite morphisms to morphisms of rings

In the sequel K, L will be field structures and J will be a map from K into L. Let us consider K, L, J. The functor ^{op}J yielding a map from K into ^{op}L is defined by:

(Def.17) ^{op}J = J.

Next we state several propositions:

(34) ^{op}(^{op}J) = J.

(35) J is linear if and only if ^{op}J is antilinear.

(36) J is antilinear if and only if ^{op}J is linear.

(37) J is monomorphism if and only if ^{op}J is antimonomorphism.

(38) J is antimonomorphism if and only if ^{op}J is monomorphism.

(39) J is epimorphism if and only if ^{op}J is antiepimorphism.

(40) J is antiepimorphism if and only if ^{op}J is epimorphism.

(41) J is isomorphism if and only if ^{op}J is antiisomorphism.

(42) J is antiisomorphism if and only if ^{op}J is isomorphism.

In the sequel J will be a map from K into K. We now state four propositions:

(43) J is endomorphism if and only if ^{op}J is antilinear.

(44) J is antiendomorphism if and only if ^{op}J is linear.

(45) J is automorphism if and only if ^{op}J is antiisomorphism.

(46) J is antiautomorphism if and only if ^{op}J is isomorphism.

6. Morphisms of groups

In the sequel G, H will denote groups. Let us consider G, H. A map from G into H is said to be a homomorphism from G to H if:

(Def.18) for all elements x, y of G holds it(x + y) = it(x) + it(y).

Then zero(G, H) is a homomorphism from G to H.

In the sequel f is a homomorphism from G to H. We now define four new constructions. Let us consider G, H. A homomorphism from G to H is monomorphism if:

(Def.19) it is one-to-one.

A homomorphism from G to H is epimorphism if: (Def.20) rng it = the carrier of H.

A homomorphism from G to H is isomorphism if:

(Def.21) it is one-to-one and rng it = the carrier of H.

Let us consider G. An endomorphism of G is a homomorphism from G to G.

We now state the proposition

(47) For every element x of G holds id_{G}(x) = x.

We now define two new constructions. Let us consider G. An endomorphism of G is automorphism-like if:

(Def.22) it is isomorphism.

An automorphism of G is an automorphism-like endomorphism of G. Then id_{G} is an automorphism of G. In the sequel x, y will be elements of G. We now state the proposition

(48) f(0_{G}) = 0_{H} and f(−x) = −f(x) and f(x −′ y) = f(x) −′ f(y). We adopt the following convention: G, H denote Abelian groups, f denotes

a homomorphism from G to H, and x, y denote elements of G. The following proposition is true

(49) f(x − y) = f(x) − f(y).

7. Semilinear morphisms

For simplicity we adopt the following rules: K, L are associative rings, J is a map from K into L, V is a left module over K, and W is a left module over L. Let us consider K, L, J, V, W. A map from V into W is said to be a homomorphism from V to W by J if:

(Def.23) for all vectors x, y of V holds it(x + y) = it(x) + it(y) and for every scalar a of K and for every vector x of V holds it(a · x) = J(a) · it(x).

The following proposition is true

(50) zero(V, W) is a homomorphism from V to W by J.

In the sequel f denotes a homomorphism from V to W by J. We now define three new predicates. Let us consider K, L, J, V, W, f. We say that f is a monomorphism wrp J if and only if:

(Def.24) f is one-to-one.

We say that f is a epimorphism wrp J if and only if: (Def.25) rng f = the carrier of W.

We say that f is a isomorphism wrp J if and only if: (Def.26) f is one-to-one and rng f = the carrier of W.

In the sequel J will denote a map from K into K and f will denote a homomorphism from V to V by J. We now define two new constructions. Let us consider K, J, V. An endomorphism of J and V is a homomorphism from V to V by J.

Let us consider K, J, V, f. We say that f is a automorphism wrp J if and only if:

(Def.27) f is one-to-one and rng f = the carrier of V.

In the sequel W is a left module over K. Let us consider K, V, W. A homomorphism from V to W is a homomorphism from V to W by id_{K}.

Next we state the proposition

(51) For every map f from V into W holds f is a homomorphism from V to W if and only if for all vectors x, y of V holds f(x + y) = f(x) + f(y) and for every scalar a of K and for every vector x of V holds f(a·x) = a·f(x).

We now define five new constructions. Let us consider K, V, W. A homomorphism from V to W is monomorphism if:

(Def.28) it is one-to-one.

A homomorphism from V to W is epimorphism if: (Def.29) rng it = the carrier of W.

A homomorphism from V to W is isomorphism if: (Def.30) it is one-to-one and rng it = the carrier of W.

Let us consider K, V. An endomorphism of V is a homomorphism from V to V.

An endomorphism of V is automorphism if: (Def.31) it is one-to-one and rng it = the carrier of V.

8. Annex

Next we state three propositions:

(52) For every skew field K holds K is a field if and only if for all scalars x, y of K holds x · y = y · x.

(53) For every K being a field structure holds K is a field if and only if K is a skew field and for all scalars x, y of K holds x · y = y · x.

(54) For every group G and for all elements x, y, z of G such that x+y = x+z holds y = z.

References

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

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

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

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

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

opposite rings, modules and their morphisms 65

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

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

[10] Michał Muzalewski. Category of rings. Formalized Mathematics, 2(5):643–648, 1991.

[11] Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3–11, 1991.

[12] Michał Muzalewski. Foundations of Metric-Affine Geometry. Dział Wydawnictw Filii UW w Białymstoku, Filia UW w Białymstoku, 1990.

[13] Michał Muzalewski. Rings and modules - part II. Formalized Mathematics, 2(4):579– 585, 1991.

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

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

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

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

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

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

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

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

66

Properties of Caratheodor’s Measure

Józef Białas University of Łódź

Summary. The paper contains definitions and basic properties of Caratheodor’s measure, with values in \overline{}, the enlarged set of real numbers, where \overline{}denotes set \overline{}= ∪ \{−∞, +∞\} - by [14]. The article includes the text being a continuation of the paper [3]. Caratheodor’s theorem and some theorems concerning basic properties of Caratheodor’s measure are proved. The work is the sixth part of the series of articles concerning the Lebesgue measure theory.

MML Identifier: MEASURE4.

The terminology and notation used in this paper have been introduced in the following papers: [16], [15], [10], [11], [8], [9], [1], [13], [2], [12], [4], [5], [7], [6], [3], and [17]. One can prove the following propositions:

(1) For all Real numbers x, y, z such that 0\overline{}≤ x and 0\overline{}≤ y and 0\overline{}≤ z holds (x + y) + z = x + (y + z).

(2) For all Real numbers x, y, z such that x ≠ −∞ and x ≠ +∞ holds y + x ≤ z if and only if y ≤ z − x.

(3) For all Real numbers x, y such that 0\overline{}≤ x and 0\overline{}≤ y holds x + y = y + x.

(4) For every set X and for every σ-field S of subsets of X and for every function F from into S and for every element A of S and for every function G from into S such that for every element n of holds G(n) = A ∩ F (n) holds ⋃ rng G = A ∩ ⋃ rng F.

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

(6) For every set X holds 2^{X} is a σ-field of subsets of X.

Let X be a set, and let F be a function from into 2^{X}. Then rng F is a non-empty family of subsets of X. Let A be a non-empty family of subsets of X. Then ⋃ A is an element of 2^{X}. Let F be a function from 2^{X} into \overline{}. We say that F is non-negative if and only if: (Def.1) for every element A of 2^{X} holds 0\overline{}≤ F (A). Let F be a function from into 2^{X}, and let M be a function from 2^{X} into \overline{}. Then M · F is a function from into \overline{}.

One can prove the following propositions:

(7) For every set X and for every Real numbers a, b there exists a function M from 2^{X} into \overline{}such that for every element A of 2^{X} holds if A = ∅, then M(A) = a but if A ≠ ∅, then M(A) = b.

(8) For every set X there exists a function M from 2^{X} into \overline{}such that for every element A of 2^{X} holds M(A) = 0\overline{}.

(9) For every set X and for every function F from into 2^{X} and for every function M from 2^{X} into \overline{}such that M is non-negative holds M · F is non-negative.

(10) For every set X and for every function F from into 2^{X} and for every function M from 2^{X} into \overline{}and for every natural number n holds (M · F)(n) = M(F (n)).

(11) Let X be a set. Then there exists a function M from 2^{X} into \overline{}such that M is non-negative and M(∅) = 0\overline{}and for all elements A, B of 2^{X} such that A ⊆ B holds M(A) ≤ M(B) and for every function F from into 2^{X} holds M(⋃ rng F) ≤ ∑(M · F).

We now define two new constructions. Let X be a set. A function from 2^{X} into \overline{}is said to be a Caratheodor’s measure on X if: (Def.2) it is non-negative and it(∅) = 0\overline{}and for all elements A, B of 2^{X} such

that A ⊆ B holds it(A) ≤ it(B) and for every function F from into 2^{X} holds it(⋃ rng F) ≤ ∑(it · F).

Let C be a Caratheodor’s measure on X. The functor σ-Field(C) yielding a non-empty family of subsets of X is defined by: (Def.3) for every element A of 2^{X} holds A ∈ σ-Field(C) if and only if for all

elements W, Z of 2^{X} such that W ⊆ A and Z ⊆ X \ A holds C(W) + C(Z) ≤ C(W ∪ Z).

The following propositions are true:

(12) For every set X and for every Caratheodor’s measure C on X and for all elements W, Z of 2^{X} holds C(W ∪ Z) ≤ C(W) + C(Z).

(13) For every set X and for every Caratheodor’s measure C on X and for all elements W, Z of 2^{X} holds C(Z) + C(W) = C(W) + C(Z).

(14) For every set X and for every Caratheodor’s measure C on X and for every element A of 2^{X} holds A ∈ σ-Field(C) if and only if for all elements W, Z of 2^{X} such that W ⊆ A and Z ⊆ X \ A holds C(W) + C(Z) = C(W ∪ Z).

(15) For every set X and for every Caratheodor’s measure C on X and for all elements W, Z of 2^{X} such that W ∈ σ-Field(C) and Z ∈ σ-Field(C) and Z ∩ W = ∅ holds C(W ∪ Z) = C(W) + C(Z).

(16) For every set X and for every Caratheodor’s measure C on X and for every set A such that A ∈ σ-Field(C) holds X \ A ∈ σ-Field(C).

(17) For every set X and for every Caratheodor’s measure C on X and for all sets A, B such that A ∈ σ-Field(C) and B ∈ σ-Field(C) holds A ∪ B ∈ σ-Field(C).

(18) For every set X and for every Caratheodor’s measure C on X and for all sets A, B such that A ∈ σ-Field(C) and B ∈ σ-Field(C) holds A ∩ B ∈ σ-Field(C).

(19) For every set X and for every Caratheodor’s measure C on X and for all sets A, B such that A ∈ σ-Field(C) and B ∈ σ-Field(C) holds A \ B ∈ σ-Field(C).

(20) For every set X and for every σ-field S of subsets of X and for every function N from into S and for every element A of S there exists a function F from into S such that for every element n of holds F (n) = A ∩ N(n).

(21) For every set X and for every Caratheodor’s measure C on X holds

σ-Field(C) is a σ-field of subsets of X. Let X be a set, and let C be a Caratheodor’s measure on X. Then σ-Field(C)

is a σ-field of subsets of X. Let S be a σ-field of subsets of X, and let A be a subfamily of S. Then ⋃ A is an element of S. The functor σ-Meas(C) yields a function from σ-Field(C) into \overline{}and is defined by: (Def.4) for every element A of 2^{X} such that A ∈ σ-Field(C) holds

(σ-Meas(C))(A) = C(A).

One can prove the following proposition

(22) For every set X and for every Caratheodor’s measure C on X holds

σ-Meas(C) is a measure on σ-Field(C). Let X be a set, and let C be a Caratheodor’s measure on X, and let A be

an element of σ-Field(C). Then C(A) is a Real number.

One can prove the following proposition

(23) For every set X and for every Caratheodor’s measure C on X holds

σ-Meas(C) is a σ-measure on σ-Field(C). Let X be a set, and let C be a Caratheodor’s measure on X. Then σ-Meas(C)

is a σ-measure on σ-Field(C).

The following propositions are true:

(24) For every set X and for every Caratheodor’s measure C on X and for every element A of 2^{X} such that C(A) = 0\overline{}holds A ∈ σ-Field(C).

(25) For every set X and for every Caratheodor’s measure C on X holds σ-Meas(C) is complete on σ-Field(C).

70 józef białas

References

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

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

[3] Józef Biał as. Completeness of the σ-additive measure. measure theory. Formalized Mathematics, 2(5):689–693, 1991.

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

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

[6] Józef Białas. Several properties of the σ-additive measure. Formalized Mathematics, 2(4):493–497, 1991.

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

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

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

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

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

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

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

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

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

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

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

Completeness of the Lattices of Domains of a Topological Space ^{1}

Zbigniew Karno Warsaw University Białystok Toshihiko Watanabe Shinshu University Nagano

Summary. Let T be a topological space and let A be a subset of T. Recall that A is said to be a domain in T provided Int \overline{A} ⊆ A ⊆ \overline{Int A} (see [24] and comp. [14]). This notion is a simple generalization of the notions of open and closed domains in T (see [24]). Our main result is concerned with an extension of the following well–known theorem (see e.g. [5], [17], [13]). For a given topological space the Boolean lattices of all its closed domains and all its open domains are complete. It is proved here, using Mizar System, that the complemented lattice of all domains of a given topological space is complete, too (comp. [23]).

It is known that both the lattice of open domains and the lattice of closed domains are sublattices of the lattice of all domains [23]. However, the following two problems remain open.

Problem 1. Let L be a sublattice of the lattice of all domains. Suppose L is complete, is smallest with respect to inclusion, and contains as sublattices the lattice of all closed domains and the lattice of all open domains. Must L be equal to the lattice of all domains ?

A domain in T is said to be a Borel domain provided it is a Borel set. Of course every open (closed) domain is a Borel domain. It can be proved that all Borel domains form a sublattice of the lattice of domains.

Problem 2. Let L be a sublattice of the lattice of all domains. Suppose L is smallest with respect to inclusion and contains as sublattices the lattice of all closed domains and the lattice of all open domains. Must L be equal to the lattice of all Borel domains ?

Note that in the beginning the closure and the interior operations for families of subsets of topological spaces are introduced and their important properties are presented (comp. [16], [15], [17]). Using these notions, certain properties of domains, closed domains and open domains are studied (comp. [15], [13]).

MML Identifier: TDLAT 2.

1This paper was done while the second author was visiting the Institute of Mathematics of Warsaw University in Białystok.

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

1. Preliminary Theorems about Subsets of Topological Spaces

In the sequel T will denote a topological space. One can prove the following propositions:

(1) For every subset A of T holds Int \overline{Int A} ⊆ Int \overline{A} and Int \overline{Int A} ⊆ \overline{Int A}.

(2) For every subset A of T holds \overline{Int A} ⊆ \overline{Int \overline{A}} and Int \overline{A} ⊆ \overline{Int \overline{A}}.

(3) For all subsets A, B of T such that B is closed holds if \overline{Int(A ∩ B)} = A, then A ⊆ B.

(4) For all subsets A, B of T such that A is open holds if Int \overline{A ∪ B} = B, then A ⊆ B.

(5) For every subset A of T such that A ⊆ \overline{Int A} holds A ∪ Int \overline{A} ⊆ \overline{Int(A ∪ Int \overline{A})}.

(6) For every subset A of T such that Int \overline{A} ⊆ A holds Int \overline{A ∩ \overline{Int A}} ⊆ A ∩ \overline{Int A}.

2. The Closure and the Interior Operations for Families of Subsets of a Topological Space

In the sequel T will be a topological space. Let us consider T, and let F be a family of subsets of T. We introduce the functor \overline{F} as a synonym of clf F.

One can prove the following propositions:

(7) For every family F of subsets of T holds \overline{F} = \{A : ⋁_{B}[A = \overline{B} ∧B ∈ F]\}, where A ranges over subsets of T, and B ranges over subsets of T.

(8) For every family F of subsets of T holds \overline{F} = \overline{\overline{F}}.

(9) For every family F of subsets of T holds F = ∅ if and only if \overline{F} = ∅.

(10) For all families F, G of subsets of T holds \overline{F ∩ G} ⊆ \overline{F} ∩ \overline{G}.

(11) For all families F, G of subsets of T holds \overline{F} \ \overline{G} ⊆ \overline{F \ G}.

(12) For every family F of subsets of T and for every subset A of T such that A ∈ F holds ⋂ \overline{F} ⊆ \overline{A} and \overline{A} ⊆ ⋃ \overline{F}.

(13) For every family F of subsets of T holds ⋂ F ⊆ ⋂ \overline{F}.

(14) For every family F of subsets of T holds \overline{⋂ F} ⊆ ⋂ \overline{F}.

(15) For every family F of subsets of T holds ⋃ \overline{F} ⊆ \overline{⋃ F}.

