### Some Properties of Binary Relations

by
Waldemar Korczynski

Copyright (c) 1992 Association of Mizar Users

MML identifier: SYSREL
[ MML identifier index ]

```environ

vocabulary RELAT_1, BOOLE, SYSREL, FUNCT_1;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1;
constructors TARSKI, RELSET_1, XBOOLE_0;
clusters RELAT_1, ZFMISC_1, XBOOLE_0;
requirements BOOLE, SUBSET;
definitions TARSKI, XBOOLE_0, RELAT_1;
theorems ZFMISC_1, RELAT_1, TARSKI, XBOOLE_0, XBOOLE_1;

begin
reserve x,y,z,t,X,Y,Z,W for set;
reserve R,S,T for Relation;

definition let X,Y;
cluster [:X,Y:] -> Relation-like;
correctness by RELAT_1:6;
end;

canceled 11;

theorem Th12:
X <> {} & Y <> {} implies dom [:X,Y:] = X & rng [:X,Y:] = Y
proof
assume A1:X <> {} & Y <> {};
thus dom [:X,Y:] c= X
proof
let x;
assume x in dom [:X,Y:];
then ex y st [x,y] in [:X,Y:] by RELAT_1:def 4;
hence thesis by ZFMISC_1:106;
end;
thus X c= dom [:X,Y:]
proof
let x;
assume A2:x in X;
consider y being Element of Y;
[x,y] in [:X,Y:] by A1,A2,ZFMISC_1:106;
hence thesis by RELAT_1:20;
end;
thus rng [:X,Y:] c= Y
proof
let y;
assume y in rng [:X,Y:];
then ex x st [x,y] in [:X,Y:] by RELAT_1:def 5;
hence thesis by ZFMISC_1:106;
end;
thus Y c= rng [:X,Y:]
proof
let y;
assume A3:y in Y;
consider x being Element of X;
[x,y] in [:X,Y:] by A1,A3,ZFMISC_1:106;
hence thesis by RELAT_1:20;
end;
end;

theorem Th13:
dom (R /\ [:X,Y:]) c= X & rng (R /\ [:X,Y:]) c= Y
proof
now per cases;
suppose X = {} or Y = {};
then R /\ [:X,Y:] = R /\ {} by ZFMISC_1:113
.= {};
hence thesis by RELAT_1:60,XBOOLE_1:2;
suppose A1: X <> {} & Y <> {};
dom (R /\ [:X,Y:]) c= dom R /\ dom [:X,Y:] by RELAT_1:14;
then A2:dom (R /\ [:X,Y:]) c= dom R /\ X by A1,Th12;
dom R /\ X c= X by XBOOLE_1:17;
hence dom (R /\ [:X,Y:]) c= X by A2,XBOOLE_1:1;
rng (R /\ [:X,Y:]) c= rng R /\ rng [:X,Y:] by RELAT_1:27;
then A3:rng (R /\ [:X,Y:]) c= rng R /\ Y by A1,Th12;
rng R /\ Y c= Y by XBOOLE_1:17;
hence rng (R /\ [:X,Y:]) c= Y by A3,XBOOLE_1:1;
end;
hence thesis;
end;

theorem
X misses Y implies dom (R /\ [:X,Y:]) misses rng (R /\ [:X,Y:]) &
dom (R~ /\ [:X,Y:]) misses rng (R~ /\ [:X,Y:])
proof
assume A1: X /\ Y = {};
A2:dom (R /\ [:X,Y:]) c= X & rng (R /\ [:X,Y:]) c= Y by Th13;
then A3:dom (R /\ [:X,Y:]) /\ rng (R /\ [:X,Y:]) c= X /\ rng (R /\ [:X,Y:])
by XBOOLE_1:26;
X /\ rng (R /\ [:X,Y:]) c= X /\ Y by A2,XBOOLE_1:26;
then dom (R /\ [:X,Y:]) /\ rng (R /\ [:X,Y:]) c= {} by A1,A3,XBOOLE_1:1;
hence dom (R /\ [:X,Y:]) /\ rng (R /\ [:X,Y:]) = {} by XBOOLE_1:3;
A4:dom (R~ /\ [:X,Y:]) c= X & rng (R~ /\ [:X,Y:]) c= Y by Th13;
then A5:dom (R~ /\ [:X,Y:]) /\ rng (R~ /\ [:X,Y:]) c= X /\ rng (R~ /\ [:X,Y:])
by XBOOLE_1:26;
X /\ rng (R~ /\ [:X,Y:]) c= X /\ Y by A4,XBOOLE_1:26;
then dom (R~ /\ [:X,Y:]) /\ rng (R~ /\ [:X,Y:]) c= {} by A1,A5,XBOOLE_1:1;
hence dom (R~ /\ [:X,Y:]) /\ rng (R~ /\ [:X,Y:]) = {} by XBOOLE_1:3;
end;

theorem Th15:
R c= [:X,Y:] implies dom R c= X & rng R c= Y
proof
assume R c= [:X,Y:];
then R /\ [:X,Y:] = R by XBOOLE_1:28;
hence thesis by Th13;
end;

theorem
R c= [:X,Y:] implies R~ c= [:Y,X:]
proof
assume A1: R c= [:X,Y:];
let z,t;
assume [z,t] in R~;
then [t,z] in R by RELAT_1:def 7;
then t in X & z in Y by A1,ZFMISC_1:106;
hence thesis by ZFMISC_1:106;
end;

canceled;

theorem
[:X,Y:]~ = [:Y,X:]
proof
let x,y;
thus [x,y] in [:X,Y:]~ implies [x,y] in [:Y,X:]
proof
assume [x,y] in [:X,Y:]~;
then [y,x] in [:X,Y:] by RELAT_1:def 7;
then y in X & x in Y by ZFMISC_1:106;
hence thesis by ZFMISC_1:106;
end;
thus [x,y] in [:Y,X:] implies [x,y] in [:X,Y:]~
proof
assume [x,y] in [:Y,X:];
then y in X & x in Y by ZFMISC_1:106;
then [y,x] in [:X,Y:] by ZFMISC_1:106;
hence thesis by RELAT_1:def 7;
end;
end;

canceled;

theorem Th20:
(R \/ S) * T = (R * T) \/ (S * T) & R * (S \/ T) = (R * S) \/ (R * T)
proof
thus (R \/ S) * T = (R * T) \/ (S * T)
proof
let x,y;
thus [x,y] in (R \/ S) * T implies [x,y] in (R * T) \/ (S * T)
proof
assume [x,y] in (R \/ S) * T;
then consider z such that A1:[x,z] in R \/ S & [z,y] in T by RELAT_1:def 8;
([x,z] in R & [z,y] in T) or ([x,z] in S & [z,y] in T) by A1,XBOOLE_0:def 2
;
then ([x,y] in R * T) or ([x,y] in S *T) by RELAT_1:def 8;
hence thesis by XBOOLE_0:def 2;
end;
