:: MCART_1 semantic presentation

theorem Th1: :: MCART_1:1
for X being set st X <> {} holds
ex Y being set st
( Y in X & Y misses X )
proof end;

theorem Th2: :: MCART_1:2
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1 being set st Y1 in Y holds
Y1 misses X ) )
proof end;

theorem Th3: :: MCART_1:3
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1, Y2 being set st Y1 in Y2 & Y2 in Y holds
Y1 misses X ) )
proof end;

theorem Th4: :: MCART_1:4
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1, Y2, Y3 being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y holds
Y1 misses X ) )
proof end;

theorem Th5: :: MCART_1:5
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4 being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y holds
Y1 misses X ) )
proof end;

theorem Th6: :: MCART_1:6
for X being set st X <> {} holds
ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5 being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y holds
Y1 misses X ) )
proof end;

definition
let x be set ;
given x1 being set , x2 being set such that E39: x = [x1,x2] ;
func c1 `1 -> set means :Def1: :: MCART_1:def 1
for y1, y2 being set st x = [y1,y2] holds
it = y1;
existence
ex b1 being set st
for y1, y2 being set st x = [y1,y2] holds
b1 = y1
proof end;
uniqueness
for b1, b2 being set st ( for y1, y2 being set st x = [y1,y2] holds
b1 = y1 ) & ( for y1, y2 being set st x = [y1,y2] holds
b2 = y1 ) holds
b1 = b2
proof end;
func c1 `2 -> set means :Def2: :: MCART_1:def 2
for y1, y2 being set st x = [y1,y2] holds
it = y2;
existence
ex b1 being set st
for y1, y2 being set st x = [y1,y2] holds
b1 = y2
proof end;
uniqueness
for b1, b2 being set st ( for y1, y2 being set st x = [y1,y2] holds
b1 = y2 ) & ( for y1, y2 being set st x = [y1,y2] holds
b2 = y2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines `1 MCART_1:def 1 :
for x being set st ex x1, x2 being set st x = [x1,x2] holds
for b2 being set holds
( b2 = x `1 iff for y1, y2 being set st x = [y1,y2] holds
b2 = y1 );

:: deftheorem Def2 defines `2 MCART_1:def 2 :
for x being set st ex x1, x2 being set st x = [x1,x2] holds
for b2 being set holds
( b2 = x `2 iff for y1, y2 being set st x = [y1,y2] holds
b2 = y2 );

theorem Th7: :: MCART_1:7
for x, y being set holds
( [x,y] `1 = x & [x,y] `2 = y ) by , ;

theorem Th8: :: MCART_1:8
for z being set st ex x, y being set st z = [x,y] holds
[(z `1 ),(z `2 )] = z
proof end;

theorem Th9: :: MCART_1:9
for X being set st X <> {} holds
ex v being set st
( v in X & ( for x, y being set holds
( ( not x in X & not y in X ) or not v = [x,y] ) ) )
proof end;

theorem Th10: :: MCART_1:10
for z, X, Y being set st z in [:X,Y:] holds
( z `1 in X & z `2 in Y )
proof end;

theorem Th11: :: MCART_1:11
for z, X, Y being set st ex x, y being set st z = [x,y] & z `1 in X & z `2 in Y holds
z in [:X,Y:]
proof end;

theorem Th12: :: MCART_1:12
for z, x, Y being set st z in [:{x},Y:] holds
( z `1 = x & z `2 in Y )
proof end;

theorem Th13: :: MCART_1:13
for z, X, y being set st z in [:X,{y}:] holds
( z `1 in X & z `2 = y )
proof end;

theorem Th14: :: MCART_1:14
for z, x, y being set st z in [:{x},{y}:] holds
( z `1 = x & z `2 = y )
proof end;

theorem Th15: :: MCART_1:15
for z, x1, x2, Y being set st z in [:{x1,x2},Y:] holds
( ( z `1 = x1 or z `1 = x2 ) & z `2 in Y )
proof end;