Let us consider T, and let F be a family of subsets of T. The functor Int F yielding a family of subsets of T is defined as follows: (Def.1) for every subset A of T holds A ∈ Int F if and only if there exists a

subset B of T such that A = Int B and B ∈ F.

The following propositions are true:

(16) For every family F of subsets of T holds Int F = {A : ⋁_{B}[A = Int B ∧ B ∈ F]}, where A ranges over subsets of T, and B ranges over subsets of T.

(17) For every family F of subsets of T holds Int F = Int Int F.

(18) For every family F of subsets of T holds Int F is open.

(19) For every family F of subsets of T holds F = ∅ if and only if Int F = ∅.

(20) For every subset A of T and for every family F of subsets of T such that F = {A} holds Int F = {Int A}.

(21) For all families F, G of subsets of T such that F ⊆ G holds Int F ⊆ Int G.

(22) For all families F, G of subsets of T holds Int(F ∪ G) = Int F ∪ Int G.

(23) For all families F, G of subsets of T holds Int(F ∩ G) ⊆ Int F ∩ Int G.

(24) For all families F, G of subsets of T holds Int F \ Int G ⊆ Int(F \ G).

(25) For every family F of subsets of T and for every subset A of T such that A ∈ F holds Int A ⊆ ⋃ Int F and ⋂ Int F ⊆ Int A.

(26) For every family F of subsets of T holds ⋃ Int F ⊆ ⋃ F.

(27) For every family F of subsets of T holds ⋂ Int F ⊆ ⋂ F.

(28) For every family F of subsets of T holds ⋃ Int F ⊆ Int ⋃ F.

(29) For every family F of subsets of T holds Int ⋂ F ⊆ ⋂ Int F.

(30) For every family F of subsets of T such that F is finite holds Int ⋂ F =

⋂ Int F. In the sequel F denotes a family of subsets of T. The following propositions

are true:

(31) \overline{Int F} = \{A : ⋁_{B}[A = \overline{Int B} ∧ B ∈ F]\}, where A ranges over subsets of T, and B ranges over subsets of T.

(32) Int \overline{F} = \{A : ⋁_{B}[A = Int \overline{B} ∧ B ∈ F]\}, where A ranges over subsets of T, and B ranges over subsets of T.

(33) \overline{Int \overline{F}} = \{A : ⋁_{B}[A = \overline{Int \overline{B}} ∧ B ∈ F]\}, where A ranges over subsets of T, and B ranges over subsets of T.

(34) Int \overline{Int F} = \{A : ⋁_{B}[A = Int \overline{Int B} ∧ B ∈ F]\}, where A ranges over subsets of T, and B ranges over subsets of T.

(35) \overline{Int \overline{Int F}} = \overline{Int F}.

(36) Int \overline{Int \overline{F}} = Int \overline{F}.

(37) ⋃ Int \overline{F} ⊆ ⋃ \overline{Int \overline{F}}.

(38) ⋂ Int \overline{F} ⊆ ⋂ \overline{Int \overline{F}}.

(39) ⋃ \overline{Int F} ⊆ ⋃ \overline{Int \overline{F}}.

(40) ⋂ \overline{Int F} ⊆ ⋂ \overline{Int \overline{F}}.

(41) ⋃ Int \overline{Int F} ⊆ ⋃ Int \overline{F}.

(42) ⋂ Int \overline{Int F} ⊆ ⋂ Int \overline{F}.

(43) ⋃ Int \overline{Int F} ⊆ ⋃ \overline{Int F}.

(44) ⋂ Int \overline{Int F} ⊆ ⋂ \overline{Int F}.

(45) ⋃ \overline{Int \overline{F}} ⊆ ⋃ \overline{F}.

(46) ⋂ \overline{Int \overline{F}} ⊆ ⋂ \overline{F}.

(47) ⋃ Int F ⊆ ⋃ Int \overline{Int F}.

(48) ⋂ Int F ⊆ ⋂ Int \overline{Int F}.

(49) ⋃ \overline{Int F} ⊆ \overline{Int ⋃ F}.

(50) \overline{Int ⋂ F} ⊆ ⋂ \overline{Int F}.

(51) ⋃ Int \overline{F} ⊆ Int \overline{⋃ F}.

(52) Int \overline{⋂ F} ⊆ ⋂ Int \overline{F}.

(53) ⋃ \overline{Int \overline{F}} ⊆ \overline{Int \overline{⋃ F}}.

(54) \overline{Int \overline{⋂ F}} ⊆ ⋂ \overline{Int \overline{F}}.

(55) ⋃ Int \overline{Int F} ⊆ Int \overline{Int ⋃ F}.

(56) Int \overline{Int ⋂ F} ⊆ ⋂ Int \overline{Int F}.

(57) For every family F of subsets of T such that for every subset A of T such that A ∈ F holds A ⊆ \overline{Int A} holds ⋃ F ⊆ \overline{Int ⋃ F} and \overline{⋃ F} = \overline{Int \overline{⋃ F}}.

(58) For every family F of subsets of T such that for every subset A of T such that A ∈ F holds Int \overline{A} ⊆ A holds Int \overline{⋂ F} ⊆ ⋂ F and Int \overline{Int ⋂ F} = Int ⋂ F.

3. Selected Properties of Domains of a Topological Space

In the sequel T is a topological space. We now state several propositions:

(59) For all subsets A, B of T such that B is a domain holds Int \overline{A ∪ B} ∪ (A ∪ B) = B if and only if A ⊆ B.

(60) For all subsets A, B of T such that A is a domain holds \overline{Int(A ∩ B)} ∩ (A ∩ B) = A if and only if A ⊆ B.

(61) For all subsets A, B of T such that A is a closed domain and B is a closed domain holds Int A ⊆ Int B if and only if A ⊆ B.

(62) For all subsets A, B of T such that A is an open domain and B is an open domain holds \overline{A} ⊆ \overline{B} if and only if A ⊆ B.

(63) For all subsets A, B of T such that A is a closed domain holds if A ⊆ B, then \overline{Int(A ∩ B)} = A.

(64) For all subsets A, B of T such that B is an open domain holds if A ⊆ B,

then Int \overline{A ∪ B} = B. Let us consider T. A family of subsets of T is domains-family if:

(Def.2) for every subset A of T such that A ∈ it holds A is a domain.

The following propositions are true:

(65) For every family F of subsets of T holds F ⊆ the domains of T if and only if F is domains-family.

(66) For every family F of subsets of T such that F is domains-family holds ⋃ F ⊆ \overline{Int ⋃ F} and \overline{⋃ F} = \overline{Int \overline{⋃ F}}.

(67) For every family F of subsets of T such that F is domains-family holds Int \overline{⋂ F} ⊆ ⋂ F and Int \overline{Int ⋂ F} = Int ⋂ F.

(68) For every family F of subsets of T such that F is domains-family holds ⋃ F ∪ Int \overline{⋃ F} is a domain.

(69) Let F be a family of subsets of T. Then for every subset B of T such that B ∈ F holds B ⊆ ⋃ F ∪ Int \overline{⋃ F} and for every subset A of T such that A is a domain holds if for every subset B of T such that B ∈ F holds B ⊆ A, then ⋃ F ∪ Int \overline{⋃ F} ⊆ A.

(70) For every family F of subsets of T such that F is domains-family holds ⋂ F ∩ \overline{Int ⋂ F} is a domain.

(71) Let F be a family of subsets of T. Then (i) for every subset B of T such that B ∈ F holds ⋂ F ∩ \overline{Int ⋂ F} ⊆ B, (ii) F = ∅ or for every subset A of T such that A is a domain holds if for

every subset B of T such that B ∈ F holds A ⊆ B, then A ⊆ ⋂ F ∩\overline{Int ⋂ F}. Let us consider T. A family of subsets of T is closed-domains-family if:

(Def.3) for every subset A of T such that A ∈ it holds A is a closed domain.

We now state several propositions:

(72) For every family F of subsets of T holds F ⊆ the closed domains of T if and only if F is closed-domains-family.

(73) For every family F of subsets of T such that F is closed-domains-family holds F is domains-family.

(74) For every family F of subsets of T such that F is closed-domains-family holds F is closed.

(75) For every family F of subsets of T such that F is domains-family holds \overline{F} is closed-domains-family.

(76) For every family F of subsets of T such that F is closed-domains-family holds \overline{⋃ F} is a closed domain and \overline{Int ⋂ F} is a closed domain.

(77) For every family F of subsets of T holds for every subset B of T such that B ∈ F holds B ⊆ \overline{⋃ F} and for every subset A of T such that A is a closed domain holds if for every subset B of T such that B ∈ F holds B ⊆ A, then \overline{⋃ F} ⊆ A.

(78) Let F be a family of subsets of T. Then if F is closed, then for every subset B of T such that B ∈ F holds \overline{Int ⋂ F} ⊆ B but F = ∅ or for every subset A of T such that A is a closed domain holds if for every subset B of T such that B ∈ F holds A ⊆ B, then A ⊆ \overline{Int ⋂ F}.

Let us consider T. A family of subsets of T is open-domains-family if:

(Def.4) for every subset A of T such that A ∈ it holds A is an open domain.

We now state several propositions:

(79) For every family F of subsets of T holds F ⊆ the open domains of T if and only if F is open-domains-family.

(80) For every family F of subsets of T such that F is open-domains-family holds F is domains-family.

(81) For every family F of subsets of T such that F is open-domains-family holds F is open.

(82) For every family F of subsets of T such that F is domains-family holds Int F is open-domains-family.

(83) For every family F of subsets of T such that F is open-domains-family holds Int ⋂ F is an open domain and Int \overline{⋃ F} is an open domain.

(84) For every family F of subsets of T holds if F is open, then for every subset B of T such that B ∈ F holds B ⊆ Int \overline{⋃ F} but for every subset A of T such that A is an open domain holds if for every subset B of T such that B ∈ F holds B ⊆ A, then Int \overline{⋃ F} ⊆ A.

(85) For every family F of subsets of T holds for every subset B of T such that B ∈ F holds Int ⋂ F ⊆ B but F = ∅ or for every subset A of T such that A is an open domain holds if for every subset B of T such that B ∈ F holds A ⊆ B, then A ⊆ Int ⋂ F.

4. Completeness of the Lattice of Domains

In the sequel T denotes a topological space. Next we state several propositions:

(86) The carrier of the lattice of domains of T = the domains of T.

(87) For all elements a, b of the lattice of domains of T and for all elements A, B of the domains of T such that a = A and b = B holds a ⊔ b = Int \overline{A ∪ B} ∪ (A ∪ B) and a ⊓ b = \overline{Int(A ∩ B)} ∩ (A ∩ B).

(88) ⊥_{the lattice of domains of T} = ∅_{T} and ⊤_{the lattice of domains of T} = Ω_{T}.

(89) For all elements a, b of the lattice of domains of T and for all elements A, B of the domains of T such that a = A and b = B holds a ⊑ b if and only if A ⊆ B.

(90) For every subset X of the lattice of domains of T there exists an element a of the lattice of domains of T such that X ⊑ a and for every element b of the lattice of domains of T such that X ⊑ b holds a ⊑ b.

(91) The lattice of domains of T is complete.

(92) For every family F of subsets of T such that F is domains-family and for every subset X of the lattice of domains of T such that X = F holds ⨆_{(the lattice of domains of T)} X = ⋃ F ∪ Int \overline{⋃ F}.

(93) For every family F of subsets of T such that F is domains-family and for every subset X of the lattice of domains of T such that X = F holds if X ≠ ∅, then ⌈−⌉_{(the lattice of domains of T)}X = ⋂ F ∩ \overline{Int ⋂ F} but if X = ∅, then ⌈−⌉_{(the lattice of domains of T)}X = Ω_{T}.

5. Completeness of the Lattices of Closed Domains and Open Domains

In the sequel T will be a topological space. The following propositions are true:

(94) The carrier of the lattice of closed domains of T = the closed domains of T.

(95) For all elements a, b of the lattice of closed domains of T and for all elements A, B of the closed domains of T such that a = A and b = B holds a ⊔ b = A ∪ B and a ⊓ b = \overline{Int(A ∩ B)}.

(96) ⊥_{the lattice of closed domains of T} = ∅_{T} and ⊤_{the lattice of closed domains of T} = Ω_{T}.

(97) For all elements a, b of the lattice of closed domains of T and for all elements A, B of the closed domains of T such that a = A and b = B holds a ⊑ b if and only if A ⊆ B.

(98) For every subset X of the lattice of closed domains of T there exists an element a of the lattice of closed domains of T such that X ⊑ a and for every element b of the lattice of closed domains of T such that X ⊑ b holds a ⊑ b.

(99) The lattice of closed domains of T is complete.

(100) For every family F of subsets of T such that F is closed-domains-family and for every subset X of the lattice of closed domains of T such that X = F holds ⨆_{(the lattice of closed domains of T)} X = \overline{⋃ F}.

(101) For every family F of subsets of T such that F is closed-domains-family and for every subset X of the lattice of closed domains of T such that X = F holds if X ≠ ∅, then ⌈−⌉_{(the lattice of closed domains of T)}X = \overline{Int ⋂ F} but if X = ∅, then ⌈−⌉_{(the lattice of closed domains of T)}X = Ω_{T}.

(102) For every family F of subsets of T such that F is closed-domains-family and for every subset X of the lattice of domains of T such that X = F holds if X ≠ ∅, then ⌈−⌉_{(the lattice of domains of T)}X = \overline{Int ⋂ F} but if X = ∅, then ⌈−⌉_{(the lattice of domains of T)}X = Ω_{T}.

(103) The carrier of the lattice of open domains of T = the open domains of T.

(104) For all elements a, b of the lattice of open domains of T and for all elements A, B of the open domains of T such that a = A and b = B holds a ⊔ b = Int \overline{A ∪ B} and a ⊓ b = A ∩ B.

(105) ⊥_{the lattice of open domains of T} = ∅_{T} and ⊤_{the lattice of open domains of T} = Ω_{T}.

(106) For all elements a, b of the lattice of open domains of T and for all elements A, B of the open domains of T such that a = A and b = B holds a ⊑ b if and only if A ⊆ B.

78 zbigniew karno and toshihiko watanabe

(107) For every subset X of the lattice of open domains of T there exists an element a of the lattice of open domains of T such that X ⊑ a and for every element b of the lattice of open domains of T such that X ⊑ b holds a ⊑ b.

(108) The lattice of open domains of T is complete.

(109) For every family F of subsets of T such that F is open-domains-family and for every subset X of the lattice of open domains of T such that X = F holds ⨆_{(the lattice of open domains of T)} X = Int \overline{⋃ F}.

(110) For every family F of subsets of T such that F is open-domains-family and for every subset X of the lattice of open domains of T such that X = F holds if X ≠ ∅, then ⌈−⌉_{(the lattice of open domains of T)}X = Int ⋂ F but if X = ∅, then ⌈−⌉_{(the lattice of open domains of T)}X = Ω_{T}.

(111) For every family F of subsets of T such that F is open-domains-family and for every subset X of the lattice of domains of T such that X = F holds ⨆_{(the lattice of domains of T)} X = Int \overline{⋃ F}.

Acknowledgments

The authors would like to thank to Professors A. Trybulec and Cz. Byliński for many helpful conversations during the preparation of this paper. The authors are also very grateful to G. Bancerek for acquainting them with the ���� TEX interface system for automated typesetting and translation.

References

[1] Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719–725, 1991.

[2] Grzegorz Bancerek. Filters - part II. Quotient lattices modulo filters and direct product of two lattices. Formalized Mathematics, 2(3):433–438, 1991.

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

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

[5] Garrett Birkhoff. Lattice Theory. Providence, Rhode Island, New York, 1967.

[6] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

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

[10] Marek Chmur. The lattice of natural numbers and the sublattice of it. The set of prime numbers. Formalized Mathematics, 2(4):453–459, 1991.

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

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

[13] Ryszard Engelking. General Topology. Volume 60 of Monografie Matematyczne, PWN - Polish Scientific Publishers, Warsaw, 1977.

[14] Yoshinori Isomichi. New concepts in the theory of topological space – supercondensed set, subcondensed set, and condensed set. Pacific Journal of Mathematics, 38(3):657– 668, 1971.

completeness of the lattices of domains … 79

[15] Kazimierz Kuratowski. Sur l’opération \overline{A} de l’analysis situs. Fundamenta Mathematicae, 3:182–199, 1922.

[16] Kazimierz Kuratowski. Topology. Volume I, PWN - Polish Scientific Publishers, Academic Press, Warsaw, New York and London, 1966.

[17] Kazimierz Kuratowski and Andrzej Mostowski. Set Theory (with an introduction to descriptive set theory). Volume 86 of Studies in Logic and The Foundations of Mathematics, PWN - Polish Scientific Publishers and North-Holland Publishing Company, Warsaw-Amsterdam, 1976.

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

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

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

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

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

[23] Toshihiko Watanabe. The lattice of domains of a topological space. Formalized Mathematics, 3(1):41–46, 1992.