thus [x,y] in (R * T) \/ (S * T) implies [x,y] in (R \/ S) * T
proof
assume A2: [x,y] in (R * T) \/ (S * T);
A3:[x,y] in R * T implies thesis
proof
assume [x,y] in R * T;
then consider z such that A4: [x,z] in R & [z,y] in T by RELAT_1:def 8;
[x,z] in R \/ S & [z,y] in T by A4,XBOOLE_0:def 2;
hence thesis by RELAT_1:def 8;
end;
[x,y] in S * T implies thesis
proof
assume [x,y] in S * T;
then consider z such that A5: [x,z] in S & [z,y] in T by RELAT_1:def 8;
[x,z] in R \/ S & [z,y] in T by A5,XBOOLE_0:def 2;
hence thesis by RELAT_1:def 8;
end;
hence thesis by A2,A3,XBOOLE_0:def 2;
end;
end;
thus R * (S \/ T) = (R * S) \/ (R * T) by RELAT_1:51;
end;

canceled;

theorem
(X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in X
implies not x in Y & not y in X & y in Y) &
(X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in Y
implies not y in X & not x in Y & x in X) &
(X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in Y
implies not x in X & not y in Y & y in X) &
(X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in X
implies not x in X & not y in Y & x in Y)
proof
thus X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in X
implies not x in Y & not y in X & y in Y
proof
assume A1:X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in X;
then A2:[:X,Y:] misses [:Y,X:] by ZFMISC_1:127;
A3:[x,y] in [:X,Y:] & not [x,y] in [:Y,X:] implies thesis
proof
assume [x,y] in [:X,Y:] & not [x,y] in [:Y,X:];
then (x in X & y in Y) & (not x in Y or not y in X) by ZFMISC_1:106;
hence thesis by A1,XBOOLE_0:3;
end;
[x,y] in [:Y,X:] implies [x,y] in [:X,Y:]
proof
assume A4: [x,y] in [:Y,X:] & not [x,y] in [:X,Y:];
not x in Y by A1,XBOOLE_0:3;
hence thesis by A4,ZFMISC_1:106;
end;
hence thesis by A1,A2,A3,XBOOLE_0:5;
end;
thus X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in Y
implies not y in X & not x in Y & x in X
proof
assume A5:X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in Y;
then A6:[:X,Y:] misses [:Y,X:] by ZFMISC_1:127;
A7:[x,y] in [:X,Y:] & not [x,y] in [:Y,X:] implies thesis
proof
assume [x,y] in [:X,Y:] & not [x,y] in [:Y,X:];
then (x in X & y in Y) & (not x in Y or not y in X) by ZFMISC_1:106;
hence thesis by A5,XBOOLE_0:3;
end;
[x,y] in [:Y,X:] implies [x,y] in [:X,Y:]
proof
assume A8: [x,y] in [:Y,X:] & not [x,y] in [:X,Y:];
not y in X by A5,XBOOLE_0:3;
hence thesis by A8,ZFMISC_1:106;
end;
hence thesis by A5,A6,A7,XBOOLE_0:3,def 2;
end;
thus X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in Y
implies not x in X & not y in Y & y in X
proof
assume A9:X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & x in Y;
then A10:[:X,Y:] misses [:Y,X:] by ZFMISC_1:127;
A11:[x,y] in [:X,Y:] implies [x,y] in [:Y,X:]
proof
assume A12: [x,y] in [:X,Y:] & not [x,y] in [:Y,X:];
not x in X by A9,XBOOLE_0:3;
hence thesis by A12,ZFMISC_1:106;
end;
[x,y] in [:Y,X:] & not [x,y] in [:X,Y:] implies thesis
proof
assume [x,y] in [:Y,X:] & not [x,y] in [:X,Y:];
then x in Y & y in X & not x in X or x in Y & y in X & not y in
Y by ZFMISC_1:106;
hence thesis by A9,XBOOLE_0:3;
end;
hence thesis by A9,A10,A11,XBOOLE_0:3,def 2;
end;
thus X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in X
implies not x in X & not y in Y & x in Y
proof
assume A13:X misses Y & R c= [:X,Y:] \/ [:Y,X:] & [x,y] in R & y in X;
then A14:[:X,Y:] misses [:Y,X:] by ZFMISC_1:127;
A15:[x,y] in [:X,Y:] implies [x,y] in [:Y,X:]
proof
assume A16: [x,y] in [:X,Y:] & not [x,y] in [:Y,X:];
not y in Y by A13,XBOOLE_0:3;
hence thesis by A16,ZFMISC_1:106;
end;
[x,y] in [:Y,X:] & not [x,y] in [:X,Y:] implies thesis
proof
assume [x,y] in [:Y,X:] & not [x,y] in [:X,Y:];
then (x in Y & y in X) & (not x in X or not y in Y) by ZFMISC_1:106;
hence thesis by A13,XBOOLE_0:3;
end;
hence thesis by A13,A14,A15,XBOOLE_0:3,def 2;
end;
end;

canceled;

theorem Th24:
(R c= [:X,Y:] & Z c= X implies (R|Z) = R /\ [:Z,Y:]) &
(R c= [:X,Y:] & Z c= Y implies (Z|R) = R /\ [:X,Z:])
proof
thus R c= [:X,Y:] & Z c= X implies (R|Z) = R /\ [:Z,Y:]
proof
assume A1:R c= [:X,Y:] & Z c= X;
let x,y;
thus [x,y] in (R|Z) implies [x,y] in R /\ [:Z,Y:]
proof
assume [x,y] in (R|Z);
then A2:x in Z & [x,y] in R by RELAT_1:def 11;
then x in Z & y in Y by A1,ZFMISC_1:106;
then [x,y] in [:Z,Y:] by ZFMISC_1:106;