theorem Th16: :: MCART_1:16
for z, X, y1, y2 being set st z in [:X,{y1,y2}:] holds
( z `1 in X & ( z `2 = y1 or z `2 = y2 ) )
proof end;

theorem Th17: :: MCART_1:17
for z, x1, x2, y being set st z in [:{x1,x2},{y}:] holds
( ( z `1 = x1 or z `1 = x2 ) & z `2 = y )
proof end;

theorem Th18: :: MCART_1:18
for z, x, y1, y2 being set st z in [:{x},{y1,y2}:] holds
( z `1 = x & ( z `2 = y1 or z `2 = y2 ) )
proof end;

theorem Th19: :: MCART_1:19
for z, x1, x2, y1, y2 being set st z in [:{x1,x2},{y1,y2}:] holds
( ( z `1 = x1 or z `1 = x2 ) & ( z `2 = y1 or z `2 = y2 ) )
proof end;

theorem Th20: :: MCART_1:20
for x being set st ex y, z being set st x = [y,z] holds
( x <> x `1 & x <> x `2 )
proof end;

theorem Th21: :: MCART_1:21
canceled;

theorem Th22: :: MCART_1:22
canceled;

theorem Th23: :: MCART_1:23
for x, X, Y being set st x in [:X,Y:] holds
x = [(x `1 ),(x `2 )]
proof end;

theorem Th24: :: MCART_1:24
for X, Y being set st X <> {} & Y <> {} holds
for x being Element of [:X,Y:] holds x = [(x `1 ),(x `2 )]
proof end;

Lemma79: for X1, X2 being set st X1 <> {} & X2 <> {} holds
for x being Element of [:X1,X2:] ex xx1 being Element of X1 ex xx2 being Element of X2 st x = [xx1,xx2]
proof end;

theorem Th25: :: MCART_1:25
for x1, x2, y1, y2 being set holds [:{x1,x2},{y1,y2}:] = {[x1,y1],[x1,y2],[x2,y1],[x2,y2]}
proof end;

theorem Th26: :: MCART_1:26
for X, Y being set st X <> {} & Y <> {} holds
for x being Element of [:X,Y:] holds
( x <> x `1 & x <> x `2 )
proof end;

definition
let x1 be set ;
let x2 be set ;
let x3 be set ;
func [c1,c2,c3] -> set equals :: MCART_1:def 3
[[x1,x2],x3];
coherence
[[x1,x2],x3] is set
;
end;

:: deftheorem Def3 defines [ MCART_1:def 3 :
for x1, x2, x3 being set holds [x1,x2,x3] = [[x1,x2],x3];

theorem Th27: :: MCART_1:27
canceled;

theorem Th28: :: MCART_1:28
for x1, x2, x3, y1, y2, y3 being set st [x1,x2,x3] = [y1,y2,y3] holds
( x1 = y1 & x2 = y2 & x3 = y3 )
proof end;

theorem Th29: :: MCART_1:29
for X being set st X <> {} holds
ex v being set st
( v in X & ( for x, y, z being set holds
( ( not x in X & not y in X ) or not v = [x,y,z] ) ) )
proof end;

definition
let x1 be set ;
let x2 be set ;
let x3 be set ;
let x4 be set ;
func [c1,c2,c3,c4] -> set equals :: MCART_1:def 4
[[x1,x2,x3],x4];
coherence
[[x1,x2,x3],x4] is set
;
end;