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

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

Received July 16, 1992 80

On Paracompactness of Metrizable Spaces

Leszek Borys Warsaw University Białystok

Summary. The aim is to prove, using Mizar System, one of the most important result in general topology, namely the Stone Theorem on paracompactness of metrizable spaces [19]. Our proof is based on [18] (and also [16]). We prove first auxiliary fact that every open cover of any metrizable space has a locally finite open refinement. We show next the main theorem that every metrizable space is paracompact. The remaining material is devoted to concepts and certain properties needed for the formulation and the proof of that theorem (see also [5]).

MML Identifier: PCOMPS 2.

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

1. Selected Properties of Real Numbers

We adopt the following rules: r, u, v, w, y are real numbers and k is a natural number. One can prove the following propositions:

(1) r0_{}= 1.

(2) r1_{}= r.

(3) If r > 0 and u > 0, then there exists a natural number k such that

\frac{u}{2k_{}}≤ r.

(4) If k ≥ n and r ≥ 1, then rk_{}≥ rn_{}.

2. Certain Functions Defined on Families of Sets

We adopt the following convention: R will be a binary relation, A, B, C will be sets, and t will be arbitrary. The following proposition is true

(5) If R well orders A, then R |^{2} A well orders A and A = field(R |^{2} A). The scheme MinSet concerns a set A, a binary relation B, and a unary

predicate P, and states that:

there exists arbitrary X such that X ∈ A and P[X] and for an arbitrary Y such that Y ∈ A and P[Y] holds ⟨⟨X, Y ⟩⟩ ∈ B provided the parameters meet the following conditions:

• B well orders A,

• there exists arbitrary X such that X ∈ A and P[X].

We now define three new functors. Let F_{1} be a family of sets, and let R be a binary relation, and let B be an element of F_{1}. The functor ⋃_{β<_{R}B} β yields a family of sets and is defined as follows: (Def.1) ⋃_{β<_{R}B} β = ⋃(R−Seg(B)). Let F_{1} be a family of sets, and let R be a binary relation. The disjoint family of F_{1}, R yielding a family of sets is defined by: (Def.2) A ∈ the disjoint family of F_{1}, R if and only if there exists an element

B of F_{1} such that B ∈ F_{1} and A = B \ ⋃_{β<_{R}B} β. Let X be a set, and let n be a natural number, and let f be a function from into 2^{X}. The functor ⋃_{κ<n} f(κ) yields a set and is defined as follows: (Def.3) ⋃_{κ<n} f(κ) = ⋃(f ^{∘} (Seg n \ {n})).

3. Paracompactness of Metrizable Spaces

We adopt the following convention: P_{1} will denote a topological space, F_{1}, G_{1} will denote families of subsets of P_{1}, and W, X will denote subsets of P_{1}. We now state several propositions:

(6) If P_{1} is a T_{3} space, then for every F_{1} such that F_{1} is a cover of P_{1} and F_{1} is open there exists H_{1} such that H_{1} is open and H_{1} is a cover of P_{1} and for every V such that V ∈ H_{1} there exists W such that W ∈ F_{1} and \overline{V} ⊆ W.

(7) For all P_{1}, F_{1} such that P_{1} is a T_{2} space and P_{1} is paracompact and F_{1} is a cover of P_{1} and F_{1} is open there exists G_{1} such that G_{1} is open and G_{1} is a cover of P_{1} and clf G_{1} is finer than F_{1} and G_{1} is locally finite.

(8) For every function f from [: the carrier of P_{1}, the carrier of P_{1} :] into such that f is a metric of the carrier of P_{1} holds if P_{2} = MetrSp((the carrier of P_{1}), f), then the carrier of P_{2} = the carrier of P_{1}.

(9) For every function f from [: the carrier of P_{1}, the carrier of P_{1} :] into such that f is a metric of the carrier of P_{1} holds if P_{2} = MetrSp((the

on paracompactness of metrizable spaces 83

carrier of P_{1}), f), then x is a point of P_{1} if and only if x is an element of the carrier of P_{2}.

(10) For every function f from [: the carrier of P_{1}, the carrier of P_{1} :] into such that f is a metric of the carrier of P_{1} holds if P_{2} = MetrSp((the carrier of P_{1}), f), then X is a subset of P_{1} if and only if X is a subset of the carrier of P_{2}.

(11) For every function f from [: the carrier of P_{1}, the carrier of P_{1} :] into such that f is a metric of the carrier of P_{1} holds if P_{2} = MetrSp((the carrier of P_{1}), f), then F_{1} is a family of subsets of P_{1} if and only if F_{1} is a family of subsets of the carrier of P_{2}.

In the sequel k is a natural number. Let P_{2} be a non-empty set, and let g be a function from into (2^{2^{P_{2}}} )^{∗}, and let us consider n. Then g(n) is a finite sequence of elements of 2^{2^{P_{2}}} .

The following propositions are true:

(12) If P_{1} is metrizable, then for every family F_{1} of subsets of P_{1} such that F_{1} is a cover of P_{1} and F_{1} is open there exists a family G_{1} of subsets of P_{1} such that G_{1} is open and G_{1} is a cover of P_{1} and G_{1} is finer than F_{1} and G_{1} is locally finite.

(13) If P_{1} is metrizable, then P_{1} is paracompact.

References

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

[2] Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123–129, 1990.

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

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

[5] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

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

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

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

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

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

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

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

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

[16] Hanna Patkowska. Wstȩp do Topologii. PWN, Warszawa, 1974.

84 leszek borys

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

[18] M. E. Rudin. A new proof that metric spaces are paracompact. Proc. Amer. Math. Soc., 20:603, 1969.

[19] A. H. Stone. Paracompactness and product spaces. Bull. Amer. Math. Soc., 54:977–982, 1948.

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

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

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

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

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

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

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

Received July 23, 1992

The Brouwer Fixed Point Theorem for Intervals ^{1}

Toshihiko Watanabe Shinshu University Nagano

Summary. The aim is to prove, using Mizar System, the following simplest version of the Brouwer Fixed Point Theorem [2]. For every continuous mapping f : → of the topological unit interval there exists a point x such that f(x) = x (see e.g. [9], [3]).

MML Identifier: TREAL 1.

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

1. Properties of Topological Intervals

The following three propositions are true:

(1) For all real numbers a, b, c, d such that a ≤ c and d ≤ b and c ≤ d holds [c, d] ⊆ [a, b].

(2) For all real numbers a, b, c, d such that a ≤ c and b ≤ d and c ≤ b holds [a, b] ∪ [c, d] = [a, d].

(3) For all real numbers a, b, c, d such that a ≤ c and b ≤ d and c ≤ b holds [a, b] ∩ [c, d] = [c, b].

In the sequel a, b, c, d are real numbers. We now state four propositions:

(4) For every subset A of ^{1} such that A = [a, b] holds A is closed.

(5) If a ≤ b, then [a, b]_{T} is a closed subspace of ^{1}.

(6) If a ≤ c and d ≤ b and c ≤ d, then [c, d]_{T} is a closed subspace of [a, b]_{T}.

1This paper was done under the supervision of Z. Karno while the author was visiting the Institute of Mathematics of Warsaw University in Białystok.

(7) If a ≤ c and b ≤ d and c ≤ b, then [a, d]_{T} = [a, b]_{T} ∪ [c, d]_{T} and [c, b]_{T} = [a, b]_{T} ∩ [c, d]_{T}.

We now define two new functors. Let a, b be real numbers. Let us assume that a ≤ b. The functor a_{[a,b]_{T}} yields a point of [a, b]_{T} and is defined by: (Def.1) a_{[a,b]_{T}} = a. The functor b_{[a,b]_{T}} yields a point of [a, b]_{T} and is defined by: (Def.2) b_{[a,b]_{T}} = b.

One can prove the following two propositions:

(8) 0_{}= 0_{[0,1]_{T}} and 1_{}= 1_{[0,1]_{T}}.

(9) If a ≤ b and b ≤ c, then a_{[a,b]_{T}} = a_{[a,c]_{T}} and c_{[b,c]_{T}} = c_{[a,c]_{T}}.

2. Continuous Mappings Between Topological Intervals

Let a, b be real numbers satisfying the condition: a ≤ b. Let t_{1}, t_{2} be points of [a, b]_{T}. The functor L_{01}(t_{1}, t_{2}) yielding a mapping from [0, 1]_{T} into [a, b]_{T} is defined as follows: (Def.3) for every point s of [0, 1]_{T} and for all real numbers r, r_{1}, r_{2} such that

s = r and r_{1} = t_{1} and r_{2} = t_{2} holds (L_{01}(t_{1}, t_{2}))(s) = (1 − r) · r_{1} + r · r_{2}. We now state four propositions:

(10) Let a, b be real numbers. Then if a ≤ b, then for all points t_{1}, t_{2} of [a, b]_{T} and for every point s of [0, 1]_{T} and for all real numbers r, r_{1}, r_{2} such that s = r and r_{1} = t_{1} and r_{2} = t_{2} holds (L_{01}(t_{1}, t_{2}))(s) = (r_{2} −r_{1})·r +r_{1}.

(11) For all real numbers a, b such that a ≤ b and for all points t_{1}, t_{2} of [a, b]_{T} holds L_{01}(t_{1}, t_{2}) is a continuous mapping from [0, 1]_{T} into [a, b]_{T}.

(12) For all real numbers a, b such that a ≤ b and for all points t_{1}, t_{2} of [a, b]_{T} holds (L_{01}(t_{1}, t_{2}))(0_{[0,1]_{T}}) = t_{1} and (L_{01}(t_{1}, t_{2}))(1_{[0,1]_{T}}) = t_{2}.

(13) L_{01}(0_{[0,1]_{T}}, 1_{[0,1]_{T}}) = id_{([0, 1]_{T})}.

Let a, b be real numbers satisfying the condition: a < b. Let t_{1}, t_{2} be points of [0, 1]_{T}. The functor P_{01}(a, b, t_{1}, t_{2}) yielding a mapping from [a, b]_{T} into [0, 1]_{T} is defined as follows: (Def.4) for every point s of [a, b]_{T} and for all real numbers r, r_{1}, r_{2} such that

s = r and r_{1} = t_{1} and r_{2} = t_{2} holds (P_{01}(a, b, t_{1}, t_{2}))(s) = \frac{(b−r)·r_{1}+(r−a)·r_{2}}{b−a}. The following propositions are true:

(14) Let a, b be real numbers. Suppose a < b. Let t_{1}, t_{2} be points of [0, 1]_{T}. Let s be a point of [a, b]_{T}. Then for all real numbers r, r_{1}, r_{2} such that s = r and r_{1} = t_{1} and r_{2} = t_{2} holds (P_{01}(a, b, t_{1}, t_{2}))(s) = \frac{r_{2}−r_{1}}{b−a} · r + \frac{b·r_{1}−a·r_{2}}{b−a}.

(15) For all real numbers a, b such that a < b and for all points t_{1}, t_{2} of [0, 1]_{T} holds P_{01}(a, b, t_{1}, t_{2}) is a continuous mapping from [a, b]_{T} into [0, 1]_{T}.

(16) For all real numbers a, b such that a < b and for all points t_{1}, t_{2} of [0, 1]_{T} holds (P_{01}(a, b, t_{1}, t_{2}))(a_{[a,b]_{T}}) = t_{1} and (P_{01}(a, b, t_{1}, t_{2}))(b_{[a,b]_{T}}) = t_{2}.

(17) P_{01}(0, 1, 0_{[0,1]_{T}}, 1_{[0,1]_{T}}) = id_{([0, 1]_{T})}.

(18) Let a, b be real numbers. Then if a < b, then id_{([a, b]_{T})} = L_{01}(a_{[a,b]_{T}}, b_{[a,b]_{T}}) · P_{01}(a, b, 0_{[0,1]_{T}}, 1_{[0,1]_{T}}) and id_{([0, 1]_{T})} = P_{01}(a, b, 0_{[0,1]_{T}}, 1_{[0,1]_{T}}) · L_{01}(a_{[a,b]_{T}}, b_{[a,b]_{T}}).

(19) Let a, b be real numbers. Then if a < b, then id_{([a, b]_{T})} = L_{01}(b_{[a,b]_{T}}, a_{[a,b]_{T}}) · P_{01}(a, b, 1_{[0,1]_{T}}, 0_{[0,1]_{T}}) and id_{([0, 1]_{T})} = P_{01}(a, b, 1_{[0,1]_{T}}, 0_{[0,1]_{T}}) · L_{01}(b_{[a,b]_{T}}, a_{[a,b]_{T}}).

(20) Let a, b be real numbers. Suppose a < b. Then (i) L_{01}(a_{[a,b]_{T}}, b_{[a,b]_{T}}) is a homeomorphism, (ii) (L_{01}(a_{[a,b]_{T}}, b_{[a,b]_{T}}))^{−1} = P_{01}(a, b, 0_{[0,1]_{T}}, 1_{[0,1]_{T}}), (iii) P_{01}(a, b, 0_{[0,1]_{T}}, 1_{[0,1]_{T}}) is a homeomorphism, (iv) (P_{01}(a, b, 0_{[0,1]_{T}}, 1_{[0,1]_{T}}))^{−1} = L_{01}(a_{[a,b]_{T}}, b_{[a,b]_{T}}).

(21) Let a, b be real numbers. Suppose a < b. Then (i) L_{01}(b_{[a,b]_{T}}, a_{[a,b]_{T}}) is a homeomorphism, (ii) (L_{01}(b_{[a,b]_{T}}, a_{[a,b]_{T}}))^{−1} = P_{01}(a, b, 1_{[0,1]_{T}}, 0_{[0,1]_{T}}), (iii) P_{01}(a, b, 1_{[0,1]_{T}}, 0_{[0,1]_{T}}) is a homeomorphism, (iv) (P_{01}(a, b, 1_{[0,1]_{T}}, 0_{[0,1]_{T}}))^{−1} = L_{01}(b_{[a,b]_{T}}, a_{[a,b]_{T}}).

3. Connectedness of Intervals and Brouwer Fixed Point Theorem for Intervals

We now state several propositions:

(22) is connected.

(23) For all real numbers a, b such that a ≤ b holds [a, b]_{T} is connected.

(24) For every continuous mapping f from into there exists a point x of such that f(x) = x.

(25) For all real numbers a, b such that a ≤ b and for every continuous mapping f from [a, b]_{T} into [a, b]_{T} there exists a point x of [a, b]_{T} such that f(x) = x.

(26) Let X, Y be subspaces of ^{1}. Then for every continuous mapping f from X into Y such that there exist real numbers a, b such that a ≤ b and [a, b] ⊆ the carrier of X and [a, b] ⊆ the carrier of Y and f ^{∘} [a, b] ⊆ [a, b] there exists a point x of X such that f(x) = x.

(27) For all subspaces X, Y of ^{1} and for every continuous mapping f from X into Y such that there exist real numbers a, b such that a ≤ b and [a, b] ⊆ the carrier of X and f ^{∘} [a, b] ⊆ [a, b] there exists a point x of X such that f(x) = x.

Acknowledgments

The author wishes to express his thanks to Professors A. Trybulec and Z. Karno for their useful suggestions and many valuable comments.

88 toshihiko watanabe

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On Powers of Cardinals

Grzegorz Bancerek IM PAN, Warsaw Warsaw University, Białystok

Summary. In the first section the results of [23, axiom (30)]^{1}, i.e. the correspondence between natural and ordinal (cardinal) numbers are shown. The next section is concerned with the concepts of infinity and cofinality (see [3]), and introduces alephs as infinite cardinal numbers. The arithmetics of alephs, i.e. some facts about addition and multiplication, is present in the third section. The concepts of regular and irregular alephs are introduced in the fourth section, and the fact that ℵ_{0} and every non-limit cardinal number are regular is proved there. Finally, for every alephs α and β

α^{β} = \{ \begin{matrix}2^{β}, & if α ≤ β, \\ ∑_{γ<α} γ^{β}, & if β < cfα and α is limit cardinal, \\ (∑_{γ<α} γ^{β})^{cfα} , & if cfα ≤ β ≤ α. \end{matrix}

Some proofs are based on [20].

MML Identifier: CARD 5.

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

1. Results of [23, axiom (30)]

One can readily check that every set which is cardinal is also ordinal-like.

For simplicity we adopt the following convention: n denotes a natural number, A, B denote ordinal numbers, X denotes a set, and x, y are arbitrary. We now state several propositions:

1Axiom (30) – n = {k ∈ : k < n} for every natural number n.

(1) 0 = ∅ and 1 = {0} and 2 = {0, 1}.

(2) succ n = n + 1.

(3) For every n holds ord(n) = n and \overline{\overline{n}} = n.

(4) 0 = 0 and 1 = 1.

(5) \overline{0} = 0 and \overline{1} = 1 and \overline{2} = 2.

(6) If X is finite, then card X = \overline{\overline{X}}.

(7) = ω and = ℵ_{0}.

(8) Seg n = (n + 1) \ {0}.

2. Infinity, alephs and cofinality

We adopt the following rules: f is a function, K, M, N are cardinal numbers, and p_{1}, p_{2} are sequences of ordinal numbers. The following propositions are true:

(9) \overline{\overline{X}}^{+} = X^{+}.

(10) y ∈ ⋃ f if and only if there exists x such that x ∈ dom f and y ∈ f(x).

(11) ℵ_{A} is not finite.