hence thesis by A2,XBOOLE_0:def 3;
end;
thus [x,y] in R /\ [:Z,Y:] implies [x,y] in (R|Z)
proof
assume [x,y] in R /\ [:Z,Y:];
then [x,y] in R & [x,y] in [:Z,Y:] by XBOOLE_0:def 3;
then x in Z & [x,y] in R by ZFMISC_1:106;
hence thesis by RELAT_1:def 11;
end;
end;
assume A3:R c= [:X,Y:] & Z c= Y;
let x,y;
thus [x,y] in (Z|R) implies [x,y] in R /\ [:X,Z:]
proof
assume [x,y] in (Z|R);
then A4:y in Z & [x,y] in R by RELAT_1:def 12;
then y in Z & x in X by A3,ZFMISC_1:106;
then [x,y] in [:X,Z:] by ZFMISC_1:106;
hence thesis by A4,XBOOLE_0:def 3;
end;
thus [x,y] in R /\ [:X,Z:] implies [x,y] in (Z|R)
proof
assume [x,y] in R /\ [:X,Z:];
then [x,y] in R & [x,y] in [:X,Z:] by XBOOLE_0:def 3;
then y in Z & [x,y] in R by ZFMISC_1:106;
hence thesis by RELAT_1:def 12;
end;
end;

theorem
R c= [:X,Y:] & X = Z \/ W implies R = (R|Z) \/ (R|W)
proof
assume A1:R c= [:X,Y:] & X = Z \/ W;
then A2:Z c= X & W c= X by XBOOLE_1:7;
thus R = R /\ [:X,Y:] by A1,XBOOLE_1:28
.= R /\ ([:Z,Y:] \/ [:W,Y:]) by A1,ZFMISC_1:120
.= (R /\ [:Z,Y:]) \/ (R /\ [:W,Y:]) by XBOOLE_1:23
.= (R|Z) \/ (R /\ [:W,Y:]) by A1,A2,Th24
.= (R|Z) \/ (R|W) by A1,A2,Th24;
end;

theorem
X misses Y & R c= [:X,Y:] \/ [:Y,X:] implies R|X c= [:X,Y:]
proof
assume that
A1: X /\ Y = {} and
A2: R c= [:X,Y:] \/ [:Y,X:];
R|X = R /\ [:X,rng R:] by RELAT_1:96;
then R|X c= ([:X,Y:] \/ [:Y,X:]) /\ [:X,rng R:] by A2,XBOOLE_1:26;
then A3: R|X c= [:X,Y:] /\ [:X,rng R:] \/ [:Y,X:] /\ [:X,rng R:] by XBOOLE_1:
23;
[:Y,X:] /\ [:X,rng R:] = [:X /\ Y, X /\ rng R:] by ZFMISC_1:123
.= {} by A1,ZFMISC_1:113;
hence thesis by A3,XBOOLE_1:18;
end;

theorem
R c= S implies R~ c= S~
proof
assume R c= S;
then R \/ S = S by XBOOLE_1:12;
then (R~) \/ (S~) = S~ by RELAT_1:40;
hence thesis by XBOOLE_1:7;
end;

:: DIAGONAL RELATION

Lm1:
id(X) c= [:X,X:]
proof
let x,y;
assume [x,y] in id(X);
then x in X & x = y by RELAT_1:def 10;
hence thesis by ZFMISC_1:106;
end;

canceled;

theorem Th29:
id(X) * id(X) = id(X)
proof
dom id(X) = X by RELAT_1:71;
hence thesis by RELAT_1:77;
end;

theorem
id({x}) = {[x,x]}
proof
thus id({x}) c= {[x,x]}
proof
[:{x},{x}:] = {[x,x]} by ZFMISC_1:35;
hence thesis by Lm1;
end;
x in {x} & x = x by TARSKI:def 1;
then [x,x] in id({x}) by RELAT_1:def 10;
hence thesis by ZFMISC_1:37;
end;

theorem
[x,y] in id(X) iff [y,x] in id(X)
proof
thus [x,y] in id(X) implies [y,x] in id(X)
proof
assume [x,y] in id(X);
then x in X & x = y by RELAT_1:def 10;
hence thesis by RELAT_1:def 10;
end;
thus [y,x] in id(X) implies [x,y] in id(X)
proof
assume [y,x] in id(X);
then y in X & y = x by RELAT_1:def 10;
hence thesis by RELAT_1:def 10;
end;
end;

theorem Th32:
id(X \/ Y) = id(X) \/ id(Y) &
id(X /\ Y) = id(X) /\ id(Y) &
id(X \ Y) = id(X) \ id(Y)
proof
thus id(X \/ Y) = id(X) \/ id(Y)
proof
let x,y;
thus [x,y] in id(X \/ Y) implies [x,y] in id(X) \/ id(Y)
proof
assume [x,y] in id(X \/ Y);
then x in X \/ Y & x = y by RELAT_1:def 10;
then (x in X or x in Y) & x = y by XBOOLE_0:def 2;
then [x,y] in id(X) or [x,y] in id(Y) by RELAT_1:def 10;
hence thesis by XBOOLE_0:def 2;
end;
assume [x,y] in id(X) \/ id(Y);
then [x,y] in id(X) or [x,y] in id(Y) by XBOOLE_0:def 2;
then (x in X or x in Y) & x = y by RELAT_1:def 10;
then x in X \/ Y & x = y by XBOOLE_0:def 2;
hence thesis by RELAT_1:def 10;
end;

thus id(X /\ Y) = id(X) /\ id(Y)
proof
let x,y;
thus [x,y] in id(X /\ Y) implies [x,y] in id(X) /\ id(Y)
proof
assume [x,y] in id(X /\ Y);
then x in X /\ Y & x = y by RELAT_1:def 10;
then x in X & x in Y & x = y by XBOOLE_0:def 3;
then [x,y] in id(X) & [x,y] in id(Y) by RELAT_1:def 10;
hence thesis by XBOOLE_0:def 3;
end;
assume [x,y] in id(X) /\ id(Y);
then [x,y] in id(X) & [x,y] in id(Y) by XBOOLE_0:def 3;
then x in X & x in Y & x = y by RELAT_1:def 10;
then x in X /\ Y & x = y by XBOOLE_0:def 3;
hence thesis by RELAT_1:def 10;
end;
thus id(X \ Y) = id(X) \ id(Y)
proof
let x,y;
thus [x,y] in id(X \ Y) implies [x,y] in id(X) \ id(Y)
proof
assume [x,y] in id(X \ Y);
then x in X \ Y & x = y by RELAT_1:def 10;
then x in X & not x in Y & x = y by XBOOLE_0:def 4;
then [x,y] in id(X) & not [x,y] in id(Y) by RELAT_1:def 10;
hence thesis by XBOOLE_0:def 4;
end;
assume [x,y] in id(X) \ id(Y);