:: deftheorem Def4 defines [ MCART_1:def 4 :
for x1, x2, x3, x4 being set holds [x1,x2,x3,x4] = [[x1,x2,x3],x4];

theorem Th30: :: MCART_1:30
canceled;

theorem Th31: :: MCART_1:31
for x1, x2, x3, x4 being set holds [x1,x2,x3,x4] = [[[x1,x2],x3],x4] ;

theorem Th32: :: MCART_1:32
for x1, x2, x3, x4 being set holds [x1,x2,x3,x4] = [[x1,x2],x3,x4] ;

theorem Th33: :: MCART_1:33
for x1, x2, x3, x4, y1, y2, y3, y4 being set st [x1,x2,x3,x4] = [y1,y2,y3,y4] holds
( x1 = y1 & x2 = y2 & x3 = y3 & x4 = y4 )
proof end;

theorem Th34: :: MCART_1:34
for X being set st X <> {} holds
ex v being set st
( v in X & ( for x1, x2, x3, x4 being set holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4] ) ) )
proof end;

theorem Th35: :: MCART_1:35
for X1, X2, X3 being set holds
( ( X1 <> {} & X2 <> {} & X3 <> {} ) iff [:X1,X2,X3:] <> {} )
proof end;

theorem Th36: :: MCART_1:36
for X1, X2, X3, Y1, Y2, Y3 being set st X1 <> {} & X2 <> {} & X3 <> {} & [:X1,X2,X3:] = [:Y1,Y2,Y3:] holds
( X1 = Y1 & X2 = Y2 & X3 = Y3 )
proof end;

theorem Th37: :: MCART_1:37
for X1, X2, X3, Y1, Y2, Y3 being set st [:X1,X2,X3:] <> {} & [:X1,X2,X3:] = [:Y1,Y2,Y3:] holds
( X1 = Y1 & X2 = Y2 & X3 = Y3 )
proof end;

theorem Th38: :: MCART_1:38
for X, Y being set st [:X,X,X:] = [:Y,Y,Y:] holds
X = Y
proof end;

Lemma91: for X1, X2, X3 being set st X1 <> {} & X2 <> {} & X3 <> {} holds
for x being Element of [:X1,X2,X3:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 st x = [xx1,xx2,xx3]
proof end;

theorem Th39: :: MCART_1:39
for x1, x2, x3 being set holds [:{x1},{x2},{x3}:] = {[x1,x2,x3]}
proof end;

theorem Th40: :: MCART_1:40
for x1, y1, x2, x3 being set holds [:{x1,y1},{x2},{x3}:] = {[x1,x2,x3],[y1,x2,x3]}
proof end;

theorem Th41: :: MCART_1:41
for x1, x2, y2, x3 being set holds [:{x1},{x2,y2},{x3}:] = {[x1,x2,x3],[x1,y2,x3]}
proof end;

theorem Th42: :: MCART_1:42
for x1, x2, x3, y3 being set holds [:{x1},{x2},{x3,y3}:] = {[x1,x2,x3],[x1,x2,y3]}
proof end;

theorem Th43: :: MCART_1:43
for x1, y1, x2, y2, x3 being set holds [:{x1,y1},{x2,y2},{x3}:] = {[x1,x2,x3],[y1,x2,x3],[x1,y2,x3],[y1,y2,x3]}
proof end;

theorem Th44: :: MCART_1:44
for x1, y1, x2, x3, y3 being set holds [:{x1,y1},{x2},{x3,y3}:] = {[x1,x2,x3],[y1,x2,x3],[x1,x2,y3],[y1,x2,y3]}
proof end;

theorem Th45: :: MCART_1:45
for x1, x2, y2, x3, y3 being set holds [:{x1},{x2,y2},{x3,y3}:] = {[x1,x2,x3],[x1,y2,x3],[x1,x2,y3],[x1,y2,y3]}
proof end;

theorem Th46: :: MCART_1:46
for x1, y1, x2, y2, x3, y3 being set holds [:{x1,y1},{x2,y2},{x3,y3}:] = {[x1,x2,x3],[x1,y2,x3],[x1,x2,y3],[x1,y2,y3],[y1,x2,x3],[y1,y2,x3],[y1,x2,y3],[y1,y2,y3]}
proof end;