(12) If M is not finite, then there exists A such that M = ℵ_{A}.

(13) There exists n such that M = \overline{\overline{n}} or there exists A such that M = ℵ_{A}. Let us consider p_{1}. Then ⋃ p_{1} is an ordinal number. Next we state a number of propositions:

(14) If X ⊆ A, then there exists p_{1} such that p_{1} = the canonical isomorphism between ^{⊆}\overline{^{⊆}X} and ^{⊆}_{X} and p_{1} is increasing and dom p_{1} = \overline{⊆X} and rng p_{1} = X.

(15) If X ⊆ A, then sup X is cofinal with \overline{⊆X}.

(16) If X ⊆ A, then \overline{\overline{X}} = \overline{\overline{\overline{⊆X}}}.

(17) There exists B such that B ⊆ \overline{\overline{A}} and A is cofinal with B.

(18) There exists M such that M ≤ \overline{\overline{A}} and A is cofinal with M and for every B such that A is cofinal with B holds M ⊆ B.

(19) If rng p_{1} = rng p_{2} and p_{1} is increasing and p_{2} is increasing, then p_{1} = p_{2}.

(20) If p_{1} is increasing, then p_{1} is one-to-one.

(21) (p_{1} ^{}p_{2}) dom p_{1} = p_{1}.

(22) If X ≠ ∅, then \overline{\overline{\{Y : \overline{\overline{Y}} < M\}}} ≤ M · \overline{\overline{X}}^{M}, where Y ranges over elements of 2^{X}.

(23) M < \overline{2}^{M}.

We now define four new constructions. A set is infinite if: (Def.1) it is not finite.

Let us observe that there exists a set which is infinite. One can readily check that there exists a cardinal number which is infinite. One can readily check that every set which is infinite is also non-empty.

An aleph is an infinite cardinal number.

Let us consider M. The functor cf M yielding a cardinal number is defined by: (Def.2) M is cofinal with cf M and for every N such that M is cofinal with N

holds cf M ≤ N. Let us consider N. The functor (α ↦ α^{N})_{α∈M} yielding a function yielding cardinal numbers is defined as follows: (Def.3) for every x holds x ∈ dom((α ↦ α^{N})_{α∈M}) if and only if x ∈ M and

x is a cardinal number and for every K such that K ∈ M holds (α ↦ α^{N})_{α∈M}(K) = K^{N}.

Let us consider A. Then ℵ_{A} is an aleph.

3. Arithmetics of alephs

In the sequel a, b will be alephs. The following propositions are true:

(24) There exists A such that a = ℵ_{A}.

(25) a ≠ \overline{0} and a ≠ \overline{1} and a ≠ \overline{2} and a ≠ \overline{\overline{n}} and \overline{\overline{n}} < a and ℵ_{0} ≤ a.

(26) If a ≤ M or a < M, then M is an aleph.

(27) If a ≤ M or a < M, then a + M = M and M + a = M and a · M = M and M · a = M.

(28) a + a = a and a · a = a.

(29) If M ≤ a or M < a, then a + M = a and M + a = a.

(30) If \overline{0} < M but M ≤ a or M < a, then a · M = a and M · a = a.

(31) M ≤ M^{a}.

(32) ⋃ a = a.

Let us consider a, M. Then a + M is an aleph. Let us consider M, a. Then M + a is an aleph. Let us consider a, b. Then a + b is an aleph. Then a · b is an aleph. Then a^{b} is an aleph.

4. Regular alephs

We now define two new attributes. An aleph is regular if: (Def.4) cf it = it. An aleph is irregular if: (Def.5) cf it < it.

Let us consider a. Then a^{+} is an aleph. We see that the element of a is an ordinal number.

One can prove the following propositions:

(33) cf M ≤ M.

(34) cf(ℵ_{0}) = ℵ_{0}.

(35) cf(a^{+}) = a^{+}.

(36) ℵ_{0} ≤ cf a.

(37) cf \overline{0} = \overline{0} and cf \overline{\overline{n + 1}} = \overline{1}.

(38) If X ⊆ M and \overline{\overline{X}} < cf M, then sup X ∈ M and ⋃ X ∈ M.

(39) If dom p_{1} = M and rng p_{1} ⊆ N and M < cf N, then sup p_{1} ∈ N and

⋃ p_{1} ∈ N. Let us consider a. Then cf a is an aleph. One can prove the following propositions:

(40) If cf a < a, then a is a limit cardinal number.

(41) If cf a < a, then there exists a sequence x_{1} of ordinal numbers such that dom x_{1} = cf a and rng x_{1} ⊆ a and x_{1} is increasing and a = sup x_{1} and x_{1} is a function yielding cardinal numbers and \overline{0} ∉ rng x_{1}.

(42) ℵ_{0} is regular and a^{+} is regular.

5. Infinite powers

In the sequel a, b will denote alephs. The following propositions are true:

(43) If a ≤ b, then a^{b} = \overline{2}^{b}.

(44) (a^{+})^{b} = a^{b} · (a^{+}).

(45) ∑((α ↦ α^{b})_{α∈a}) ≤ a^{b}.

(46) If a is a limit cardinal number and b < cf a, then a^{b} = ∑((α ↦ α^{b})_{α∈a}).

(47) If cf a ≤ b and b < a, then a^{b} = (∑((α ↦ α^{b})_{α∈a}))^{cf a}.

References

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

[3] Grzegorz Bancerek. Consequences of the reflection theorem. Formalized Mathematics, 1(5):989–993, 1990.

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on powers of cardinals 93

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94

Basic Properties of Connecting Points with Line Segments in E_{T}^{2}

Yatsuka Nakamura Shinshu University Nagano Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. Some properties of line segments in 2-dimensional Euclidean space and some relations between line segments and balls are proved.

MML Identifier: TOPREAL3.

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

1. Real Numbers Preliminaries

For simplicity we follow the rules: p, p_{1}, p_{2}, p_{3}, q will denote points of E_{T}^{2}, f, h will denote finite sequences of elements of E_{T}^{2}, r, r_{1}, r_{2}, s, s_{1}, s_{2} will denote real numbers, u, u_{1}, u_{2} will denote points of E^{2}, n, m, i, j, k will denote natural numbers, and x, y, z will be arbitrary. One can prove the following propositions:

(1) 3 − 2 = 1 and 3 − 1 = 2 and \frac{1}{2} = 1 − \frac{1}{2}.

(2) 0 ≤ \frac{1}{2} and \frac{1}{2} ≤ 1.

(3) If r < s, then r < \frac{r+s}{2} and \frac{r+s}{2} < s and r < \frac{s+r}{2} and \frac{s+r}{2} < s.

(4) If r ≠ s, then r ≠ \frac{r+s}{2} and \frac{r+s}{2} ≠ s.

(5) If r_{1} > s_{1} and r_{2} ≥ s_{2} or r_{1} ≥ s_{1} and r_{2} > s_{2}, then r_{1} + r_{2} > s_{1} + s_{2}.

1The article was written during my visit at Shinshu University in 1992.

2. Properties of Line Segments

We now state a number of propositions: (6) 1 ∈ Seg len⟨x, y, z⟩ and 2 ∈ Seg len⟨x, y, z⟩ and 3 ∈ Seg len⟨x, y, z⟩.

(7) (p_{1} + p_{2})_{1} = p_{1}_{1} + p_{2}_{1} and (p_{1} + p_{2})_{2} = p_{1}_{2} + p_{2}_{2}.

(8) (p_{1} − p_{2})_{1} = p_{1}_{1} − p_{2}_{1} and (p_{1} − p_{2})_{2} = p_{1}_{2} − p_{2}_{2}.

(9) (r · p)_{1} = r · p_{1} and (r · p)_{2} = r · p_{2}.

(10) If p_{1} = ⟨r_{1}, s_{1}⟩ and p_{2} = ⟨r_{2}, s_{2}⟩, then p_{1} + p_{2} = ⟨r_{1} + r_{2}, s_{1} + s_{2}⟩ and p_{1} − p_{2} = ⟨r_{1} − r_{2}, s_{1} − s_{2}⟩.

(11) p = q if and only if p_{1} = q_{1} and p_{2} = q_{2}.

(12) If u_{1} = p_{1} and u_{2} = p_{2}, then ρ^{2}(u_{1}, u_{2}) = \sqrt{(p_{1}_{1} − p_{2}_{1})^{2} + (p_{1}_{2} − p_{2}_{2})^{2}}.

(13) The carrier of E_{T}^{n} = the carrier of E^{n}.

(14) x is a point of E^{2} if and only if x is a point of E_{T}^{2}.

(15) If r_{1} < s_{1}, then {p_{1} : p_{1}_{1} = r ∧ r_{1} ≤ p_{1}_{2} ∧ p_{1}_{2} ≤ s_{1}} = L([r, r_{1}], [r, s_{1}]).

(16) If r_{1} < s_{1}, then {p_{1} : p_{1}_{2} = r ∧ r_{1} ≤ p_{1}_{1} ∧ p_{1}_{1} ≤ s_{1}} = L([r_{1}, r], [s_{1}, r]).

(17) If p ∈ L([r, r_{1}], [r, s_{1}]), then p_{1} = r.

(18) If p ∈ L([r_{1}, r], [s_{1}, r]), then p_{2} = r.

(19) If p_{1} ≠ q_{1} and p_{2} = q_{2}, then [\frac{p1+q1}{2} , p_{2}] ∈ L(p, q).

(20) If p_{1} = q_{1} and p_{2} ≠ q_{2}, then [p_{1}, \frac{p2+q2}{2} ] ∈ L(p, q).

(21) If f = ⟨p, p_{1}, q⟩ and i ≠ 0 and j − i > 1, then L(f, j, j + 1) = ∅.

(22) If i = 0, then L(f, i, i + 1) = ∅.

(23) If f = ⟨p_{1}, p_{2}, p_{3}⟩, then L̃(f) = L(p_{1}, p_{2}) ∪ L(p_{2}, p_{3}).

(24) If i ∈ dom f and j ∈ dom(f i) and k ∈ dom(f i), then L(f, j, k) = L(f i, j, k).

(25) If j ∈ dom f and i ∈ dom f, then L(f ^{}h, j, i) = L(f, j, i).

(26) L(f, i, i + 1) ⊆ L̃(f).

(27) L̃(f i) ⊆ L̃(f).

(28) For all r, p_{1}, p_{2}, u such that r > 0 and p_{1} ∈ Ball(u, r) and p_{2} ∈ Ball(u, r) holds L(p_{1}, p_{2}) ⊆ Ball(u, r).

(29) If u = p_{1} and p_{1} = [r_{1}, s_{1}] and p_{2} = [r_{2}, s_{2}] and p = [r_{2}, s_{1}] and p_{2} ∈ Ball(u, r), then p ∈ Ball(u, r).

(30) If r_{1} ≠ s_{1} and r > 0 and [s, r_{1}] ∈ Ball(u, r) and [s, s_{1}] ∈ Ball(u, r), then [s, \frac{r_{1}+s_{1}}{2} ] ∈ Ball(u, r).

(31) If r_{1} ≠ s_{1} and r > 0 and [r_{1}, s] ∈ Ball(u, r) and [s_{1}, s] ∈ Ball(u, r), then [\frac{r_{1}+s_{1}}{2} , s] ∈ Ball(u, r).

(32) If r_{1} ≠ s_{1} and s_{2} ≠ r_{2} and r > 0 and [r_{1}, r_{2}] ∈ Ball(u, r) and [s_{1}, s_{2}] ∈ Ball(u, r), then [r_{1}, s_{2}] ∈ Ball(u, r) or [s_{1}, r_{2}] ∈ Ball(u, r).

(33) Suppose that (i) f(1) ∉ Ball(u, r), (ii) 1 ≤ m, (iii) m ≤ len f − 1, (iv) L(f, m, m + 1) ∩ Ball(u, r) ≠ ∅,

(v) for every i such that 1 ≤ i and i ≤ len f − 1 and L(f, i, i + 1) ∩ Ball(u, r) ≠ ∅ holds m ≤ i. Then f(m) ∉ Ball(u, r).

(34) For all q, p_{2}, p such that q_{2} = p_{2}_{2} and p_{2} ≠ p_{2}_{2} holds (L(p_{2}, [p_{2}_{1}, p_{2}]) ∪ L([p_{2}_{1}, p_{2}], p)) ∩ L(q, p_{2}) = {p_{2}}.

(35) For all q, p_{2}, p such that q_{1} = p_{2}_{1} and p_{1} ≠ p_{2}_{1} holds (L(p_{2}, [p_{1}, p_{2}_{2}]) ∪ L([p_{1}, p_{2}_{2}], p)) ∩ L(q, p_{2}) = {p_{2}}.

(36) If p_{1} ≠ q_{1} and p_{2} ≠ q_{2}, then L(p, [p_{1}, q_{2}]) ∩ L([p_{1}, q_{2}], q) = {[p_{1}, q_{2}]}. One can prove the following propositions:

(37) If p_{1} ≠ q_{1} and p_{2} ≠ q_{2}, then L(p, [q_{1}, p_{2}]) ∩ L([q_{1}, p_{2}], q) = {[q_{1}, p_{2}]}.

(38) If p_{1} = q_{1} and p_{2} ≠ q_{2}, then L(p, [p_{1}, \frac{p2+q2}{2} ])∩L([p_{1}, \frac{p2+q2}{2} ], q) = \{[p_{1}, \frac{p2+q2}{2} ]\}.

(39) If p_{1} ≠ q_{1} and p_{2} = q_{2}, then L(p, [\frac{p1+q1}{2} , p_{2}]) ∩ L([\frac{p1+q1}{2} , p_{2}], q) = \{[\frac{p1+q1}{2} , p_{2}]\}.

(40) If i > 2 and i ∈ dom f and f is a special sequence, then f i is a special sequence.

(41) If p_{1} ≠ q_{1} and p_{2} ≠ q_{2} and f = ⟨p, [p_{1}, q_{2}], q⟩, then f(1) = p and f(len f) = q and f is a special sequence.

(42) If p_{1} ≠ q_{1} and p_{2} ≠ q_{2} and f = ⟨p, [q_{1}, p_{2}], q⟩, then f(1) = p and f(len f) = q and f is a special sequence.

(43) If p_{1} = q_{1} and p_{2} ≠ q_{2} and f = ⟨p, [p_{1}, \frac{p2+q2}{2} ], q⟩, then f(1) = p and f(len f) = q and f is a special sequence.

(44) If p_{1} ≠ q_{1} and p_{2} = q_{2} and f = ⟨p, [\frac{p1+q1}{2} , p_{2}], q⟩, then f(1) = p and f(len f) = q and f is a special sequence.

(45) If i ∈ dom f and i + 1 ∈ dom f and f(i) = p and f(i + 1) = q, then L̃(f (i + 1)) = L̃(f i) ∪ L(p, q).

(46) If len f ≥ 2 and p ∉ L̃(f), then for every n such that 1 ≤ n and n ≤ len f holds f(n) ≠ p.

(47) If q ≠ p and L(q, p) ∩ L̃(f) = {q}, then p ∉ L̃(f).

(48) Suppose that (i) f is a special sequence, (ii) f(1) = p, (iii) f(len f) = q, (iv) p ∉ Ball(u, r),

(v) q ∈ Ball(u, r), (vi) q ∈ L(f, m, m + 1), (vii) 1 ≤ m, (viii) m ≤ len f − 1, (ix) L(f, m, m + 1) ∩ Ball(u, r) ≠ ∅.

Then m = len f − 1.

(49) Suppose that (i) r > 0, (ii) p_{1} ∉ Ball(u, r), (iii) q ∈ Ball(u, r), (iv) p ∈ Ball(u, r),

(v) p ∉ L(p_{1}, q), (vi) q_{1} = p_{1} and q_{2} ≠ p_{2} or q_{1} ≠ p_{1} and q_{2} = p_{2}, (vii) p_{1}_{1} = q_{1} or p_{1}_{2} = q_{2}.

Then L(p_{1}, q) ∩ L(q, p) = {q}.

(50) Suppose that (i) r > 0, (ii) p_{1} ∉ Ball(u, r), (iii) p ∈ Ball(u, r), (iv) [p_{1}, q_{2}] ∈ Ball(u, r),

(v) q ∈ Ball(u, r), (vi) [p_{1}, q_{2}] ∉ L(p_{1}, p), (vii) p_{1}_{1} = p_{1}, (viii) p_{1} ≠ q_{1}, (ix) p_{2} ≠ q_{2}.

Then (L(p, [p_{1}, q_{2}]) ∪ L([p_{1}, q_{2}], q)) ∩ L(p_{1}, p) = {p}.

(51) Suppose that (i) r > 0, (ii) p_{1} ∉ Ball(u, r), (iii) p ∈ Ball(u, r), (iv) [q_{1}, p_{2}] ∈ Ball(u, r),

(v) q ∈ Ball(u, r), (vi) [q_{1}, p_{2}] ∉ L(p_{1}, p), (vii) p_{1}_{2} = p_{2}, (viii) p_{1} ≠ q_{1}, (ix) p_{2} ≠ q_{2}.