then [x,y] in id(X) & not [x,y] in id(Y) by XBOOLE_0:def 4;
then x in X & x = y & not (x in Y & x = y) by RELAT_1:def 10;
then x in X \ Y & x = y by XBOOLE_0:def 4;
hence thesis by RELAT_1:def 10;
end;
end;

theorem Th33:
X c= Y implies id(X) c= id(Y)
proof
assume X c= Y;
then X \/ Y = Y by XBOOLE_1:12;
then id(X) \/ id(Y) = id(Y) by Th32;
hence thesis by XBOOLE_1:7;
end;

theorem
id(X \ Y) \ id(X) = {}
proof
X \ Y c= X by XBOOLE_1:36;
then id(X \ Y) c= id(X) by Th33;
hence thesis by XBOOLE_1:37;
end;

theorem
R c= id(dom R) implies R = id(dom R)
proof
assume A1:R c= id(dom R);
let x,y;
thus [x,y] in R implies [x,y] in id(dom R) by A1;
assume [x,y] in id(dom R);
then A2:x in dom R & x = y by RELAT_1:def 10;
then consider z such that A3:[x,z] in R by RELAT_1:def 4;
thus thesis by A1,A2,A3,RELAT_1:def 10;
end;

theorem
id(X) c= R \/ R~ implies id(X) c= R & id(X) c= R~
proof
assume A1:id(X) c= R \/ R~;
for x holds x in X implies [x,x] in R & [x,x] in R~
proof
let x;
assume x in X;
then [x,x] in id(X) by RELAT_1:def 10;
then A2: [x,x] in R or [x,x] in R~ by A1,XBOOLE_0:def 2;
hence [x,x] in R by RELAT_1:def 7;
thus thesis by A2,RELAT_1:def 7;
end;
then (for x holds x in X implies [x,x] in R) &
(for x holds x in X implies [x,x] in R~);
hence thesis by RELAT_1:73;
end;

theorem
id(X) c= R implies id(X) c= R~
proof
assume A1:id(X) c= R;
for x holds x in X implies [x,x] in R~
proof
let x;
assume x in X;
then [x,x] in id(X) by RELAT_1:def 10;
hence thesis by A1,RELAT_1:def 7;
end;
hence thesis by RELAT_1:73;
end;

:: CLOSURE RELATION

theorem
R c= [:X,X:] implies R \ id(dom R) = R \ id(X) &
R \ id(rng R) = R \ id(X)
proof
assume A1:R c= [:X,X:];
then dom R c= X by Th15;
then id(dom R) c= id(X) by Th33;
then A2: R \ id(X) c= R \ id(dom R) by XBOOLE_1:34;
R \ id(dom R) c= R \ id(X)
proof
let x,y;
assume [x,y] in R \ id(dom R);
then A3:[x,y] in R & not [x,y] in id(dom R) by XBOOLE_0:def 4;
not [x,y] in id(X)
proof
assume [x,y] in id(X);
then x in dom R & x = y by A3,RELAT_1:20,def 10;
end;
hence thesis by A3,XBOOLE_0:def 4;
end;
hence R \ id(dom R) = R \ id(X) by A2,XBOOLE_0:def 10;
rng R c= X by A1,Th15;
then id(rng R) c= id(X) by Th33;
then A4: R \ id(X) c= R \ id(rng R) by XBOOLE_1:34;
R \ id(rng R) c= R \ id(X)
proof
let x,y;
assume [x,y] in R \ id(rng R);
then A5:[x,y] in R & not [x,y] in id(rng R) by XBOOLE_0:def 4;
not [x,y] in id(X)
proof
assume [x,y] in id(X);
then y in rng R & x = y by A5,RELAT_1:20,def 10;
end;
hence thesis by A5,XBOOLE_0:def 4;
end;
hence R \ id(rng R) = R \ id(X) by A4,XBOOLE_0:def 10;
end;

theorem
(id(X) * (R \ id(X)) = {} implies
dom (R \ id(X)) = dom R \ dom (id(X))) &
((R \ id(X)) * id(X) = {} implies
rng (R \ id(X)) = rng R \ rng (id(X)))
proof
thus (id(X) * (R \ id(X)) = {} implies
dom (R \ id(X)) = dom R \ dom (id(X)))
proof
assume A1:id(X) * (R \ id(X)) = {};
A2:dom R \ dom(id(X)) c= dom (R \ id(X)) by RELAT_1:15;
for x holds x in dom(R \ id(X)) implies x in (dom R \ dom(id(X)
)
)
proof
let x;
assume x in dom(R \ id(X));
then consider y such that A3: [x,y] in (R \ id(X)) by RELAT_1:def 4;
A4:   [x,y] in R & not [x,y] in id(X) by A3,XBOOLE_0:def 4;
A5:x in dom R & not x in X implies thesis
proof
assume x in dom R & not x in X;
then x in dom R & not x in dom(id(X)) by RELAT_1:71;
hence thesis by XBOOLE_0:def 4;
end;
x in dom R & not x = y implies thesis
proof
not x in X
proof
assume x in X;
then [x,x] in id(X) & [x,y] in (R \ id(X)) by A3,RELAT_1:def 10;
hence thesis by A1,RELAT_1:def 8;
end;
hence thesis by A5;
end;
hence thesis by A4,A5,RELAT_1:def 4,def 10;
end;
then dom (R \ id(X)) c= dom R \ dom (id(X)) by TARSKI:def 3;
hence thesis by A2,XBOOLE_0:def 10;
end;
thus (R \ id(X)) * id(X) = {} implies
rng (R \ id(X)) = rng R \ rng (id(X))
proof
assume A6:(R \ id(X)) * id(X) = {};
A7:rng R \ rng(id(X)) c= rng (R \ id(X)) by RELAT_1:28;
rng (R \ id(X)) c= rng R \ rng (id(X))
proof
let y;
assume y in rng(R \ id(X));
then consider x such that A8: [x,y] in (R \ id(X)) by RELAT_1:def 5;
A9:   [x,y] in R & not [x,y] in id(X) by A8,XBOOLE_0:def 4;

A10:y in rng R & not y in X implies thesis
proof
assume y in rng R & not y in X;
then y in rng R & not y in rng(id(X)) by RELAT_1:71;
hence thesis by XBOOLE_0:def 4;
end;