definition
let X1 be set ;
let X2 be set ;
let X3 be set ;
assume E39: ( X1 <> {} & X2 <> {} & X3 <> {} ) ;
let x be Element of [:X1,X2,X3:];
func c4 `1 -> Element of a1 means :Def5: :: MCART_1:def 5
for x1, x2, x3 being set st x = [x1,x2,x3] holds
it = x1;
existence
ex b1 being Element of X1 st
for x1, x2, x3 being set st x = [x1,x2,x3] holds
b1 = x1
proof end;
uniqueness
for b1, b2 being Element of X1 st ( for x1, x2, x3 being set st x = [x1,x2,x3] holds
b1 = x1 ) & ( for x1, x2, x3 being set st x = [x1,x2,x3] holds
b2 = x1 ) holds
b1 = b2
proof end;
func c4 `2 -> Element of a2 means :Def6: :: MCART_1:def 6
for x1, x2, x3 being set st x = [x1,x2,x3] holds
it = x2;
existence
ex b1 being Element of X2 st
for x1, x2, x3 being set st x = [x1,x2,x3] holds
b1 = x2
proof end;
uniqueness
for b1, b2 being Element of X2 st ( for x1, x2, x3 being set st x = [x1,x2,x3] holds
b1 = x2 ) & ( for x1, x2, x3 being set st x = [x1,x2,x3] holds
b2 = x2 ) holds
b1 = b2
proof end;
func c4 `3 -> Element of a3 means :Def7: :: MCART_1:def 7
for x1, x2, x3 being set st x = [x1,x2,x3] holds
it = x3;
existence
ex b1 being Element of X3 st
for x1, x2, x3 being set st x = [x1,x2,x3] holds
b1 = x3
proof end;
uniqueness
for b1, b2 being Element of X3 st ( for x1, x2, x3 being set st x = [x1,x2,x3] holds
b1 = x3 ) & ( for x1, x2, x3 being set st x = [x1,x2,x3] holds
b2 = x3 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines `1 MCART_1:def 5 :
for X1, X2, X3 being set st X1 <> {} & X2 <> {} & X3 <> {} holds
for x being Element of [:X1,X2,X3:]
for b5 being Element of X1 holds
( b5 = x `1 iff for x1, x2, x3 being set st x = [x1,x2,x3] holds
b5 = x1 );

:: deftheorem Def6 defines `2 MCART_1:def 6 :
for X1, X2, X3 being set st X1 <> {} & X2 <> {} & X3 <> {} holds
for x being Element of [:X1,X2,X3:]
for b5 being Element of X2 holds
( b5 = x `2 iff for x1, x2, x3 being set st x = [x1,x2,x3] holds
b5 = x2 );

:: deftheorem Def7 defines `3 MCART_1:def 7 :
for X1, X2, X3 being set st X1 <> {} & X2 <> {} & X3 <> {} holds
for x being Element of [:X1,X2,X3:]
for b5 being Element of X3 holds
( b5 = x `3 iff for x1, x2, x3 being set st x = [x1,x2,x3] holds
b5 = x3 );

theorem Th47: :: MCART_1:47
for X1, X2, X3 being set st X1 <> {} & X2 <> {} & X3 <> {} holds
for x being Element of [:X1,X2,X3:]
for x1, x2, x3 being set st x = [x1,x2,x3] holds
( x `1 = x1 & x `2 = x2 & x `3 = x3 ) by , , ;

theorem Th48: :: MCART_1:48
for X1, X2, X3 being set st X1 <> {} & X2 <> {} & X3 <> {} holds
for x being Element of [:X1,X2,X3:] holds x = [(x `1 ),(x `2 ),(x `3 )]
proof end;

theorem Th49: :: MCART_1:49
for X, Y, Z being set st ( X c= [:X,Y,Z:] or X c= [:Y,Z,X:] or X c= [:Z,X,Y:] ) holds
X = {}
proof end;

theorem Th50: :: MCART_1:50
for X1, X2, X3 being set st X1 <> {} & X2 <> {} & X3 <> {} holds
for x being Element of [:X1,X2,X3:] holds
( x `1 = (x `1 ) `1 & x `2 = (x `1 ) `2 & x `3 = x `2 )
proof end;