Then (L(p, [q_{1}, p_{2}]) ∪ L([q_{1}, p_{2}], q)) ∩ L(p_{1}, p) = {p}.

References

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

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

[3] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

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

basic properties of connecting points with … 99

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

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

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

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

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

[12] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

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

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

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

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

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

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

100

Connectedness Conditions Using Polygonal Arcs

Yatsuka Nakamura Shinshu University Nagano Jarosław Kotowicz^{1} Warsaw University Białystok

Summary. A concept of special polygonal arc joining two different points is defined. Any two points in a ball can be connected by this kind of arc, and that is also true for any region in E_{T}^{2}.

MML Identifier: TOPREAL4.

The notation and terminology used here have been introduced in the following articles: [13], [9], [1], [4], [2], [12], [11], [14], [10], [5], [3], [6], [7], and [8]. For simplicity we follow a convention: P, P_{1}, P_{2}, R will denote subsets of E_{T}^{2}, p, p_{1}, p_{2}, q will denote points of E_{T}^{2}, f, h will denote finite sequences of elements of E_{T}^{2}, r will denote a real number, u will denote a point of E^{2}, and n, i will denote natural numbers. We now define three new predicates. Let us consider P, p, q. We say that P is a special polygonal arc joining p and q if and only if: (Def.1) there exists f such that f is a special sequence and P = L̃(f) and

p = f(1) and q = f(len f). Let us consider P. We say that P is a special polygon if and only if the conditions (Def.2) is satisfied. (Def.2) (i) There exist p_{1}, p_{2} such that p_{1} ≠ p_{2} and p_{1} ∈ P and p_{2} ∈ P,

(ii) for all p, q such that p ∈ P and q ∈ P and p ≠ q there exist P_{1}, P_{2} such that P_{1} is a special polygonal arc joining p and q and P_{2} is a special polygonal arc joining p and q and P_{1} ∩ P_{2} = {p, q} and P = P_{1} ∪ P_{2}.

We say that P is a region if and only if: (Def.3) P is open and P is connected.

The following propositions are true:

1The article was written during my visit at Shinshu University in 1992.

(1) If P is a special polygonal arc joining p and q, then P is a special polygonal arc.

(2) If P is a special polygonal arc joining p and q, then P is an arc from p to q.

(3) If P is a special polygonal arc joining p and q, then p ∈ P and q ∈ P.

(4) If P is a special polygonal arc joining p and q, then p ≠ q.

(5) If P is a special polygon, then P is a simple closed curve.

(6) Suppose p_{1} = q_{1} and p_{2} ≠ q_{2} and r > 0 and p ∈ Ball(u, r) and q ∈ Ball(u, r) and f = ⟨p, [p_{1}, \frac{p2+q2}{2} ], q⟩. Then f is a special sequence and f(1) = p and f(len f) = q and L̃(f) is a special polygonal arc joining p and q and L̃(f) ⊆ Ball(u, r).

(7) Suppose p_{1} ≠ q_{1} and p_{2} = q_{2} and r > 0 and p ∈ Ball(u, r) and q ∈ Ball(u, r) and f = ⟨p, [\frac{p1+q1}{2} , p_{2}], q⟩. Then f is a special sequence and f(1) = p and f(len f) = q and L̃(f) is a special polygonal arc joining p and q and L̃(f) ⊆ Ball(u, r).

(8) Suppose p_{1} ≠ q_{1} and p_{2} ≠ q_{2} and r > 0 and p ∈ Ball(u, r) and q ∈ Ball(u, r) and [p_{1}, q_{2}] ∈ Ball(u, r) and f = ⟨p, [p_{1}, q_{2}], q⟩. Then f is a special sequence and f(1) = p and f(len f) = q and L̃(f) is a special polygonal arc joining p and q and L̃(f) ⊆ Ball(u, r).

(9) Suppose p_{1} ≠ q_{1} and p_{2} ≠ q_{2} and r > 0 and p ∈ Ball(u, r) and q ∈ Ball(u, r) and [q_{1}, p_{2}] ∈ Ball(u, r) and f = ⟨p, [q_{1}, p_{2}], q⟩. Then f is a special sequence and f(1) = p and f(len f) = q and L̃(f) is a special polygonal arc joining p and q and L̃(f) ⊆ Ball(u, r).

(10) If r > 0 and p ≠ q and p ∈ Ball(u, r) and q ∈ Ball(u, r), then there exists P such that P is a special polygonal arc joining p and q and P ⊆ Ball(u, r).

(11) Suppose p ≠ p_{1} and p_{1}_{2} = p_{2} and f is a special sequence and f(1) = p_{1} and f(len f) = p_{2} and p ∈ L(f, 1, 2) and h = ⟨p_{1}, [\frac{p_{1}1+p1}{2} , p_{1}_{2}], p⟩. Then h is a special sequence and h(1) = p_{1} and h(len h) = p and L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) and L̃(h) = L̃(f 1) ∪ L(p_{1}, p).

(12) Suppose p ≠ p_{1} and p_{1}_{1} = p_{1} and f is a special sequence and f(1) = p_{1} and f(len f) = p_{2} and p ∈ L(f, 1, 2) and h = ⟨p_{1}, [p_{1}_{1}, \frac{p_{1}2+p2}{2} ], p⟩. Then h is a special sequence and h(1) = p_{1} and h(len h) = p and L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) and L̃(h) = L̃(f 1) ∪ L(p_{1}, p).

(13) Suppose that (i) p ≠ p_{1}, (ii) f is a special sequence, (iii) f(1) = p_{1}, (iv) f(len f) = p_{2},

(v) i ∈ dom f, (vi) i + 1 ∈ dom f, (vii) i > 1, (viii) p ∈ L(f, i, i + 1), (ix) p ≠ f(i),

(x) p ≠ f(i + 1), (xi) h = (f i)^{}⟨p⟩, (xii) q = f(i).

Then h is a special sequence and h(1) = p_{1} and h(len h) = p and L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) and L̃(h) = L̃(f i) ∪ L(q, p).

(14) Suppose p ≠ p_{1} and f is a special sequence and f(1) = p_{1} and f(len f) =

p_{2} and f(2) = p and p_{2} = p_{1}_{2} and h = ⟨p_{1}, [\frac{p_{1}1+p1}{2} , p_{1}_{2}], p⟩. Then

(i) h is a special sequence, (ii) h(1) = p_{1}, (iii) h(len h) = p, (iv) L̃(h) is a special polygonal arc joining p_{1} and p,

(v) L̃(h) ⊆ L̃(f), (vi) L̃(h) = L̃(f 1) ∪ L(p_{1}, p), (vii) L̃(h) = L̃(f 2) ∪ L(p, p).

(15) Suppose p ≠ p_{1} and f is a special sequence and f(1) = p_{1} and f(len f) =

p_{2} and f(2) = p and p_{1} = p_{1}_{1} and h = ⟨p_{1}, [p_{1}_{1}, \frac{p_{1}2+p2}{2} ], p⟩. Then

(i) h is a special sequence, (ii) h(1) = p_{1}, (iii) h(len h) = p, (iv) L̃(h) is a special polygonal arc joining p_{1} and p,

(v) L̃(h) ⊆ L̃(f), (vi) L̃(h) = L̃(f 1) ∪ L(p_{1}, p), (vii) L̃(h) = L̃(f 2) ∪ L(p, p).

(16) Suppose p ≠ p_{1} and f is a special sequence and f(1) = p_{1} and f(len f) = p_{2} and f(i) = p and i > 2 and i ∈ dom f and h = f i. Then h is a special sequence and h(1) = p_{1} and h(len h) = p and L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) and L̃(h) = L̃(f i) ∪ L(p, p).

(17) Suppose p ≠ p_{1} and f is a special sequence and f(1) = p_{1} and f(len f) = p_{2} and p ∈ L(f, n, n + 1) and q = f(n). Then there exists h such that h is a special sequence and h(1) = p_{1} and h(len h) = p and L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) and L̃(h) = L̃(f n) ∪ L(q, p).

(18) Suppose p ≠ p_{1} and f is a special sequence and f(1) = p_{1} and f(len f) = p_{2} and p ∈ L̃(f). Then there exists h such that h is a special sequence and h(1) = p_{1} and h(len h) = p and L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f).

(19) Suppose that (i) p_{1} = p_{2}_{1} and p_{2} ≠ p_{2}_{2} or p_{1} ≠ p_{2}_{1} and p_{2} = p_{2}_{2}, (ii) r > 0, (iii) p_{1} ∉ Ball(u, r), (iv) p_{2} ∈ Ball(u, r),

(v) p ∈ Ball(u, r), (vi) f is a special sequence, (vii) f(1) = p_{1}, (viii) f(len f) = p_{2}, (ix) L(p_{2}, p) ∩ L̃(f) = {p_{2}},

(x) h = f ^{}⟨p⟩.

Then h is a special sequence and L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) ∪ Ball(u, r).

(20) Suppose that (i) r > 0, (ii) p_{1} ∉ Ball(u, r), (iii) p_{2} ∈ Ball(u, r), (iv) p ∈ Ball(u, r),

(v) [p_{1}, p_{2}_{2}] ∈ Ball(u, r), (vi) f is a special sequence, (vii) f(1) = p_{1}, (viii) f(len f) = p_{2}, (ix) p_{1} ≠ p_{2}_{1},

(x) p_{2} ≠ p_{2}_{2}, (xi) h = f ^{}⟨[p_{1}, p_{2}_{2}], p⟩, (xii) (L(p_{2}, [p_{1}, p_{2}_{2}]) ∪ L([p_{1}, p_{2}_{2}], p)) ∩ L̃(f) = {p_{2}}.

Then L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) ∪ Ball(u, r).

(21) Suppose that (i) r > 0, (ii) p_{1} ∉ Ball(u, r), (iii) p_{2} ∈ Ball(u, r), (iv) p ∈ Ball(u, r),

(v) [p_{2}_{1}, p_{2}] ∈ Ball(u, r), (vi) f is a special sequence, (vii) f(1) = p_{1}, (viii) f(len f) = p_{2}, (ix) p_{1} ≠ p_{2}_{1},

(x) p_{2} ≠ p_{2}_{2}, (xi) h = f ^{}⟨[p_{2}_{1}, p_{2}], p⟩, (xii) (L(p_{2}, [p_{2}_{1}, p_{2}]) ∪ L([p_{2}_{1}, p_{2}], p)) ∩ L̃(f) = {p_{2}}.

Then L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) ∪ Ball(u, r).

connectedness conditions using polygonal arcs 105

(22) Suppose r > 0 and p_{1} ∉ Ball(u, r) and p_{2} ∈ Ball(u, r) and p ∈ Ball(u, r) and f is a special sequence and f(1) = p_{1} and f(len f) = p_{2} and p ∉ L̃(f). Then there exists h such that L̃(h) is a special polygonal arc joining p_{1} and p and L̃(h) ⊆ L̃(f) ∪ Ball(u, r).

(23) Given R, p, p_{1}, p_{2}, P, r, u. Then if p ≠ p_{1} and P is a special polygonal arc joining p_{1} and p_{2} and P ⊆ R and r > 0 and p ∈ Ball(u, r) and p_{2} ∈ Ball(u, r) and Ball(u, r) ⊆ R, then there exists P_{1} such that P_{1} is a special polygonal arc joining p_{1} and p and P_{1} ⊆ R.

(24) For every p such that R is a region and P = {q : q ≠ p∧q ∈ R∧¬ ⋁_{P_{1}}[P_{1} is a special polygonal arc joining p and q ∧ P_{1} ⊆ R]} holds P is open.

(25) If R is a region and p ∈ R and P = {q : q = p ∨ ⋁_{P_{1}}[P_{1} is a special polygonal arc joining p and q ∧ P_{1} ⊆ R]}, then P is open.

(26) If p ∈ R and P = {q : q = p ∨ ⋁_{P_{1}}[P_{1} is a special polygonal arc joining p and q ∧ P_{1} ⊆ R]}, then P ⊆ R.

(27) If R is a region and p ∈ R and P = {q : q = p ∨ ⋁_{P_{1}}[P_{1} is a special polygonal arc joining p and q ∧ P_{1} ⊆ R]}, then R ⊆ P.

(28) If R is a region and p ∈ R and P = {q : q = p ∨ ⋁_{P_{1}}[P_{1} is a special polygonal arc joining p and q ∧ P_{1} ⊆ R]}, then R = P.

(29) If R is a region and p ∈ R and q ∈ R and p ≠ q, then there exists P such that P is a special polygonal arc joining p and q and P ⊆ R.

References

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

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

[3] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

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

[7] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

[8] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Simple closed curves. Formalized Mathematics, 2(5):663–664, 1991.

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

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

[11] Beata Padlewska. Connected spaces. Formalized Mathematics, 1(1):239–244, 1990.

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

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

106 yatsuka nakamura and jarosław kotowicz

[14] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized

Mathematics, 1(3):445–449, 1990.

Received August 24, 1992

Introduction to Go-Board - Part I

Jarosław Kotowicz^{1} Warsaw University Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. In the article we introduce Go-board as some kinds of matrix which elements belong to topological space E_{T}^{2}. We define the functor of delaying column in Go-board and relation between Go-board and finite sequence of point from E_{T}^{2}. Basic facts about those notations are proved. The concept of the article is based on [16].

MML Identifier: GOBOARD1.

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

1. Real Numbers Preliminaries

For simplicity we follow the rules: p denotes a point of E_{T}^{2}, f, f_{1}, f_{2}, g denote finite sequences of elements of E_{T}^{2}, v denotes a finite sequence of elements of , r, s denote real numbers, n, m, i, j, k denote natural numbers, and x is arbitrary. One can prove the following three propositions:

(1) |r − s| = 1 if and only if r > s and r = s + 1 or r < s and s = r + 1.

(2) |i − j| + |n − m| = 1 if and only if |i − j| = 1 and n = m or |n − m| = 1 and i = j.

(3) n > 1 if and only if there exists m such that n = m + 1 and m > 0.

1This article was written during my visit at Shinshu University in 1992.

2. Finite Sequences Preliminaries

The scheme FinSeqDChoice concerns a non-empty set A, a natural number B, and a binary predicate P, and states that:

there exists a finite sequence f of elements of A such that len f = B and for every n such that n ∈ Seg B holds P[n, f(n)] provided the parameters have the following property:

• for every n such that n ∈ Seg B there exists an element d of A such that P[n, d].

One can prove the following propositions:

(4) If n = m + 1 and i ∈ Seg n, then len Sgm(Seg n \ {i}) = m.

(5) Suppose n = m + 1 and k ∈ Seg n and i ∈ Seg m. Then if 1 ≤ i and i < k, then (Sgm(Seg n \ {k}))(i) = i but if k ≤ i and i ≤ m, then (Sgm(Seg n \ {k}))(i) = i + 1.

(6) For every finite sequence f and for all n, m such that len f = m+1 and n ∈ Seg len f holds len(f_{n}) = m.

(7) For every finite sequence f and for all n, m, k such that len f = m + 1 and n ∈ Seg len f and k ∈ Seg m holds f_{n}(k) = f(k) or f_{n}(k) = f(k+1).

(8) For every finite sequence f and for all n, m, k such that len f = m + 1 and n ∈ Seg len f and 1 ≤ k and k < n holds f_{n}(k) = f(k).

(9) For every finite sequence f and for all n, m, k such that len f = m + 1 and n ∈ Seg len f and n ≤ k and k ≤ m holds f_{n}(k) = f(k + 1).

(10) If n ∈ dom f and m ∈ Seg n, then (f n)(m) = f(m) and m ∈ dom f. We now define four new constructions. A finite sequence of elements of is

increasing if: (Def.1) for all n, m such that n ∈ dom it and m ∈ dom it and n < m and for

all r, s such that r = it(n) and s = it(m) holds r < s. A finite sequence is constant if: (Def.2) for all n, m such that n ∈ dom it and m ∈ dom it holds it(n) = it(m). Let us observe that there exists a finite sequence of elements of which is increasing. Note also that there exists a finite sequence of elements of which is constant.

Let us consider f. The functor X-coordinate(f) yields a finite sequence of elements of and is defined by: (Def.3) len X-coordinate(f) = len f

and for every n such that n ∈ dom X-coordinate(f) and for every p such that p = f(n) holds (X-coordinate(f))(n) = p_{1}.

The functor Y-coordinate(f) yielding a finite sequence of elements of is defined as follows: (Def.4) len Y-coordinate(f) = len f

and for every n such that n ∈ dom Y-coordinate(f) and for every p such that p = f(n) holds (Y-coordinate(f))(n) = p_{2}.

One can prove the following propositions:

(11) Suppose that (i) v ≠ ε, (ii) rng v ⊆ Seg n, (iii) v(len v) = n, (iv) for every k such that 1 ≤ k and k ≤ len v − 1 and for all r, s such that

r = v(k) and s = v(k + 1) holds |r − s| = 1 or r = s,

(v) i ∈ Seg n, (vi) i + 1 ∈ Seg n, (vii) m ∈ dom v, (viii) v(m) = i, (ix) for every k such that k ∈ dom v and v(k) = i holds k ≤ m.

Then m + 1 ∈ dom v and v(m + 1) = i + 1.