y in rng R & not x = y implies thesis
proof
not y in X
proof
assume y in X;
then [y,y] in id(X) & [x,y] in (R \ id(X)) by A8,RELAT_1:def 10;
hence thesis by A6,RELAT_1:def 8;
end;
hence thesis by A10;
end;
hence thesis by A9,A10,RELAT_1:def 5,def 10;
end;
hence thesis by A7,XBOOLE_0:def 10;
end;
end;

theorem Th40:
(R c= R * R & R * (R \ id(rng(R))) = {} implies
id(rng(R)) c= R) &
(R c= R * R & (R \ id(dom(R))) * R = {} implies
id(dom(R)) c= R)
proof
thus (R c= R * R & R * (R \ id(rng(R))) = {} implies
id(rng(R)) c= R)
proof
assume A1:R c= R * R & R * (R \ id(rng(R))) = {};
for y holds y in rng(R) implies [y,y] in R
proof
let y;
assume y in rng R;
then consider x such that A2:[x,y] in R by RELAT_1:def 5;
consider z such that A3:[x,z] in R & [z,y] in R by A1,A2,RELAT_1:def 8;
z = y
proof
assume z <> y;
then not [z,y] in id(rng(R)) by RELAT_1:def 10;
then [z,y] in (R \ id(rng(R))) by A3,XBOOLE_0:def 4;
end;
hence thesis by A3;
end;
hence thesis by RELAT_1:73;
end;
thus R c= R * R & (R \ id(dom(R))) * R = {} implies
id(dom(R)) c= R
proof
assume A4:R c= R * R & (R \ id(dom(R))) * R = {};

for x holds x in dom(R) implies [x,x] in R
proof
let x;
assume x in dom R;
then consider y such that A5:[x,y] in R by RELAT_1:def 4;
consider z such that A6:[x,z] in R & [z,y] in R by A4,A5,RELAT_1:def 8;
z = x
proof
assume z <> x;
then not [x,z] in id(dom(R)) by RELAT_1:def 10;
then [x,z] in (R \ id(dom(R))) by A6,XBOOLE_0:def 4;
end;
hence thesis by A6;
end;
hence thesis by RELAT_1:73;
end;
end;

theorem
(R c= R * R & R * (R \ id(rng(R))) = {} implies
R /\ (id(rng(R))) = id(rng(R))) &
(R c= R * R & (R \ id(dom(R))) * R = {} implies
R /\ (id(dom(R))) = id(dom(R)))
proof
thus R c= R * R & R * (R \ id(rng(R))) = {} implies
R /\ (id(rng(R))) = id(rng(R))
proof
assume R c= R * R & R * (R \ id(rng(R))) = {};
then id(rng(R)) c= R by Th40;
hence thesis by XBOOLE_1:28;
end;
thus R c= R * R & (R \ id(dom(R))) * R = {} implies
R /\ (id(dom(R))) = id(dom(R))
proof
assume R c= R * R & (R \ id(dom(R))) * R = {};
then id(dom(R)) c= R by Th40;
hence thesis by XBOOLE_1:28;
end;
end;

theorem
(R * (R \ id(X)) = {} & rng R c= X implies
R * (R \ id(rng R)) = {}) &
((R \ id(X)) * R = {} & dom R c= X implies
(R \ id(dom R)) * R = {})
proof
thus (R * (R \ id(X)) = {} & rng R c= X implies
R * (R \ id(rng R)) = {})
proof
assume that A1:R * (R \ id(X)) = {} & rng R c= X and
A2:not R * (R \ id(rng R)) = {};
consider x,y such that A3:[x,y] in R * (R \ id(rng R))
by A2,RELAT_1:56;
consider z such that A4:[x,z] in R & [z,y] in (R \ id(rng R))
by A3,RELAT_1:def 8;
A5:z in rng R by A4,RELAT_1:20;
[z,y] in R & not [z,y] in id(rng R) by A4,XBOOLE_0:def 4;
then [z,y] in R & z <> y by A5,RELAT_1:def 10;
then [z,y] in R & not [z,y] in id(X) by RELAT_1:def 10;
then [z,y] in R \ id(X) by XBOOLE_0:def 4;
end;
thus ((R \ id(X)) * R = {} & dom R c= X implies
(R \ id(dom R)) * R = {})
proof
assume that A6:(R \ id(X)) * R = {} & dom R c= X and
A7:not (R \ id(dom R)) * R = {};
consider x,y such that A8:[x,y] in (R \ id(dom R)) * R
by A7,RELAT_1:56;
consider z such that A9: [x,z] in (R \ id(dom R)) & [z,y] in R
by A8,RELAT_1:def 8;
[x,z] in R & not [x,z] in id(dom R) by A9,XBOOLE_0:def 4;
then [x,z] in R & (not z in dom R or x <> z) by RELAT_1:def 10;
then [x,z] in R & not [x,z] in id(X) by A9,RELAT_1:20,def 10;
then [x,z] in R \ id(X) by XBOOLE_0:def 4;
end;
end;

:: OPERATION CL

definition let R;
func CL(R) -> Relation equals
:Def1: R /\ id(dom R);
correctness;
end;

theorem Th43:
CL(R) c= R & CL(R) c= id(dom R)
proof
R /\ id(dom R) c= R & R /\ id(dom R) c= id(dom R)
by XBOOLE_1:17;
hence thesis by Def1;
end;

theorem Th44:
[x,y] in CL(R) implies x in dom( CL(R)) & x = y
proof
assume [x,y] in CL(R);
then x in dom( CL(R)) & [x,y] in R /\ id(dom R)
by Def1,RELAT_1:20;
then x in dom( CL(R)) & [x,y] in id(dom R) by XBOOLE_0:def 3;
hence thesis by RELAT_1:def 10;
end;

theorem Th45:
dom(CL(R)) = rng(CL(R))
proof
thus dom(CL(R)) c= rng(CL(R))
proof
let x;
assume x in dom(CL(R));
then consider y such that A1: [x,y] in CL(R) by RELAT_1:def 4;
[x,y] in R /\ id(dom R) by A1,Def1;
then [x,y] in R & [x,y] in id(dom R) by XBOOLE_0:def 3;
then [x,x] in CL(R) by A1,RELAT_1:def 10;