theorem Th51: :: MCART_1:51
for X1, X2, X3 being set st X1 <> {} & X2 <> {} & X3 <> {} holds
for x being Element of [:X1,X2,X3:] holds
( x <> x `1 & x <> x `2 & x <> x `3 )
proof end;

theorem Th52: :: MCART_1:52
for X1, X2, X3, Y1, Y2, Y3 being set st [:X1,X2,X3:] meets [:Y1,Y2,Y3:] holds
( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 )
proof end;

theorem Th53: :: MCART_1:53
for X1, X2, X3, X4 being set holds [:X1,X2,X3,X4:] = [:[:[:X1,X2:],X3:],X4:]
proof end;

theorem Th54: :: MCART_1:54
for X1, X2, X3, X4 being set holds [:[:X1,X2:],X3,X4:] = [:X1,X2,X3,X4:]
proof end;

theorem Th55: :: MCART_1:55
for X1, X2, X3, X4 being set holds
( ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} ) iff [:X1,X2,X3,X4:] <> {} )
proof end;

theorem Th56: :: MCART_1:56
for X1, X2, X3, X4, Y1, Y2, Y3, Y4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] holds
( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 )
proof end;

theorem Th57: :: MCART_1:57
for X1, X2, X3, X4, Y1, Y2, Y3, Y4 being set st [:X1,X2,X3,X4:] <> {} & [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] holds
( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 )
proof end;

theorem Th58: :: MCART_1:58
for X, Y being set st [:X,X,X,X:] = [:Y,Y,Y,Y:] holds
X = Y
proof end;

Lemma109: for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:] ex xx1 being Element of X1 ex xx2 being Element of X2 ex xx3 being Element of X3 ex xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4]
proof end;

definition
let X1 be set ;
let X2 be set ;
let X3 be set ;
let X4 be set ;
assume E39: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} ) ;
let x be Element of [:X1,X2,X3,X4:];
func c5 `1 -> Element of a1 means :Def8: :: MCART_1:def 8
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
it = x1;
existence
ex b1 being Element of X1 st
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b1 = x1
proof end;
uniqueness
for b1, b2 being Element of X1 st ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b1 = x1 ) & ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b2 = x1 ) holds
b1 = b2
proof end;
func c5 `2 -> Element of a2 means :Def9: :: MCART_1:def 9
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
it = x2;
existence
ex b1 being Element of X2 st
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b1 = x2
proof end;
uniqueness
for b1, b2 being Element of X2 st ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b1 = x2 ) & ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b2 = x2 ) holds
b1 = b2
proof end;
func c5 `3 -> Element of a3 means :Def10: :: MCART_1:def 10
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
it = x3;
existence
ex b1 being Element of X3 st
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b1 = x3
proof end;
uniqueness
for b1, b2 being Element of X3 st ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b1 = x3 ) & ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b2 = x3 ) holds
b1 = b2
proof end;
func c5 `4 -> Element of a4 means :Def11: :: MCART_1:def 11
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
it = x4;
existence
ex b1 being Element of X4 st
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b1 = x4
proof end;
uniqueness
for b1, b2 being Element of X4 st ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b1 = x4 ) & ( for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b2 = x4 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines `1 MCART_1:def 8 :
for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:]
for b6 being Element of X1 holds
( b6 = x `1 iff for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b6 = x1 );

:: deftheorem Def9 defines `2 MCART_1:def 9 :
for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:]
for b6 being Element of X2 holds
( b6 = x `2 iff for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b6 = x2 );

:: deftheorem Def10 defines `3 MCART_1:def 10 :
for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:]
for b6 being Element of X3 holds
( b6 = x `3 iff for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b6 = x3 );