(12) Suppose that (i) v ≠ ε, (ii) rng v ⊆ Seg n, (iii) v(1) = 1, (iv) v(len v) = n,

(v) for every k such that 1 ≤ k and k ≤ len v − 1 and for all r, s such that r = v(k) and s = v(k + 1) holds |r − s| = 1 or r = s. Then

(vi) for every i such that i ∈ Seg n there exists k such that k ∈ dom v and

v(k) = i, (vii) for all m, k, i, r such that m ∈ dom v and v(m) = i and for every j

such that j ∈ dom v and v(j) = i holds j ≤ m and m < k and k ∈ dom v and r = v(k) holds i < r.

(13) If i ∈ dom f and 2 ≤ len f, then f(i) ∈ L̃(f).

3. Matrix Preliminaries

Next we state two propositions:

(14) For every non-empty set D and for every matrix M over D and for all i, j such that j ∈ Seg len M and i ∈ Seg width M holds M_{,i} (j) = Line(M, j)(i).

(15) For every non-empty set D and for every matrix M over D and for

every k such that k ∈ Seg len M holds M(k) = Line(M, k). We now define several new constructions. Let T be a topological space. A

matrix over T is a matrix over the carrier of T.

A matrix over E_{T}^{2} is non-trivial if: (Def.5) 0 < len it and 0 < width it. A matrix over E_{T}^{2} is line X-constant if:

(Def.6) for every n such that n ∈ Seg len it holds X-coordinate(Line(it, n)) is

constant. A matrix over E_{T}^{2} is column Y-constant if: (Def.7) for every n such that n ∈ Seg width it holds Y-coordinate(it_{,n} ) is con-

stant. A matrix over E_{T}^{2} is line Y-increasing if: (Def.8) for every n such that n ∈ Seg len it holds Y-coordinate(Line(it, n)) is

increasing. A matrix over E_{T}^{2} is column X-increasing if: (Def.9) for every n such that n ∈ Seg width it holds X-coordinate(it_{,n} ) is in-

creasing. One can readily verify that there exists a matrix over E_{T}^{2} which is non-trivial, line X-constant, column Y-constant, line Y-increasing and column X-increasing.

We now state two propositions:

(16) For every column X-increasing line X-constant matrix M over E_{T}^{2} and for all x, n, m such that x ∈ rng Line(M, n) and x ∈ rng Line(M, m) and n ∈ Seg len M and m ∈ Seg len M holds n = m.

(17) For every line Y-increasing column Y-constant matrix M over E_{T}^{2} and for all x, n, m such that x ∈ rng(M_{,n} ) and x ∈ rng(M_{,m} ) and n ∈ Seg width M and m ∈ Seg width M holds n = m.

4. Basic Go-Board‘s Notation

A Go-board is a non-trivial line X-constant column Y-constant line Y-increasing column X-increasing matrix over E_{T}^{2}.

In the sequel G denotes a Go-board. The following four propositions are true:

(18) If x = G_{m,k} and x = G_{i,j} and ⟨⟨m, k⟩⟩ ∈ the indices of G and ⟨⟨i, j⟩⟩ ∈ the indices of G, then m = i and k = j.

(19) If m ∈ dom f and f(1) ∈ rng(G_{,1} ), then (f m)(1) ∈ rng(G_{,1} ).

(20) If m ∈ dom f and f(m) ∈ rng(G_{,width G} ), then (f m)(len(f m)) ∈ rng(G_{,width G} ).

(21) If rng f ∩ rng(G_{,i} ) = ∅ and f(n) = G_{m,k} and n ∈ dom f and m ∈

Seg len G, then i ≠ k. Let us consider G, i. Let us assume that i ∈ Seg width G and width G > 1.

The deleting of i-column in G yielding a Go-board is defined by: (Def.10) len(the deleting of i-column in G) = len G and for every k such that

k ∈ Seg len G holds (the deleting of i-column in G)(k) = Line(G, k)_{i}. One can prove the following propositions:

(22) If i ∈ Seg width G and width G > 1 and k ∈ Seg len G, then Line(the deleting of i-column in G, k) = Line(G, k)_{i}.

(23) If i ∈ Seg width G and width G = m + 1 and m > 0, then width(the deleting of i-column in G) = m.

(24) If i ∈ Seg width G and width G > 1, then width G = width(the deleting of i-column in G) + 1.

(25) If i ∈ Seg width G and width G > 1 and n ∈ Seg len G and m ∈ Seg width(the deleting of i-column in G), then (the deleting of i-column in G)_{n,m} = Line(G, n)_{i}(m).

(26) If i ∈ Seg width G and width G = m+1 and m > 0 and 1 ≤ k and k < i, then (the deleting of i-column in G)_{,k} = G_{,k} and k ∈ Seg width(the deleting of i-column in G) and k ∈ Seg width G.

(27) Suppose i ∈ Seg width G and width G = m + 1 and m > 0 and i ≤ k and k ≤ m. Then (the deleting of i-column in G)_{,k} = G_{,k+1} and k ∈ Seg width(the deleting of i-column in G) and k + 1 ∈ Seg width G.

(28) If i ∈ Seg width G and width G = m + 1 and m > 0 and n ∈ Seg len G and 1 ≤ k and k < i, then (the deleting of i-column in G)_{n,k} = G_{n,k} and k ∈ Seg width G.

(29) Suppose i ∈ Seg width G and width G = m + 1 and m > 0 and n ∈ Seg len G and i ≤ k and k ≤ m. Then (the deleting of i-column in G)_{n,k} = G_{n,k+1} and k + 1 ∈ Seg width G.

(30) If width G = m + 1 and m > 0 and k ∈ Seg m, then (the deleting of 1-column in G)_{,k} = G_{,k+1} and k ∈ Seg width(the deleting of 1-column in G) and k + 1 ∈ Seg width G.

(31) If width G = m + 1 and m > 0 and k ∈ Seg m and n ∈ Seg len G, then (the deleting of 1-column in G)_{n,k} = G_{n,k+1} and 1 ∈ Seg width G.

(32) If width G = m + 1 and m > 0 and k ∈ Seg m, then (the deleting of width G-column in G)_{,k} = G_{,k} and k ∈ Seg width(the deleting of width G-column in G).

(33) If width G = m + 1 and m > 0 and k ∈ Seg m and n ∈ Seg len G, then k ∈ Seg width G and (the deleting of width G-column in G)_{n,k} = G_{n,k} and width G ∈ Seg width G.

(34) Suppose rng f ∩ rng(G_{,i} ) = ∅ and f(n) ∈ rng Line(G, m) and n ∈ dom f and i ∈ Seg width G and m ∈ Seg len G and width G > 1. Then f(n) ∈ rng Line(the deleting of i-column in G, m).

Let us consider f, G. We say that f is a sequence which elements belong to G if and only if the conditions (Def.11) is satisfied.

(Def.11) (i) For every n such that n ∈ dom f there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the

indices of G and f(n) = G_{i,j}, (ii) for every n such that n ∈ dom f and n + 1 ∈ dom f and for all m, k,

i, j such that ⟨⟨m, k⟩⟩ ∈ the indices of G and ⟨⟨i, j⟩⟩ ∈ the indices of G and f(n) = G_{m,k} and f(n + 1) = G_{i,j} holds |m − i| + |k − j| = 1.

One can prove the following propositions:

(35) If f is a sequence which elements belong to G and m ∈ dom f, then 1 ≤ len(f m) and f m is a sequence which elements belong to G.

(36) Suppose that

(i) for every n such that n ∈ dom f_{1} there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of G and f_{1}(n) = G_{i,j},

(ii) for every n such that n ∈ dom f_{2} there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of G and f_{2}(n) = G_{i,j}. Then for every n such that n ∈ dom(f_{1} ^{}f_{2}) there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of G and (f_{1} ^{}f_{2})(n) = G_{i,j}.

(37) Suppose that

(i) for every n such that n ∈ dom f_{1} and n + 1 ∈ dom f_{1} and for all m, k, i, j such that ⟨⟨m, k⟩⟩ ∈ the indices of G and ⟨⟨i, j⟩⟩ ∈ the indices of G and f_{1}(n) = G_{m,k} and f_{1}(n + 1) = G_{i,j} holds |m − i| + |k − j| = 1,

(ii) for every n such that n ∈ dom f_{2} and n + 1 ∈ dom f_{2} and for all m, k, i, j such that ⟨⟨m, k⟩⟩ ∈ the indices of G and ⟨⟨i, j⟩⟩ ∈ the indices of G and f_{2}(n) = G_{m,k} and f_{2}(n + 1) = G_{i,j} holds |m − i| + |k − j| = 1,

(iii) for all m, k, i, j such that ⟨⟨m, k⟩⟩ ∈ the indices of G and ⟨⟨i, j⟩⟩ ∈ the indices of G and f_{1}(len f_{1}) = G_{m,k} and f_{2}(1) = G_{i,j} and len f_{1} ∈ dom f_{1} and 1 ∈ dom f_{2} holds |m − i| + |k − j| = 1. Given n. Suppose n ∈ dom(f_{1}^{}f_{2}) and n+1 ∈ dom(f_{1}^{}f_{2}). Given m, k, i, j. Then if ⟨⟨m, k⟩⟩ ∈ the indices of G and ⟨⟨i, j⟩⟩ ∈ the indices of G and (f_{1} ^{}f_{2})(n) = G_{m,k} and (f_{1} ^{}f_{2})(n + 1) = G_{i,j}, then |m − i| + |k − j| = 1.

(38) If f is a sequence which elements belong to G and i ∈ Seg width G and rng f ∩ rng(G_{,i} ) = ∅ and width G > 1, then f is a sequence which elements belong to the deleting of i-column in G.

(39) If f is a sequence which elements belong to G and i ∈ dom f, then there exists n such that n ∈ Seg len G and f(i) ∈ rng Line(G, n).

(40) Suppose f is a sequence which elements belong to G and i ∈ dom f and i + 1 ∈ dom f and n ∈ Seg len G and f(i) ∈ rng Line(G, n). Then f(i+1) ∈ rng Line(G, n) or for every k such that f(i+1) ∈ rng Line(G, k) and k ∈ Seg len G holds |n − k| = 1.

(41) Suppose that (i) 1 ≤ len f, (ii) f(len f) ∈ rng Line(G, len G), (iii) f is a sequence which elements belong to G, (iv) i ∈ Seg len G,

(v) i + 1 ∈ Seg len G, (vi) m ∈ dom f, (vii) f(m) ∈ rng Line(G, i), (viii) for every k such that k ∈ dom f and f(k) ∈ rng Line(G, i) holds k ≤ m.

Then m + 1 ∈ dom f and f(m + 1) ∈ rng Line(G, i + 1).

(42) Suppose 1 ≤ len f and f(1) ∈ rng Line(G, 1) and f(len f) ∈ rng Line(G, len G) and f is a sequence which elements belong to G. Then (i) for every i such that 1 ≤ i and i ≤ len G there exists k such that

k ∈ dom f and f(k) ∈ rng Line(G, i), (ii) for every i such that 1 ≤ i and i ≤ len G and 2 ≤ len f holds L̃(f) ∩

rng Line(G, i) ≠ ∅, (iii) for all i, j, k, m such that 1 ≤ i and i ≤ len G and 1 ≤ j and j ≤ len G

and k ∈ dom f and m ∈ dom f and f(k) ∈ rng Line(G, i) and for every n such that n ∈ dom f and f(n) ∈ rng Line(G, i) holds n ≤ k and k < m and f(m) ∈ rng Line(G, j) holds i < j.

(43) If f is a sequence which elements belong to G and i ∈ dom f, then there exists n such that n ∈ Seg width G and f(i) ∈ rng(G_{,n} ).

(44) Suppose f is a sequence which elements belong to G and i ∈ dom f and i + 1 ∈ dom f and n ∈ Seg width G and f(i) ∈ rng(G_{,n} ). Then f(i + 1) ∈ rng(G_{,n} ) or for every k such that f(i + 1) ∈ rng(G_{,k} ) and k ∈ Seg width G holds |n − k| = 1.

(45) Suppose that (i) 1 ≤ len f, (ii) f(len f) ∈ rng(G_{,width G} ), (iii) f is a sequence which elements belong to G, (iv) i ∈ Seg width G,

(v) i + 1 ∈ Seg width G, (vi) m ∈ dom f, (vii) f(m) ∈ rng(G_{,i} ), (viii) for every k such that k ∈ dom f and f(k) ∈ rng(G_{,i} ) holds k ≤ m.

Then m + 1 ∈ dom f and f(m + 1) ∈ rng(G_{,i+1} ).

(46) Suppose 1 ≤ len f and f(1) ∈ rng(G_{,1} ) and f(len f) ∈ rng(G_{,width G} )

and f is a sequence which elements belong to G. Then

(i) for every i such that 1 ≤ i and i ≤ width G there exists k such that k ∈ dom f and f(k) ∈ rng(G_{,i} ),

(ii) for every i such that 1 ≤ i and i ≤ width G and 2 ≤ len f holds

L̃(f) ∩ rng(G_{,i} ) ≠ ∅, (iii) for all i, j, k, m such that 1 ≤ i and i ≤ width G and 1 ≤ j and

j ≤ width G and k ∈ dom f and m ∈ dom f and f(k) ∈ rng(G_{,i} ) and for every n such that n ∈ dom f and f(n) ∈ rng(G_{,i} ) holds n ≤ k and k < m and f(m) ∈ rng(G_{,j} ) holds i < j.

(47) Suppose that (i) n ∈ dom f, (ii) f(n) ∈ rng(G_{,k} ), (iii) k ∈ Seg width G, (iv) f(1) ∈ rng(G_{,1} ),

(v) f is a sequence which elements belong to G, (vi) for every i such that i ∈ dom f and f(i) ∈ rng(G_{,k} ) holds n ≤ i.

Then for every i such that i ∈ dom f and i ≤ n and for every m such that m ∈ Seg width G and f(i) ∈ rng(G_{,m} ) holds m ≤ k.

114 jarosław kotowicz and yatsuka nakamura

(48) Suppose f is a sequence which elements belong to G and f(1) ∈ rng(G_{,1} ) and f(len f) ∈ rng(G_{,width G} ) and width G > 1 and 1 ≤ len f. Then there exists g such that g(1) ∈ rng((the deleting of width G-column in G)_{,1} ) and g(len g) ∈ rng((the deleting of width G-column in G)_{,width(the deleting of width G−column in G)} ) and 1 ≤ len g and g is a sequence which elements belong to the deleting of width G-column in G and rng g ⊆ rng f.

(49) Suppose f is a sequence which elements belong to G and rng f ∩ rng(G_{,1} ) ≠ ∅ and rng f ∩ rng(G_{,width G} ) ≠ ∅. Then there exists g such that rng g ⊆ rng f and g(1) ∈ rng(G_{,1} ) and g(len g) ∈ rng(G_{,width G} ) and 1 ≤ len g and g is a sequence which elements belong to G.

(50) Suppose k ∈ Seg len G and f is a sequence which elements belong to G and f(len f) ∈ rng Line(G, len G) and n ∈ dom f and f(n) ∈ rng Line(G, k). Then

(i) for every i such that k ≤ i and i ≤ len G there exists j such that j ∈ dom f and n ≤ j and f(j) ∈ rng Line(G, i),

(ii) for every i such that k < i and i ≤ len G there exists j such that j ∈ dom f and n < j and f(j) ∈ rng Line(G, i).

References

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116

Introduction to Go-Board - Part II

Jarosław Kotowicz^{1} Warsaw University Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. In article we define Go-board determined by finite sequence of points from topological space E_{T}^{2}. A few facts about this notation are proved.

MML Identifier: GOBOARD2.

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

1. Real Numbers Preliminaries

For simplicity we follow the rules: p, q denote points of E_{T}^{2}, f, f_{1}, f_{2}, g denote finite sequences of elements of E_{T}^{2}, R denotes a subset of , r, s denote real numbers, v, v_{1}, v_{2} denote finite sequences of elements of , n, m, i, j, k denote natural numbers, and G denotes a Go-board. We now state the proposition

(1) If R is finite and R ≠ ∅, then R is upper bounded and sup R ∈ R and R is lower bounded and inf R ∈ R.

2. Properties of Finite Sequences of Points from E_{T}^{2}

One can prove the following propositions:

(2) For every finite sequence f holds f is one-to-one if and only if for all n, m such that n ∈ dom f and m ∈ dom f and n ≠ m holds f(n) ≠ f(m).

(3) For every n holds 1 ≤ n and n ≤ len f − 1 if and only if n ∈ dom f and n + 1 ∈ dom f.

1This article was written during my visit at Shinshu University in 1992.

(4) For every n holds 1 ≤ n and n ≤ len f − 2 if and only if n ∈ dom f and n + 1 ∈ dom f and n + 2 ∈ dom f.

(5) The following conditions are equivalent:

(i) for all n, m such that n − m > 1 or m − n > 1 holds L(f, n, n + 1) ∩ L(f, m, m + 1) = ∅,

(ii) for all n, m such that n − m > 1 or m − n > 1 but n ∈ dom f and n + 1 ∈ dom f and m ∈ dom f and m + 1 ∈ dom f holds L(f, n, n + 1) ∩ L(f, m, m + 1) = ∅.