hence thesis by RELAT_1:20;
end;
let x;
assume x in rng (CL(R));
then consider y such that A2:[y,x] in CL(R) by RELAT_1:def 5;
[y,x] in R /\ id(dom R) by A2,Def1;
then [y,x] in R & [y,x] in id(dom R) by XBOOLE_0:def 3;
then [x,x] in CL(R) by A2,RELAT_1:def 10;
hence thesis by RELAT_1:20;
end;

theorem Th46:
(x in dom(CL(R)) iff x in dom R & [x,x] in R) &
(x in rng(CL(R)) iff x in dom R & [x,x] in R) &
(x in rng(CL(R)) iff x in rng R & [x,x] in R) &
(x in dom(CL(R)) iff x in rng R & [x,x] in R)
proof
thus A1: x in dom(CL(R)) iff x in dom R & [x,x] in R
proof
thus x in dom(CL(R)) implies x in dom R & [x,x] in R
proof
assume x in dom(CL(R));
then consider y such that A2: [x,y] in CL(R) by RELAT_1:def 4;
[x,y] in R /\ id(dom R) by A2,Def1;
then [x,y] in R & [x,y] in id(dom R) by XBOOLE_0:def 3;
hence thesis by RELAT_1:def 10;
end;
thus x in dom R & [x,x] in R implies x in dom(CL(R))
proof
assume A3:x in dom R & [x,x] in R;
then [x,x] in id(dom R) by RELAT_1:def 10;
then [x,x] in (R /\ id(dom R)) by A3,XBOOLE_0:def 3;
then [x,x] in CL(R) by Def1;
hence thesis by RELAT_1:def 4;
end;
end;
hence x in rng(CL(R)) iff x in dom R & [x,x] in R by Th45;
thus x in rng(CL(R)) iff x in rng R & [x,x] in R by A1,Th45,RELAT_1:20;
thus x in dom(CL(R)) iff x in rng R & [x,x] in R by A1,RELAT_1:20;
end;

theorem Th47:
CL(R) = id(dom CL(R))
proof
let x,y;
thus [x,y] in CL(R) implies [x,y] in id(dom CL(R))
proof
assume [x,y] in CL(R);
then x in dom CL(R) & [x,y] in R /\ id(dom R) by Def1,RELAT_1:20;
then x in dom CL(R) & [x,y] in id(dom R) by XBOOLE_0:def 3;
then x in dom CL(R) & x = y by RELAT_1:def 10;
hence thesis by RELAT_1:def 10;
end;
assume [x,y] in id(dom CL(R));
then A1:x in dom CL(R) & x = y by RELAT_1:def 10;
then ex z st [x,z] in CL(R) by RELAT_1:def 4;
hence thesis by A1,Th44;
end;

theorem Th48:
(R * R = R & R * (R \ CL(R)) = {} & [x,y] in R & x <> y implies
x in (dom R \ dom(CL(R))) & y in dom(CL(R))) &
(R * R = R & (R \ CL(R)) * R = {} & [x,y] in R & x <> y implies
y in (rng R \ dom(CL(R))) & x in dom(CL(R)))
proof
thus R * R = R & R * (R \ CL(R)) = {} & [x,y] in R & x <> y implies
x in (dom R \ dom(CL(R))) & y in dom(CL(R))
proof
assume A1:R * R = R & R * (R \ CL(R)) = {} & [x,y] in R & x <> y;
then not [x,y] in id(dom R) by RELAT_1:def 10;
then not [x,y] in R /\ id(dom R) by XBOOLE_0:def 3;
then not [x,y] in CL(R) by Def1;
then A2:[x,y] in R \ CL(R) by A1,XBOOLE_0:def 4;
consider z such that A3: [x,z] in R & [z,y] in R by A1,RELAT_1:def 8;
z = y
proof
assume z <> y;
then not [z,y] in id(dom R) by RELAT_1:def 10;
then not [z,y] in R /\ id(dom R) by XBOOLE_0:def 3;
then not [z,y] in CL(R) by Def1;
then [x,z] in R & [z,y] in R \ CL(R) by A3,XBOOLE_0:def 4;
hence thesis by A1,RELAT_1:def 8;
end;
then A4: y in rng R & y = z by A1,RELAT_1:20;
A5: not x in dom(CL(R))
proof
assume x in dom(CL(R));
then [x,x] in R by Th46;
hence thesis by A1,A2,RELAT_1:def 8;
end;
x in dom R by A3,RELAT_1:20;
hence thesis by A3,A4,A5,Th46,XBOOLE_0:def 4;
end;
thus R * R = R & (R \ CL(R)) * R = {} & [x,y] in R & x <> y implies
y in (rng R \ dom(CL(R))) & x in dom(CL(R))
proof
assume A6:R * R = R & (R \ CL(R)) * R = {} & [x,y] in R & x <> y;
then not [x,y] in id(dom R) by RELAT_1:def 10;
then not [x,y] in R /\ id(dom R) by XBOOLE_0:def 3;
then not [x,y] in CL(R) by Def1;
then A7:[x,y] in R \ CL(R) by A6,XBOOLE_0:def 4;
consider z such that A8: [x,z] in R & [z,y] in R by A6,RELAT_1:def 8;
z = x
proof
assume z <> x;
then not [x,z] in id(dom R) by RELAT_1:def 10;
then not [x,z] in R /\ id(dom R) by XBOOLE_0:def 3;
then not [x,z] in CL(R) by Def1;
then [z,y] in R & [x,z] in R \ CL(R) by A8,XBOOLE_0:def 4;
hence thesis by A6,RELAT_1:def 8;
end;
then A9: x in dom R & x = z by A6,RELAT_1:20;
A10: not y in dom(CL(R))
proof
assume y in dom(CL(R));
then [y,y] in R by Th46;
hence thesis by A6,A7,RELAT_1:def 8;
end;
y in rng R by A8,RELAT_1:20;
hence thesis by A8,A9,A10,Th46,XBOOLE_0:def 4;
end;
end;

theorem
(R * R = R & R * (R \ id(dom R)) = {} & [x,y] in R & x <> y
implies x in ((dom R) \ dom(CL(R))) & y in dom(CL(R))) &
(R * R = R & (R \ id(dom R)) * R = {} & [x,y] in R & x <> y
implies y in ((rng R) \ dom(CL(R))) & x in dom(CL(R)))
proof
(R \ CL(R)) = (R \ R /\ id(dom R)) by Def1