:: deftheorem Def11 defines `4 MCART_1:def 11 :
for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:]
for b6 being Element of X4 holds
( b6 = x `4 iff for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b6 = x4 );

theorem Th59: :: MCART_1:59
for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:]
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
( x `1 = x1 & x `2 = x2 & x `3 = x3 & x `4 = x4 ) by , , Th1, ;

theorem Th60: :: MCART_1:60
for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:] holds x = [(x `1 ),(x `2 ),(x `3 ),(x `4 )]
proof end;

theorem Th61: :: MCART_1:61
for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:] holds
( x `1 = ((x `1 ) `1 ) `1 & x `2 = ((x `1 ) `1 ) `2 & x `3 = (x `1 ) `2 & x `4 = x `2 )
proof end;

theorem Th62: :: MCART_1:62
for X1, X2, X3, X4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x being Element of [:X1,X2,X3,X4:] holds
( x <> x `1 & x <> x `2 & x <> x `3 & x <> x `4 )
proof end;

theorem Th63: :: MCART_1:63
for X1, X2, X3, X4 being set st ( X1 c= [:X1,X2,X3,X4:] or X1 c= [:X2,X3,X4,X1:] or X1 c= [:X3,X4,X1,X2:] or X1 c= [:X4,X1,X2,X3:] ) holds
X1 = {}
proof end;

theorem Th64: :: MCART_1:64
for X1, X2, X3, X4, Y1, Y2, Y3, Y4 being set st [:X1,X2,X3,X4:] meets [:Y1,Y2,Y3,Y4:] holds
( X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 )
proof end;

theorem Th65: :: MCART_1:65
for x1, x2, x3, x4 being set holds [:{x1},{x2},{x3},{x4}:] = {[x1,x2,x3,x4]}
proof end;

theorem Th66: :: MCART_1:66
for X, Y being set st [:X,Y:] <> {} holds
for x being Element of [:X,Y:] holds
( x <> x `1 & x <> x `2 )
proof end;

theorem Th67: :: MCART_1:67
for x, X, Y being set st x in [:X,Y:] holds
( x <> x `1 & x <> x `2 ) by ;

theorem Th68: :: MCART_1:68
for X1, X2, X3 being set
for x being Element of [:X1,X2,X3:] st X1 <> {} & X2 <> {} & X3 <> {} holds
for x1, x2, x3 being set st x = [x1,x2,x3] holds
( x `1 = x1 & x `2 = x2 & x `3 = x3 ) by , , ;

theorem Th69: :: MCART_1:69
for X1, X2, X3, y1 being set
for x being Element of [:X1,X2,X3:] st X1 <> {} & X2 <> {} & X3 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3 st x = [xx1,xx2,xx3] holds
y1 = xx1 ) holds
y1 = x `1
proof end;

theorem Th70: :: MCART_1:70
for X1, X2, X3, y2 being set
for x being Element of [:X1,X2,X3:] st X1 <> {} & X2 <> {} & X3 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3 st x = [xx1,xx2,xx3] holds
y2 = xx2 ) holds
y2 = x `2
proof end;

theorem Th71: :: MCART_1:71
for X1, X2, X3, y3 being set
for x being Element of [:X1,X2,X3:] st X1 <> {} & X2 <> {} & X3 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3 st x = [xx1,xx2,xx3] holds
y3 = xx3 ) holds
y3 = x `3
proof end;

theorem Th72: :: MCART_1:72
for z, X1, X2, X3 being set st z in [:X1,X2,X3:] holds
ex x1, x2, x3 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] )
proof end;

theorem Th73: :: MCART_1:73
for x1, x2, x3, X1, X2, X3 being set holds
( [x1,x2,x3] in [:X1,X2,X3:] iff ( x1 in X1 & x2 in X2 & x3 in X3 ) )
proof end;

theorem Th74: :: MCART_1:74
for Z, X1, X2, X3 being set st ( for z being set holds
( z in Z iff ex x1, x2, x3 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] ) ) ) holds
Z = [:X1,X2,X3:]
proof end;

theorem Th75: :: MCART_1:75
for X1, X2, X3, Y1, Y2, Y3 being set st X1 <> {} & X2 <> {} & X3 <> {} & Y1 <> {} & Y2 <> {} & Y3 <> {} holds
for x being Element of [:X1,X2,X3:]
for y being Element of [:Y1,Y2,Y3:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 )
proof end;