(6) Suppose that

(i) for every n such that 1 ≤ n and n ≤ len f − 2 holds L(f, n, n + 1) ∩ L(f, n + 1, n + 2) = {f(n + 1)},

(ii) for all n, m such that n − m > 1 or m − n > 1 holds L(f, n, n + 1) ∩

L(f, m, m + 1) = ∅, (iii) f is one-to-one, (iv) f(len f) ∈ L(f, i, i + 1),

(v) i ∈ dom f, (vi) i + 1 ∈ dom f.

Then i + 1 = len f.

(7) If k ≠ 0 and len f = k + 1, then L̃(f) = L̃(f k) ∪ L(f, k, k + 1).

(8) Suppose that

(i) 1 < k, (ii) len f = k + 1, (iii) for every n such that 1 ≤ n and n ≤ len f − 2 holds L(f, n, n + 1) ∩

L(f, n + 1, n + 2) = {f(n + 1)}, (iv) for all n, m such that n − m > 1 or m − n > 1 holds L(f, n, n + 1) ∩

L(f, m, m + 1) = ∅. Then L̃(f k) ∩ L(f, k, k + 1) = {f(k)}.

(9) If len f_{1} < n and n ≤ len(f_{1} ^{}f_{2}) − 1 and m = n − len f_{1}, then L(f_{1} ^{} f_{2}, n, n + 1) = L(f_{2}, m, m + 1).

(10) L̃(f) ⊆ L̃(f ^{}g).

(11) Suppose for all n, m such that n−m > 1 or m−n > 1 holds L(f, n, n+ 1) ∩ L(f, m, m + 1) = ∅. Then for all n, m such that n − m > 1 or m − n > 1 holds L(f i, n, n + 1) ∩ L(f i, m, m + 1) = ∅.

(12) Suppose that

(i) for all n, p, q such that 1 ≤ n and n ≤ len f_{1} − 1 and f_{1}(n) = p and f_{1}(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2},

(ii) for all n, p, q such that 1 ≤ n and n ≤ len f_{2} − 1 and f_{2}(n) = p and

f_{2}(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2}, (iii) for all p, q such that f_{1}(len f_{1}) = p and f_{2}(1) = q holds p_{1} = q_{1} or

p_{2} = q_{2}. Then for all n, p, q such that 1 ≤ n and n ≤ len(f_{1} ^{}f_{2}) − 1 and (f_{1} ^{}f_{2})(n) = p and (f_{1} ^{}f_{2})(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2}.

(13) If f ≠ ε, then X-coordinate(f) ≠ ε.

(14) If f ≠ ε, then Y-coordinate(f) ≠ ε.

(15) Suppose for all n, p, q such that n ∈ dom f and n + 1 ∈ dom f and f(n) = p and f(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2}. Given n. Suppose n ∈ dom f and n + 1 ∈ dom f. Then for all i, j, m, k such that ⟨⟨i, j⟩⟩ ∈ the indices of G and ⟨⟨m, k⟩⟩ ∈ the indices of G and f(n) = G_{i,j} and f(n + 1) = G_{m,k} holds i = m or k = j.

(16) Suppose that

(i) for every n such that n ∈ dom f there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of G and f(n) = G_{i,j},

(ii) for all n, p, q such that n ∈ dom f and n + 1 ∈ dom f and f(n) = p

and f(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2}, (iii) for every n such that n ∈ dom f and n + 1 ∈ dom f holds f(n) ≠

f(n + 1). Then there exists g such that g is a sequence which elements belong to G and L̃(f) = L̃(g) and g(1) = f(1) and g(len g) = f(len f) and len f ≤ len g.

(17) If v is increasing, then for all n, m such that n ∈ dom v and m ∈ dom v and n ≤ m and for all r, s such that r = v(n) and s = v(m) holds r ≤ s.

(18) If v is increasing, then for all n, m such that n ∈ dom v and m ∈ dom v and n ≠ m holds v(n) ≠ v(m).

(19) If v is increasing and v_{1} = v Seg n, then v_{1} is increasing.

(20) For every v there exists v_{1} such that rng v_{1} = rng v and len v_{1} = card rng v and v_{1} is increasing.

(21) For all v_{1}, v_{2} such that len v_{1} = len v_{2} and rng v_{1} = rng v_{2} and v_{1} is increasing and v_{2} is increasing holds v_{1} = v_{2}.

3. Go-Board Determined by Finite Sequence

We now define three new functors. Let v_{1}, v_{2} be increasing finite sequences of elements of . Let us assume that v_{1} ≠ ε and v_{2} ≠ ε. The Go-board of v_{1}, v_{2} yields a Go-board and is defined by: (Def.1) len the Go-board of v_{1}, v_{2} = len v_{1} and width the Go-board of v_{1}, v_{2} =

len v_{2} and for all n, m such that ⟨⟨n, m⟩⟩ ∈ the indices of the Go-board of v_{1}, v_{2} and for all r, s such that v_{1}(n) = r and v_{2}(m) = s holds (the Go-board of v_{1}, v_{2})_{n,m} = [r, s].

Let us consider v. The functor Inc(v) yielding an increasing finite sequence of elements of is defined by: (Def.2) rng Inc(v) = rng v and len Inc(v) = card rng v. Let us consider f. Let us assume that f ≠ ε. The Go-board of f yielding a Go-board is defined by: (Def.3) the Go-board of f = the Go-board of Inc(X-coordinate(f)),

Inc(Y-coordinate(f)).

One can prove the following propositions:

(22) If v ≠ ε, then Inc(v) ≠ ε.

(23) If f ≠ ε, then len the Go-board of f = card rng X-coordinate(f) and width the Go-board of f = card rng Y-coordinate(f).

(24) If f ≠ ε, then for every n such that n ∈ dom f there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of the Go-board of f and f(n) = (the Go-board of f)_{i,j}.

(25) If f ≠ ε and n ∈ dom f and r = (X-coordinate(f))(n) and for every m such that m ∈ dom f and for every s such that s = (X-coordinate(f))(m) holds r ≤ s, then f(n) ∈ rng Line(the Go-board of f, 1).

(26) If f ≠ ε and n ∈ dom f and r = (X-coordinate(f))(n) and for every m such that m ∈ dom f and for every s such that s = (X-coordinate(f))(m) holds s ≤ r, then f(n) ∈ rng Line(the Go-board of f, len the Go-board of f).

(27) If f ≠ ε and n ∈ dom f and r = (Y-coordinate(f))(n) and for every m such that m ∈ dom f and for every s such that s = (Y-coordinate(f))(m) holds r ≤ s, then f(n) ∈ rng((the Go-board of f)_{,1} ).

(28) If f ≠ ε and n ∈ dom f and r = (Y-coordinate(f))(n) and for every m such that m ∈ dom f and for every s such that s = (Y-coordinate(f))(m) holds s ≤ r, then f(n) ∈ rng((the Go-board of f)_{,width the Go−board of f} ).

References

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

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

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

[4] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

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

[9] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

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

[11] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

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

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

[14] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part I. Formalized Mathematics, 3(1):107–115, 1992.

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

introduction to go-board - part ii 121

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

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

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

122

Properties of Go-Board - Part III

Jarosław Kotowicz^{1} Warsaw University Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. Two useful facts about Go-board are proved.

MML Identifier: GOBOARD3.

The terminology and notation used in this paper have been introduced in the following articles: [16], [8], [1], [5], [2], [14], [15], [17], [4], [10], [9], [3], [6], [7], [13], [11], and [12]. For simplicity we follow the rules: p, q are points of E_{T}^{2}, f, g are finite sequences of elements of E_{T}^{2}, n, m, i, j are natural numbers, and G is a Go-board. One can prove the following two propositions:

(1) Suppose that

(i) for every n such that n ∈ dom f there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of G and f(n) = G_{i,j},

(ii) f is one-to-one, (iii) for every n such that 1 ≤ n and n ≤ len f − 2 holds L(f, n, n + 1) ∩

L(f, n + 1, n + 2) = {f(n + 1)}, (iv) for all n, m such that n − m > 1 or m − n > 1 holds L(f, n, n + 1) ∩

L(f, m, m + 1) = ∅,

(v) for all n, p, q such that 1 ≤ n and n ≤ len f − 1 and f(n) = p and f(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2}. Then there exists g such that g is a sequence which elements belong to G and g is one-to-one and for every n such that 1 ≤ n and n ≤ len g − 2 holds L(g, n, n + 1) ∩ L(g, n + 1, n + 2) = {g(n + 1)} and for all n, m such that n − m > 1 or m − n > 1 holds L(g, n, n + 1) ∩ L(g, m, m + 1) = ∅ and for all n, p, q such that 1 ≤ n and n ≤ len g − 1 and g(n) = p and g(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2} and L̃(f) = L̃(g) and f(1) = g(1) and f(len f) = g(len g) and len f ≤ len g.

1This article was written during my visit at Shinshu University in 1992.

(2) Suppose for every n such that n ∈ dom f there exist i, j such that ⟨⟨i, j⟩⟩ ∈ the indices of G and f(n) = G_{i,j} and f is a special sequence. Then there exists g such that g is a sequence which elements belong to G and g is a special sequence and L̃(f) = L̃(g) and f(1) = g(1) and f(len f) = g(len g) and len f ≤ len g.

References

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

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

[3] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481– 485, 1991.

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

[7] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

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

[9] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

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

[11] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part I. Formalized Mathematics, 3(1):107–115, 1992.

[12] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part II. Formalized Mathematics, 3(1):117–121, 1992.

[13] Yatsuka Nakamura and Jarosław Kotowicz. Connectedness conditions using polygonal arcs. Formalized Mathematics, 3(1):101–106, 1992.

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

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

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

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

Go-Board Theorem

Jarosław Kotowicz^{1} Warsaw University Białystok Yatsuka Nakamura Shinshu University Nagano

Summary. We prove the Go-board theorem which is a special case of Hex Theorem. The article is based on [15].

MML Identifier: GOBOARD4.

The terminology and notation used in this paper are introduced in the following articles: [16], [7], [1], [4], [2], [13], [14], [17], [3], [8], [5], [6], [9], [12], [10], and [11]. For simplicity we adopt the following convention: p, p_{1}, p_{2}, q, q_{1}, q_{2} will be points of E_{T}^{2}, P_{1}, P_{2} will be subsets of E_{T}^{2}, f_{1}, f_{2} will be finite sequences of elements of E_{T}^{2}, r, s will be real numbers, n will be a natural number, and G will be a Go-board. We now state several propositions:

(1) Given G, f_{1}, f_{2}. Suppose that

(i) 1 ≤ len f_{1}, (ii) 1 ≤ len f_{2}, (iii) f_{1} is a sequence which elements belong to G, (iv) f_{2} is a sequence which elements belong to G,

(v) f_{1}(1) ∈ rng Line(G, 1), (vi) f_{1}(len f_{1}) ∈ rng Line(G, len G), (vii) f_{2}(1) ∈ rng(G_{,1} ), (viii) f_{2}(len f_{2}) ∈ rng(G_{,width G} ).

Then rng f_{1} ∩ rng f_{2} ≠ ∅.

(2) Given G, f_{1}, f_{2}. Suppose that

(i) 2 ≤ len f_{1}, (ii) 2 ≤ len f_{2}, (iii) f_{1} is a sequence which elements belong to G, (iv) f_{2} is a sequence which elements belong to G,

(v) f_{1}(1) ∈ rng Line(G, 1), (vi) f_{1}(len f_{1}) ∈ rng Line(G, len G), (vii) f_{2}(1) ∈ rng(G_{,1} ), (viii) f_{2}(len f_{2}) ∈ rng(G_{,width G} ).

1This article was written during my visit at Shinshu University in 1992.

Then L̃(f_{1}) ∩ L̃(f_{2}) ≠ ∅.

(3) Given G, f_{1}, f_{2}. Suppose that

(i) f_{1} is a special sequence, (ii) f_{2} is a special sequence, (iii) f_{1} is a sequence which elements belong to G, (iv) f_{2} is a sequence which elements belong to G,

(v) f_{1}(1) ∈ rng Line(G, 1), (vi) f_{1}(len f_{1}) ∈ rng Line(G, len G), (vii) f_{2}(1) ∈ rng(G_{,1} ), (viii) f_{2}(len f_{2}) ∈ rng(G_{,width G} ).

Then L̃(f_{1}) ∩ L̃(f_{2}) ≠ ∅.

(4) Given f_{1}, f_{2}. Suppose that

(i) 2 ≤ len f_{1}, (ii) 2 ≤ len f_{2}, (iii) for all n, p, q such that n ∈ dom f_{1} and n + 1 ∈ dom f_{1} and f_{1}(n) = p

and f_{1}(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2}, (iv) for all n, p, q such that n ∈ dom f_{2} and n + 1 ∈ dom f_{2} and f_{2}(n) = p

and f_{2}(n + 1) = q holds p_{1} = q_{1} or p_{2} = q_{2},

(v) for every n such that n ∈ dom f_{1} and n + 1 ∈ dom f_{1} holds f_{1}(n) ≠ f_{1}(n + 1),

(vi) for every n such that n ∈ dom f_{2} and n + 1 ∈ dom f_{2} holds f_{2}(n) ≠

f_{2}(n + 1), (vii) for every r such that r = (X-coordinate(f_{1}))(1) and for all n, s such

that n ∈ dom f_{1} and s = (X-coordinate(f_{1}))(n) holds r ≤ s, (viii) for every r such that r = (X-coordinate(f_{1}))(1) and for all n, s such

that n ∈ dom f_{2} and s = (X-coordinate(f_{2}))(n) holds r ≤ s, (ix) for every r such that r = (X-coordinate(f_{1}))(len f_{1}) and for all n, s

such that n ∈ dom f_{1} and s = (X-coordinate(f_{1}))(n) holds s ≤ r,

(x) for every r such that r = (X-coordinate(f_{1}))(len f_{1}) and for all n, s such that n ∈ dom f_{2} and s = (X-coordinate(f_{2}))(n) holds s ≤ r,

(xi) for every r such that r = (Y-coordinate(f_{2}))(1) and for all n, s such

that n ∈ dom f_{1} and s = (Y-coordinate(f_{1}))(n) holds r ≤ s, (xii) for every r such that r = (Y-coordinate(f_{2}))(1) and for all n, s such

that n ∈ dom f_{2} and s = (Y-coordinate(f_{2}))(n) holds r ≤ s, (xiii) for every r such that r = (Y-coordinate(f_{2}))(len f_{2}) and for all n, s

such that n ∈ dom f_{1} and s = (Y-coordinate(f_{1}))(n) holds s ≤ r, (xiv) for every r such that r = (Y-coordinate(f_{2}))(len f_{2}) and for all n, s

such that n ∈ dom f_{2} and s = (Y-coordinate(f_{2}))(n) holds s ≤ r. Then L̃(f_{1}) ∩ L̃(f_{2}) ≠ ∅.

(5) Given f_{1}, f_{2}. Suppose that

(i) f_{1} is a special sequence, (ii) f_{2} is a special sequence, (iii) for every r such that r = (X-coordinate(f_{1}))(1) and for all n, s such

that n ∈ dom f_{1} and s = (X-coordinate(f_{1}))(n) holds r ≤ s, (iv) for every r such that r = (X-coordinate(f_{1}))(1) and for all n, s such

that n ∈ dom f_{2} and s = (X-coordinate(f_{2}))(n) holds r ≤ s,

(v) for every r such that r = (X-coordinate(f_{1}))(len f_{1}) and for all n, s such that n ∈ dom f_{1} and s = (X-coordinate(f_{1}))(n) holds s ≤ r,

(vi) for every r such that r = (X-coordinate(f_{1}))(len f_{1}) and for all n, s

such that n ∈ dom f_{2} and s = (X-coordinate(f_{2}))(n) holds s ≤ r, (vii) for every r such that r = (Y-coordinate(f_{2}))(1) and for all n, s such

that n ∈ dom f_{1} and s = (Y-coordinate(f_{1}))(n) holds r ≤ s, (viii) for every r such that r = (Y-coordinate(f_{2}))(1) and for all n, s such

that n ∈ dom f_{2} and s = (Y-coordinate(f_{2}))(n) holds r ≤ s, (ix) for every r such that r = (Y-coordinate(f_{2}))(len f_{2}) and for all n, s

such that n ∈ dom f_{1} and s = (Y-coordinate(f_{1}))(n) holds s ≤ r,

(x) for every r such that r = (Y-coordinate(f_{2}))(len f_{2}) and for all n, s such that n ∈ dom f_{2} and s = (Y-coordinate(f_{2}))(n) holds s ≤ r. Then L̃(f_{1}) ∩ L̃(f_{2}) ≠ ∅.