.= R \ id(dom R) by XBOOLE_1:47;
hence thesis by Th48;
end;

theorem
(R * R = R & R * (R \ id(dom R)) = {}
implies dom(CL(R)) = rng(R) & rng(CL(R)) = rng(R)) &
(R * R = R & (R \ id(dom R)) * R = {}
implies dom(CL(R)) = dom(R) & rng(CL(R)) = dom(R))

proof
thus R * R = R & R * (R \ id(dom R)) = {}
implies dom(CL(R)) = rng(R) & rng(CL(R)) = rng(R)
proof
assume A1: R * R = R & R * (R \ id(dom R)) = {};
CL(R) c= R by Th43;
then rng(CL(R)) c= rng(R) by RELAT_1:25;
then A2: dom(CL(R)) c= rng(R) by Th45;
rng R c= dom(CL(R))
proof
for y holds y in rng R implies y in dom(CL(R))
proof
let y;
assume y in rng R;
then consider x such that A3:[x,y] in R by RELAT_1:def 5;
consider z such that A4: [x,z] in R & [z,y] in R by A1,A3,RELAT_1:def 8;
A5:    z = y
proof
assume z <> y;
then not [z,y] in id(dom R) by RELAT_1:def 10;
then [x,z] in R & [z,y] in R \ id(dom R) by A4,XBOOLE_0:def 4;
hence thesis by A1,RELAT_1:def 8;
end;
z in dom R by A4,RELAT_1:20;
then [z,y] in id(dom R) by A5,RELAT_1:def 10;
then [z,y] in R /\ id(dom R) by A4,XBOOLE_0:def 3;
then [z,y] in CL(R) by Def1;
hence thesis by A5,RELAT_1:def 4;
end;
hence thesis by TARSKI:def 3;
end;
then dom(CL(R)) = rng R by A2,XBOOLE_0:def 10;
hence thesis by Th45;
end;
thus R * R = R & (R \ id(dom R)) * R = {}
implies dom(CL(R)) = dom(R) & rng(CL(R)) = dom R
proof
assume A6: R * R = R & (R \ id(dom R)) * R = {};
CL(R) c= R by Th43;
then A7:dom(CL(R)) c= dom(R) by RELAT_1:25;
dom R c= dom(CL(R))
proof
for x holds x in dom R implies x in dom(CL(R))
proof
let x;
assume A8:x in dom R;
then consider y such that A9:[x,y] in R by RELAT_1:def 4;
consider z such that A10: [x,z] in R & [z,y] in R by A6,A9,RELAT_1:def 8;
A11:    z = x
proof
assume z <> x;
then not [x,z] in id(dom R) by RELAT_1:def 10;
then [z,y] in R & [x,z] in R \ id(dom R) by A10,XBOOLE_0:def 4;
hence thesis by A6,RELAT_1:def 8;
end;
then [x,z] in id(dom R) by A8,RELAT_1:def 10;
then [x,z] in R /\ id(dom R) by A10,XBOOLE_0:def 3;
then [x,z] in CL(R) by Def1;
then z in rng(CL(R)) by RELAT_1:def 5;
hence thesis by A11,Th45;
end;
hence thesis by TARSKI:def 3;
end;
then dom(CL(R)) = dom R by A7,XBOOLE_0:def 10;
hence thesis by Th45;
end;
end;

theorem Th51:
dom(CL(R)) c= dom R & rng(CL(R)) c= rng R &
rng(CL(R)) c= dom R & dom(CL(R)) c= rng R
proof
thus dom(CL(R)) c= dom R & rng(CL(R)) c= rng R
proof
CL(R) c= R by Th43;
hence thesis by RELAT_1:25;
end;
hence thesis by Th45;
end;

theorem
id(dom(CL(R))) c= id(dom R) &
id(rng(CL(R))) c= id(dom R)
proof
dom(CL(R)) c= dom R by Th51;
then id(dom(CL(R))) c= id(dom R) by Th33;
hence thesis by Th45;
end;

theorem Th53:
id(dom(CL(R))) c= R & id(rng(CL(R))) c= R
proof
thus id(dom(CL(R))) c= R
proof
let x,y;
assume [x,y] in id(dom(CL(R)));
then x in dom(CL(R)) & x = y by RELAT_1:def 10;
hence thesis by Th46;
end;
hence thesis by Th45;
end;

theorem Th54:
(id(X) c= R & id(X) * (R \ id(X)) = {}
implies R|X = id(X)) &
(id(X) c= R & (R \ id(X)) * id(X) = {}
implies X|R = id(X))
proof
thus id(X) c= R & id(X) * (R \ id(X)) = {}
implies R|X = id(X)
proof
assume A1: id(X) c= R & id(X) * (R \ id(X)) = {};
R|X = id(X) * R by RELAT_1:94
.= id(X) * (R \/ id(X)) by A1,XBOOLE_1:12
.= id(X) * ((R \ id(X)) \/ id(X)) by XBOOLE_1:39
.= {} \/ id(X) * id(X) by A1,RELAT_1:51
.= id(X) by Th29;
hence thesis;
end;
thus id(X) c= R & (R \ id(X)) * id(X) = {}
implies X|R = id(X)
proof
assume A2: id(X) c= R & (R \ id(X)) * id(X) = {};
X|R = R * id(X) by RELAT_1:123
.= (R \/ id(X)) * id(X) by A2,XBOOLE_1:12
.= ((R \ id(X)) \/ id(X)) * id(X) by XBOOLE_1:39
.= {} \/ id(X) * id(X) by A2,Th20
.= id(X) by Th29;
hence thesis;
end;
end;

theorem
(id(dom(CL(R))) * (R \ id(dom(CL(R)))) = {} implies
R|(dom(CL(R))) = id(dom(CL(R))) &
R|(rng(CL(R))) = id(dom(CL(R)))) &
((R \ id(rng(CL(R)))) * id(rng(CL(R))) = {} implies
(dom(CL(R)))|R = id(dom(CL(R))) &
(rng(CL(R)))|R = id(rng(CL(R))))
proof
thus id(dom(CL(R))) * (R \ id(dom(CL(R)))) = {} implies
R|(dom(CL(R))) = id(dom(CL(R))) &
R|(rng(CL(R))) = id(dom(CL(R)))
proof
assume A1: id(dom(CL(R))) * (R \ id(dom(CL(R)))) = {};
id(dom(CL(R))) c= R by Th53;
then R|(dom(CL(R))) = id(dom(CL(R))) by A1,Th54;
hence thesis by Th45;
end;
thus (R \ id(rng(CL(R)))) * id(rng(CL(R))) = {} implies