theorem Th76: :: MCART_1:76
for X1, X2, X3 being set
for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3
for x being Element of [:X1,X2,X3:] st x in [:A1,A2,A3:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 )
proof end;

theorem Th77: :: MCART_1:77
for X1, Y1, X2, Y2, X3, Y3 being set st X1 c= Y1 & X2 c= Y2 & X3 c= Y3 holds
[:X1,X2,X3:] c= [:Y1,Y2,Y3:]
proof end;

theorem Th78: :: MCART_1:78
for X1, X2, X3, X4 being set
for x being Element of [:X1,X2,X3,X4:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} holds
for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
( x `1 = x1 & x `2 = x2 & x `3 = x3 & x `4 = x4 ) by , , Th1, ;

theorem Th79: :: MCART_1:79
for X1, X2, X3, X4, y1 being set
for x being Element of [:X1,X2,X3,X4:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4] holds
y1 = xx1 ) holds
y1 = x `1
proof end;

theorem Th80: :: MCART_1:80
for X1, X2, X3, X4, y2 being set
for x being Element of [:X1,X2,X3,X4:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4] holds
y2 = xx2 ) holds
y2 = x `2
proof end;

theorem Th81: :: MCART_1:81
for X1, X2, X3, X4, y3 being set
for x being Element of [:X1,X2,X3,X4:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4] holds
y3 = xx3 ) holds
y3 = x `3
proof end;

theorem Th82: :: MCART_1:82
for X1, X2, X3, X4, y4 being set
for x being Element of [:X1,X2,X3,X4:] st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & ( for xx1 being Element of X1
for xx2 being Element of X2
for xx3 being Element of X3
for xx4 being Element of X4 st x = [xx1,xx2,xx3,xx4] holds
y4 = xx4 ) holds
y4 = x `4
proof end;

theorem Th83: :: MCART_1:83
for z, X1, X2, X3, X4 being set st z in [:X1,X2,X3,X4:] holds
ex x1, x2, x3, x4 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & z = [x1,x2,x3,x4] )
proof end;

theorem Th84: :: MCART_1:84
for x1, x2, x3, x4, X1, X2, X3, X4 being set holds
( [x1,x2,x3,x4] in [:X1,X2,X3,X4:] iff ( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 ) )
proof end;

theorem Th85: :: MCART_1:85
for Z, X1, X2, X3, X4 being set st ( for z being set holds
( z in Z iff ex x1, x2, x3, x4 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & z = [x1,x2,x3,x4] ) ) ) holds
Z = [:X1,X2,X3,X4:]
proof end;

theorem Th86: :: MCART_1:86
for X1, X2, X3, X4, Y1, Y2, Y3, Y4 being set st X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & Y1 <> {} & Y2 <> {} & Y3 <> {} & Y4 <> {} holds
for x being Element of [:X1,X2,X3,X4:]
for y being Element of [:Y1,Y2,Y3,Y4:] st x = y holds
( x `1 = y `1 & x `2 = y `2 & x `3 = y `3 & x `4 = y `4 )
proof end;

theorem Th87: :: MCART_1:87
for X1, X2, X3, X4 being set
for A1 being Subset of X1
for A2 being Subset of X2
for A3 being Subset of X3
for A4 being Subset of X4
for x being Element of [:X1,X2,X3,X4:] st x in [:A1,A2,A3,A4:] holds
( x `1 in A1 & x `2 in A2 & x `3 in A3 & x `4 in A4 )
proof end;

theorem Th88: :: MCART_1:88
for X1, Y1, X2, Y2, X3, Y3, X4, Y4 being set st X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 holds
[:X1,X2,X3,X4:] c= [:Y1,Y2,Y3,Y4:]
proof end;

definition
let X1 be set ;
let X2 be set ;
let A1 be Subset of X1;
let A2 be Subset of X2;
redefine func [: as [:c3,c4:] -> Subset of [:a1,a2:];
coherence
[:A1,A2:] is Subset of [:X1,X2:]
by ZFMISC_1:119;
end;