(6) Given P_{1}, P_{2}. Suppose P_{1} is a special polygonal arc and P_{2} is a special polygonal arc. Given G, f_{1}, f_{2}. Suppose that

(i) f_{1} is a special sequence, (ii) P_{1} = L̃(f_{1}), (iii) f_{2} is a special sequence, (iv) P_{2} = L̃(f_{2}),

(v) f_{1} is a sequence which elements belong to G, (vi) f_{2} is a sequence which elements belong to G, (vii) f_{1}(1) ∈ rng Line(G, 1), (viii) f_{1}(len f_{1}) ∈ rng Line(G, len G), (ix) f_{2}(1) ∈ rng(G_{,1} ),

(x) f_{2}(len f_{2}) ∈ rng(G_{,width G} ).

Then P_{1} ∩ P_{2} ≠ ∅.

(7) Given P_{1}, P_{2}. Suppose P_{1} is a special polygonal arc and P_{2} is a special polygonal arc. Given f_{1}, f_{2}. Suppose that

(i) f_{1} is a special sequence, (ii) P_{1} = L̃(f_{1}), (iii) f_{2} is a special sequence, (iv) P_{2} = L̃(f_{2}),

(v) for every r such that r = (X-coordinate(f_{1}))(1) and for all n, s such that n ∈ dom f_{1} and s = (X-coordinate(f_{1}))(n) holds r ≤ s,

(vi) for every r such that r = (X-coordinate(f_{1}))(1) and for all n, s such

that n ∈ dom f_{2} and s = (X-coordinate(f_{2}))(n) holds r ≤ s, (vii) for every r such that r = (X-coordinate(f_{1}))(len f_{1}) and for all n, s

such that n ∈ dom f_{1} and s = (X-coordinate(f_{1}))(n) holds s ≤ r, (viii) for every r such that r = (X-coordinate(f_{1}))(len f_{1}) and for all n, s

such that n ∈ dom f_{2} and s = (X-coordinate(f_{2}))(n) holds s ≤ r, (ix) for every r such that r = (Y-coordinate(f_{2}))(1) and for all n, s such

that n ∈ dom f_{1} and s = (Y-coordinate(f_{1}))(n) holds r ≤ s,

(x) for every r such that r = (Y-coordinate(f_{2}))(1) and for all n, s such that n ∈ dom f_{2} and s = (Y-coordinate(f_{2}))(n) holds r ≤ s,

(xi) for every r such that r = (Y-coordinate(f_{2}))(len f_{2}) and for all n, s such that n ∈ dom f_{1} and s = (Y-coordinate(f_{1}))(n) holds s ≤ r,

(xii) for every r such that r = (Y-coordinate(f_{2}))(len f_{2}) and for all n, s such that n ∈ dom f_{2} and s = (Y-coordinate(f_{2}))(n) holds s ≤ r. Then P_{1} ∩ P_{2} ≠ ∅.

(8) Given P_{1}, P_{2}, p_{1}, p_{2}, q_{1}, q_{2}. Suppose that

(i) P_{1} is a special polygonal arc joining p_{1} and q_{1}, (ii) P_{2} is a special polygonal arc joining p_{2} and q_{2}, (iii) for every p such that p ∈ P_{1} ∪ P_{2} holds p_{1}_{1} ≤ p_{1} and p_{1} ≤ q_{1}_{1}, (iv) for every p such that p ∈ P_{1} ∪ P_{2} holds p_{2}_{2} ≤ p_{2} and p_{2} ≤ q_{2}_{2}.

Then P_{1} ∩ P_{2} ≠ ∅.

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

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

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

[6] Agata Darmochwał and Yatsuka Nakamura. The topological space E_{T}^{2}. Arcs, line segments and special polygonal arcs. Formalized Mathematics, 2(5):617–621, 1991.

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

[8] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475–480, 1991.

[9] Katarzyna Jankowska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5):711–717, 1991.

[10] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part I. Formalized Mathematics, 3(1):107–115, 1992.

[11] Jarosław Kotowicz and Yatsuka Nakamura. Introduction to Go-board - part II. Formalized Mathematics, 3(1):117–121, 1992.

[12] Yatsuka Nakamura and Jarosław Kotowicz. Connectedness conditions using polygonal arcs. Formalized Mathematics, 3(1):101–106, 1992.

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

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

[15] Yukio Takeuchi and Yatsuka Nakamura. On the Jordan curve theorem. Technical Report 19804, Dept. of Information Eng., Shinshu University, 500 Wakasato, Nagano city, Japan, April 1980.

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

go-board theorem 129

[17] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized

Mathematics, 1(3):445–449, 1990.

Received August 24, 1992 130

Some Properties of Binary Relations

Waldemar Korczyński Pedagogical University Kielce

Summary. The article contains some theorems on binary relations, which are used in papers [2], [3], [1], and other.

MML Identifier: SYSREL.

The articles [5], [6], [7], and [4] provide the terminology and notation for this paper. We adopt the following rules: x, y are arbitrary, X, Y, Z, W are sets, and R, S, T are binary relations. We now state a number of propositions:

(1) If X ∩ Y = ∅ and x ∈ X ∪ Y, then x ∈ X and x ∉ Y or x ∈ Y and x ∉ X.

(2) (X ∪ Y) ∪ Z = X ∪ Z ∪ (Y ∪ Z).

(3) X ∪ (X ∪ Y) = X ∪ Y.

(4) If X ⊆ Y ∩ Z, then X ⊆ Y and X ⊆ Z.

(5) = ∅.

(6) \ R = .

(7) R ⊆ S if and only if R \ S = .

(8) R ∩ S = if and only if R \ S = R.

(9) R \ R = .

(10) If R ⊆ , then R = .

(11) ∪ R = R and R ∪ = R and ∩ R = and R ∩ = .

Let us consider X, Y. Then [: X, Y :] is a binary relation. Next we state several propositions:

(12) If X ≠ ∅ and Y ≠ ∅, then dom[: X, Y :] = X and rng[: X, Y :] = Y.

(13) dom(R ∩ [: X, Y :]) ⊆ X and rng(R ∩ [: X, Y :]) ⊆ Y.

(14) If X ∩ Y = ∅, then dom(R ∩ [: X, Y :]) ∩ rng(R ∩ [: X, Y :]) = ∅ and dom(R^{}∩ [: X, Y :]) ∩ rng(R^{}∩ [: X, Y :]) = ∅.

(15) If R ⊆ [: X, Y :], then dom R ⊆ X and rng R ⊆ Y.

(16) If R ⊆ [: X, Y :], then R^{}⊆ [: Y, X:].

(17) If X ∩ Y = ∅, then [: X, Y :] ∩ [: Y, X:] = ∅.

(18) [: X, Y :]^{}= [: Y, X:].

Next we state a number of propositions:

(19) (R ∪ S) · T = R · T ∪ S · T and R · (S ∪ T) = R · S ∪ R · T.

(20) If R ⊆ [: X, Y :] and ⟨⟨x, y⟩⟩ ∈ R, then x ∈ X and y ∈ Y.

(21) (i) If X ∩ Y = ∅ and R ⊆ [: X, Y :] ∪ [: Y, X:] and ⟨⟨x, y⟩⟩ ∈ R and x ∈ X,

then x ∉ Y and y ∉ X and y ∈ Y, (ii) if X ∩ Y = ∅ and R ⊆ [: X, Y :] ∪ [: Y, X:] and ⟨⟨x, y⟩⟩ ∈ R and y ∈ Y,

then y ∉ X and x ∉ Y and x ∈ X, (iii) if X ∩ Y = ∅ and R ⊆ [: X, Y :] ∪ [: Y, X:] and ⟨⟨x, y⟩⟩ ∈ R and x ∈ Y,

then x ∉ X and y ∉ Y and y ∈ X, (iv) if X ∩ Y = ∅ and R ⊆ [: X, Y :] ∪ [: Y, X:] and ⟨⟨x, y⟩⟩ ∈ R and y ∈ X,

then x ∉ X and y ∉ Y and x ∈ Y.

(22) If rng R ∩ dom S = ∅ or dom S ∩ rng R = ∅, then R · S = .

(23) If R ⊆ [: X, Y :] and Z ⊆ X, then R Z = R ∩ [: Z, Y :] but if R ⊆ [: X, Y :] and Z ⊆ Y, then Z R = R ∩ [: X, Z:].

(24) If R ⊆ [: X, Y :] and X = Z ∪ W, then R = R Z ∪ R W.

(25) If X ∩ Y = ∅ and R ⊆ [: X, Y :] ∪ [: Y, X:], then R X ⊆ [: X, Y :].

(26) If R ⊆ S, then R^{}⊆ S^{}.

(27) △_{X} ⊆ [: X, X:].

(28) △_{X} · △_{X} = △_{X}.

(29) △_{{x}} = {⟨⟨x, x⟩⟩}.

(30) ⟨⟨x, y⟩⟩ ∈ △_{X} if and only if ⟨⟨y, x⟩⟩ ∈ △_{X}.

(31) △_{X∪Y} = △_{X} ∪ △_{Y} and △_{X∩Y} = △_{X} ∩ △_{Y} and △_{X\Y} = △_{X} \ △_{Y}.

(32) If X ⊆ Y, then △_{X} ⊆ △_{Y}.

(33) △_{X\Y} \ △_{X} = .

(34) If R ⊆ △_{dom R}, then R = △_{dom R}.

(35) If △_{X} ⊆ R ∪ R^{}, then △_{X} ⊆ R and △_{X} ⊆ R^{}.

(36) If △_{X} ⊆ R, then △_{X} ⊆ R^{}.

(37) If R ⊆ [: X, X:], then R \ △_{dom R} = R \ △_{X} and R \ △_{rng R} = R \ △_{X}.

(38) If △_{X} · (R \ △_{X}) = , then dom(R \ △_{X}) = dom R \ dom(△_{X}) but if (R \ △_{X}) · △_{X} = , then rng(R \ △_{X}) = rng R \ rng(△_{X}).

(39) If R ⊆ R · R and R · (R \ △_{rng R}) = , then △_{rng R} ⊆ R but if R ⊆ R · R and (R \ △_{dom R}) · R = , then △_{dom R} ⊆ R.

(40) (i) If R ⊆ R · R and R · (R \ △_{rng R}) = , then R ∩ △_{rng R} = △_{rng R}, (ii) if R ⊆ R · R and (R \ △_{dom R}) · R = , then R ∩ △_{dom R} = △_{dom R}.

(41) If R · (R \ △_{X}) = and rng R ⊆ X, then R · (R \ △_{rng R}) = but if

(R \ △_{X}) · R = and dom R ⊆ X, then (R \ △_{dom R}) · R = . Let us consider R. The functor CL(R) yielding a binary relation is defined

as follows: (Def.1) CL(R) = R ∩ △_{dom R}.

One can prove the following propositions:

(42) CL(R) ⊆ R and CL(R) ⊆ △_{dom R}.

(43) If ⟨⟨x, y⟩⟩ ∈ CL(R), then x ∈ dom CL(R) and x = y.

(44) dom CL(R) = rng CL(R).

(45) (i) x ∈ dom CL(R) if and only if x ∈ dom R and ⟨⟨x, x⟩⟩ ∈ R, (ii) x ∈ rng CL(R) if and only if x ∈ dom R and ⟨⟨x, x⟩⟩ ∈ R, (iii) x ∈ rng CL(R) if and only if x ∈ rng R and ⟨⟨x, x⟩⟩ ∈ R, (iv) x ∈ dom CL(R) if and only if x ∈ rng R and ⟨⟨x, x⟩⟩ ∈ R.

(46) CL(R) = △_{dom CL(R)}.

(47) (i) If R · R = R and R · (R \ CL(R)) = and ⟨⟨x, y⟩⟩ ∈ R and x ≠ y,

then x ∈ dom R \ dom CL(R) and y ∈ dom CL(R), (ii) if R · R = R and (R \ CL(R)) · R = and ⟨⟨x, y⟩⟩ ∈ R and x ≠ y, then

y ∈ rng R \ dom CL(R) and x ∈ dom CL(R).

(48) (i) If R · R = R and R · (R \ △_{dom R}) = and ⟨⟨x, y⟩⟩ ∈ R and x ≠ y,

then x ∈ dom R \ dom CL(R) and y ∈ dom CL(R), (ii) if R · R = R and (R \ △_{dom R}) · R = and ⟨⟨x, y⟩⟩ ∈ R and x ≠ y, then

y ∈ rng R \ dom CL(R) and x ∈ dom CL(R).

(49) (i) If R · R = R and R · (R \ △_{dom R}) = , then dom CL(R) = rng R

and rng CL(R) = rng R, (ii) if R · R = R and (R \ △_{dom R}) · R = , then dom CL(R) = dom R and

rng CL(R) = dom R.

(50) dom CL(R) ⊆ dom R and rng CL(R) ⊆ rng R and rng CL(R) ⊆ dom R and dom CL(R) ⊆ rng R.

(51) △_{dom CL(R)} ⊆ △_{dom R} and △_{rng CL(R)} ⊆ △_{dom R}.

(52) △_{dom CL(R)} ⊆ R and △_{rng CL(R)} ⊆ R.

(53) If △_{X} ⊆ R and △_{X} · (R \ △_{X}) = , then R X = △_{X} but if △_{X} ⊆ R and (R \ △_{X}) · △_{X} = , then X R = △_{X}.

(54) (i) If △_{dom CL(R)}·(R\△_{dom CL(R)}) = , then R dom CL(R) = △_{dom CL(R)}

and R rng CL(R) = △_{dom CL(R)}, (ii) if (R \ △_{rng CL(R)}) · △_{rng CL(R)} = , then dom CL(R) R = △_{dom CL(R)}

and rng CL(R) R = △_{rng CL(R)}.

(55) If R · (R \ △_{dom R}) = , then △_{dom CL(R)} · (R \ △_{dom CL(R)}) = but if (R \ △_{dom R}) · R = , then (R \ △_{dom CL(R)}) · △_{dom CL(R)} = .

(56) (i) If S · R = S and R · (R \ △_{dom R}) = , then S · (R \ △_{dom R}) = , (ii) if R · S = S and (R \ △_{dom R}) · R = , then (R \ △_{dom R}) · S = .

(57) If S · R = S and R · (R \ △_{dom R}) = , then CL(S) ⊆ CL(R) but if R · S = S and (R \ △_{dom R}) · R = , then CL(S) ⊆ CL(R).

(58) (i) If S · R = S and R · (R \ △_{dom R}) = and R · S = R and S · (S \

△_{dom S}) = , then CL(S) = CL(R), (ii) if R·S = S and (R\△_{dom R})·R = and S·R = R and (S\△_{dom S})·S =

, then CL(S) = CL(R).

134 waldemar korczyński

References

[1] Waldemar Korczyński. Definitions of Petri net - part III - S SIEC. Main Mizar Library, 1992.

[2] Waldemar Korczyński. Definitons of Petri net - part I - FF SIEC. Main Mizar Library, 1992.

[3] Waldemar Korczyński. Definitons of Petri net - part II - E SIEC. Main Mizar Library, 1992.

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

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

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

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

Received January 17, 1992

some properties of binary relations 135

alalalalalal alalalalalalal

Index of MML Identifiers

CARD 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 DIRORT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 GOBOARD1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 GOBOARD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 GOBOARD3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 GOBOARD4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 GR CY 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ISOCAT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 LMOD 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47 MEASURE4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 MIDSP 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 MOD 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 PCOMPS 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 SEQFUNC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 SYSREL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 TDLAT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 TDLAT 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 TMAP 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 TOPREAL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 TOPREAL4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 TREAL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Contents Form. Math. 3 (1)

Continuity of Mappings over the Union of Subspaces

By Zbigniew Karno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Functional Sequence from a Domain to a Domain

By Beata Perkowska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Reper Algebras

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Isomorphisms of Cyclic Groups. Some Properties of Cyclic Groups

By Dariusz Surowik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Some Isomorphisms Between Functor Categories

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

The Lattice of Domains of a Topological Space

By Toshihiko Watanabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Submodules

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Oriented Metric-Affine Plane - Part II

By Jarosław Zajkowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Opposite Rings, Modules and their Morphisms

By Michał Muzalewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Properties of Caratheodor’s Measure

By Józef Białas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Completeness of the Lattices of Domains of a Topological Space

By Zbigniew Karno and Toshihiko Watanabe . . . . . . . . . . . 71

On Paracompactness of Metrizable Spaces

By Leszek Borys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Continued on inside back cover

The Brouwer Fixed Point Theorem for Intervals

By Toshihiko Watanabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

On Powers of Cardinals

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Basic Properties of Connecting Points with Line Segments in E_{T}^{2}

By Yatsuka Nakamura and Jarosław Kotowicz . . . . . . . . 95

Connectedness Conditions Using Polygonal Arcs

By Yatsuka Nakamura and Jarosław Kotowicz . . . . . . . . 101

Introduction to Go-Board - Part I

By Jarosław Kotowicz and Yatsuka Nakamura . . . . . . . . 107

Introduction to Go-Board - Part II

By Jarosław Kotowicz and Yatsuka Nakamura . . . . . . . . 117

Properties of Go-Board - Part III

By Jarosław Kotowicz and Yatsuka Nakamura . . . . . . . . 123

Go-Board Theorem

By Jarosław Kotowicz and Yatsuka Nakamura . . . . . . . . 125

Some Properties of Binary Relations

By Waldemar Korczyński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136