(dom(CL(R)))|R = id(dom(CL(R))) &
(rng(CL(R)))|R = id(rng(CL(R)))
proof
assume A2: (R \ id(rng(CL(R)))) * id(rng(CL(R))) = {};
id(rng(CL(R))) c= R by Th53;
then (rng(CL(R)))|R = id(rng(CL(R))) by A2,Th54;
then (dom(CL(R)))|R = id(rng(CL(R))) by Th45;
hence thesis by Th45;
end;
end;

theorem
(R * (R \ id(dom R)) = {} implies
id(dom(CL(R))) * (R \ id(dom(CL(R)))) = {}) &
((R \ id(dom R)) * R = {} implies
(R \ id(dom(CL(R)))) * id(dom(CL(R))) = {})
proof
thus R * (R \ id(dom R)) = {} implies
id(dom(CL(R))) * (R \ id(dom(CL(R)))) = {}
proof
assume A1:R * (R \ id(dom R)) = {};
A2: R \ id(dom R) = R \ (R /\ id(dom R)) by XBOOLE_1:47
.= R \ CL(R) by Def1
.= R \ id(dom(CL(R))) by Th47;
id(dom(CL(R))) c= R by Th53;
then id(dom(CL(R))) * (R \ id(dom(CL(R)))) c=
R * (R \ id(dom R)) by A2,RELAT_1:49;
hence thesis by A1,XBOOLE_1:3;
end;
thus (R \ id(dom R)) * R = {} implies
(R \ id(dom(CL(R)))) * id(dom(CL(R))) = {}
proof
assume A3:(R \ id(dom R)) * R = {};
A4: R \ id(dom R) = R \ (R /\ id(dom R)) by XBOOLE_1:47
.= R \ CL(R) by Def1
.= R \ id(dom(CL(R))) by Th47;
id(dom(CL(R))) c= R by Th53;
then (R \ id(dom(CL(R)))) * id(dom(CL(R))) c=
(R \ id(dom R)) * R by A4,RELAT_1:48;
hence thesis by A3,XBOOLE_1:3;
end;
end;

theorem Th57:
(S * R = S & R * (R \ id(dom R)) = {} implies
S * (R \ id(dom R)) = {}) &
(R * S = S & (R \ id(dom R)) * R = {} implies
(R \ id(dom R)) * S = {})
proof
thus S * R = S & R * (R \ id(dom R)) = {} implies
S * (R \ id(dom R)) = {}
proof
assume S * R = S & R * (R \ id(dom R)) = {};
then S * (R \ id(dom R)) = S * {} by RELAT_1:55
.= {};
hence thesis;
end;
thus R * S = S & (R \ id(dom R)) * R = {} implies
(R \ id(dom R)) * S = {}
proof
assume R * S = S & (R \ id(dom R)) * R = {};
then (R \ id(dom R)) * S = {} * S by RELAT_1:55
.= {};
hence thesis;
end;
end;

theorem Th58:
(S * R = S & R * (R \ id(dom R)) = {} implies CL(S) c= CL(R)) &
(R * S = S & (R \ id(dom R)) * R = {} implies
CL(S) c= CL(R))
proof
thus S * R = S & R * (R \ id(dom R)) = {} implies CL(S) c= CL(R)
proof
assume A1:S * R = S & R * (R \ id(dom R)) = {};
then A2: S * (R \ id(dom R)) = {} by Th57;
for x,y holds [x,y] in CL(S) implies [x,y] in CL(R)
proof
let x,y;
assume [x,y] in CL(S);
then [x,y] in S /\ id(dom S) by Def1;
then A3: [x,y] in S & [x,y] in id(dom S) by XBOOLE_0:def 3;
then A4:[x,y] in S & x in dom S & x = y by RELAT_1:def 10;
consider z such that A5: [x,z] in S & [z,y] in R by A1,A3,RELAT_1:def 8;
z = y
proof
assume z <> y;
then not [z,y] in id(dom R) by RELAT_1:def 10;
then [z,y] in R \ id(dom R) by A5,XBOOLE_0:def 4;
end;
then [x,y] in R & x in dom R & x = y by A4,A5,RELAT_1:20;
then [x,y] in R & [x,y] in id(dom R) by RELAT_1:def 10;
then [x,y] in R /\ id(dom R) by XBOOLE_0:def 3;
hence thesis by Def1;
end;
hence thesis by RELAT_1:def 3;
end;
assume A6:R * S = S & (R \ id(dom R)) * R = {};
then A7: (R \ id(dom R)) * S = {} by Th57;
for x,y holds [x,y] in CL(S) implies [x,y] in CL(R)
proof
let x,y;
assume [x,y] in CL(S);
then [x,y] in S /\ id(dom S) by Def1;
then A8: [x,y] in S & [x,y] in id(dom S) by XBOOLE_0:def 3;
then consider z such that A9: [x,z] in R & [z,y] in S by A6,RELAT_1:def 8;
z = x
proof
assume z <> x;
then not [x,z] in id(dom R) by RELAT_1:def 10;
then [x,z] in R \ id(dom R) by A9,XBOOLE_0:def 4;
end;
then [x,y] in R & x in dom R & x = y by A8,A9,RELAT_1:20,def 10;
then [x,y] in R & [x,y] in id(dom R) by RELAT_1:def 10;
then [x,y] in R /\ id(dom R) by XBOOLE_0:def 3;
hence thesis by Def1;
end;
hence thesis by RELAT_1:def 3;
end;

theorem
(S * R = S & R * (R \ id(dom R)) = {} &
R * S = R & S * (S \ id(dom S)) = {} implies
CL(S) = CL(R)) & (R * S = S & (R \ id(dom R)) * R = {} &
S * R = R & (S \ id(dom S)) * S = {} implies CL(S) = CL(R))
proof
thus (S * R = S & R * (R \ id(dom R)) = {} &
R * S = R & S * (S \ id(dom S)) = {}) implies CL(S) = CL(R)
proof
assume (S * R = S & R * (R \ id(dom R)) = {}) &
(R * S = R & S * (S \ id(dom S)) = {});
then CL(S) c= CL(R) & CL(R) c= CL(S) by Th58;
hence thesis by XBOOLE_0:def 10;
end;
assume (R * S = S & (R \ id(dom R)) * R = {}) &
(S * R = R & (S \ id(dom S)) * S = {});
then CL(S) c= CL(R) & CL(R) c= CL(S) by Th58;
hence thesis by XBOOLE_0:def 10;
end;
```