definition
let X1 be set ;
let X2 be set ;
let X3 be set ;
let A1 be Subset of X1;
let A2 be Subset of X2;
let A3 be Subset of X3;
redefine func [: as [:c4,c5,c6:] -> Subset of [:a1,a2,a3:];
coherence
[:A1,A2,A3:] is Subset of [:X1,X2,X3:]
by ;
end;

definition
let X1 be set ;
let X2 be set ;
let X3 be set ;
let X4 be set ;
let A1 be Subset of X1;
let A2 be Subset of X2;
let A3 be Subset of X3;
let A4 be Subset of X4;
redefine func [: as [:c5,c6,c7,c8:] -> Subset of [:a1,a2,a3,a4:];
coherence
[:A1,A2,A3,A4:] is Subset of [:X1,X2,X3,X4:]
by ;
end;

definition
let f be Function;
func pr1 c1 -> Function means :: MCART_1:def 12
( dom it = dom f & ( for x being set st x in dom f holds
it . x = (f . x) `1 ) );
existence
ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom f holds
b1 . x = (f . x) `1 ) )
proof end;
uniqueness
for b1, b2 being Function st dom b1 = dom f & ( for x being set st x in dom f holds
b1 . x = (f . x) `1 ) & dom b2 = dom f & ( for x being set st x in dom f holds
b2 . x = (f . x) `1 ) holds
b1 = b2
proof end;
func pr2 c1 -> Function means :: MCART_1:def 13
( dom it = dom f & ( for x being set st x in dom f holds
it . x = (f . x) `2 ) );
existence
ex b1 being Function st
( dom b1 = dom f & ( for x being set st x in dom f holds
b1 . x = (f . x) `2 ) )
proof end;
uniqueness
for b1, b2 being Function st dom b1 = dom f & ( for x being set st x in dom f holds
b1 . x = (f . x) `2 ) & dom b2 = dom f & ( for x being set st x in dom f holds
b2 . x = (f . x) `2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines pr1 MCART_1:def 12 :
for f, b2 being Function holds
( b2 = pr1 f iff ( dom b2 = dom f & ( for x being set st x in dom f holds
b2 . x = (f . x) `1 ) ) );

:: deftheorem Def13 defines pr2 MCART_1:def 13 :
for f, b2 being Function holds
( b2 = pr2 f iff ( dom b2 = dom f & ( for x being set st x in dom f holds
b2 . x = (f . x) `2 ) ) );

definition
let x be set ;
func c1 `11 -> set equals :: MCART_1:def 14
(x `1 ) `1 ;
coherence
(x `1 ) `1 is set
;
func c1 `12 -> set equals :: MCART_1:def 15
(x `1 ) `2 ;
coherence
(x `1 ) `2 is set
;
func c1 `21 -> set equals :: MCART_1:def 16
(x `2 ) `1 ;
coherence
(x `2 ) `1 is set
;
func c1 `22 -> set equals :: MCART_1:def 17
(x `2 ) `2 ;
coherence
(x `2 ) `2 is set
;
end;

:: deftheorem Def14 defines `11 MCART_1:def 14 :
for x being set holds x `11 = (x `1 ) `1 ;

:: deftheorem Def15 defines `12 MCART_1:def 15 :
for x being set holds x `12 = (x `1 ) `2 ;

:: deftheorem Def16 defines `21 MCART_1:def 16 :
for x being set holds x `21 = (x `2 ) `1 ;

:: deftheorem Def17 defines `22 MCART_1:def 17 :
for x being set holds x `22 = (x `2 ) `2 ;

theorem Th89: :: MCART_1:89
for x1, x2, y, y1, y2, x being set holds
( [[x1,x2],y] `11 = x1 & [[x1,x2],y] `12 = x2 & [x,[y1,y2]] `21 = y1 & [x,[y1,y2]] `22 = y2 )
proof end;