:: Tuples, Projections and Cartesian Products
:: by Andrzej Trybulec
::
:: Received March 30, 1989
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

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 :: 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 :: 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, x2 being set such that A1: x = [x1,x2] ;
func x `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 x `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 b<sub>2</sub> being set holds
( b<sub>2</sub> = 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 b<sub>2</sub> being set holds
( b<sub>2</sub> = x `2 iff for y1, y2 being set st x = [y1,y2] holds
b₂ = y2 );

theorem :: MCART_1:7

for x, y being set holds
( [x,y] `1 = x & [x,y] `2 = y ) by Def1, Def2;

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 :: 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 :: 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 :: 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 :: 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 :: MCART_1:14

for z, x, y being set st z in [:{x},{y}:] holds
( z `1 = x & z `2 = y )

proof end;

theorem :: 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 :: 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 :: 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 :: 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 :: 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 :: MCART_1:21

canceled;

theorem :: MCART_1:22

canceled;

theorem Th23: :: MCART_1:23

for x being set
for R being Relation st x in R 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;

Lm1: 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, x2, x3 be set ;
func [x1,x2,x3] -> set equals :: MCART_1:def 3
[[x1,x2],x3];
coherence
[[x1,x2],x3] is set ;

end;

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

theorem :: 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, x2, x3, x4 be set ;
func [x1,x2,x3,x4] -> set equals :: MCART_1:def 4
[[x1,x2,x3],x4];
coherence
[[x1,x2,x3],x4] is set ;

end;

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

theorem :: MCART_1:30

canceled;

theorem :: MCART_1:31

for x1, x2, x3, x4 being set holds [x1,x2,x3,x4] = [[[x1,x2],x3],x4] ;

theorem :: 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 :: 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 :: MCART_1:38

for X, Y being set st [:X,X,X:] = [:Y,Y,Y:] holds
X = Y

proof end;

Lm2: 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 :: MCART_1:40

for x1, y1, x2, x3 being set holds [:{x1,y1},{x2},{x3}:] = {[x1,x2,x3],[y1,x2,x3]}

proof end;

theorem :: MCART_1:41

for x1, x2, y2, x3 being set holds [:{x1},{x2,y2},{x3}:] = {[x1,x2,x3],[x1,y2,x3]}

proof end;

theorem :: MCART_1:42

for x1, x2, x3, y3 being set holds [:{x1},{x2},{x3,y3}:] = {[x1,x2,x3],[x1,x2,y3]}

proof end;

theorem :: 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 :: 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 :: 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 :: 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, X2, X3 be set ;
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} ) ;
let x be Element of [:X1,X2,X3:];
func x `1 -> Element of X1 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 x `2 -> Element of X2 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 x `3 -> Element of X3 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 b<sub>5</sub> being Element of X1 holds
( b<sub>5</sub> = 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 b<sub>5</sub> being Element of X2 holds
( b<sub>5</sub> = 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 b<sub>5</sub> being Element of X3 holds
( b<sub>5</sub> = x `3 iff for x1, x2, x3 being set st x = [x1,x2,x3] holds
b₅ = x3 );

theorem :: 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 Def5, Def6, Def7;

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 :: 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 :: 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 :: MCART_1:58

for X, Y being set st [:X,X,X,X:] = [:Y,Y,Y,Y:] holds
X = Y

proof end;

Lm3: 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, X2, X3, X4 be set ;
assume A1: ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} ) ;
let x be Element of [:X1,X2,X3,X4:];
func x `1 -> Element of X1 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 x `2 -> Element of X2 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 x `3 -> Element of X3 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 x `4 -> Element of X4 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 b<sub>6</sub> being Element of X1 holds
( b<sub>6</sub> = 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 b<sub>6</sub> being Element of X2 holds
( b<sub>6</sub> = 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 b<sub>6</sub> being Element of X3 holds
( b<sub>6</sub> = 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 b<sub>6</sub> being Element of X4 holds
( b<sub>6</sub> = x `4 iff for x1, x2, x3, x4 being set st x = [x1,x2,x3,x4] holds
b₆ = x4 );

theorem :: 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 Def8, Def9, Def10, Def11;

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 :: 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 :: 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 :: 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 :: 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 :: MCART_1:67

for x, X, Y being set st x in [:X,Y:] holds
( x <> x `1 & x <> x `2 ) by Th66;

theorem :: 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 Def5, Def6, Def7;

theorem :: 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 :: 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 :: 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 :: 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 :: 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 :: 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 Def8, Def9, Def10, Def11;

theorem :: 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 :: 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 :: 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 :: 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 :: 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 :: 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 :: 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 :: 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, X2 be set ;
let A1 be Subset of X1;
let A2 be Subset of X2;
:: original: [:
redefine func [:A1,A2:] -> Subset of [:X1,X2:];
coherence
[:A1,A2:] is Subset of [:X1,X2:] by ZFMISC_1:119;

end;

definition

let X1, X2, X3 be set ;
let A1 be Subset of X1;
let A2 be Subset of X2;
let A3 be Subset of X3;
:: original: [:
redefine func [:A1,A2,A3:] -> Subset of [:X1,X2,X3:];
coherence
[:A1,A2,A3:] is Subset of [:X1,X2,X3:] by Th77;

end;

definition

let X1, X2, X3, 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;
:: original: [:
redefine func [:A1,A2,A3,A4:] -> Subset of [:X1,X2,X3,X4:];
coherence
[:A1,A2,A3,A4:] is Subset of [:X1,X2,X3,X4:] by Th88;

end;

begin

definition

let f be Function;
func pr1 f -> 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 b<sub>1</sub> being Function st
( dom b<sub>1</sub> = dom f & ( for x being set st x in dom f holds
b<sub>1</sub> . 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 b<sub>2</sub> = dom f & ( for x being set st x in dom f holds
b<sub>2</sub> . x = (f . x) `1 ) holds
b₁ = b₂

proof end;

func pr2 f -> 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 b<sub>1</sub> being Function st
( dom b<sub>1</sub> = dom f & ( for x being set st x in dom f holds
b<sub>1</sub> . 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 b<sub>2</sub> = dom f & ( for x being set st x in dom f holds
b<sub>2</sub> . x = (f . x) `2 ) holds
b₁ = b₂

proof end;

end;

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

end;

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

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

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

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

theorem :: 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;

theorem :: MCART_1:90

canceled;

theorem :: MCART_1:91

for R being Relation
for x being set st x in R holds
( x `1 in dom R & x `2 in rng R )

proof end;

theorem Th92: :: MCART_1:92

for R being non empty Relation
for x being set holds Im (R,x) = { (I `2) where I is Element of R : I `1 = x }

proof end;

theorem :: MCART_1:93

for R being Relation
for x being set st x in R holds
x `2 in Im (R,(x `1))

proof end;

theorem Th94: :: MCART_1:94

for y being set
for R being Relation
for x being set st x in R & y in R & x `1 = y `1 & x `2 = y `2 holds
x = y

proof end;

theorem :: MCART_1:95

for R being non empty Relation
for x, y being Element of R st x `1 = y `1 & x `2 = y `2 holds
x = y by Th94;

theorem :: MCART_1:96

for x1, x2, x3, y1, y2, y3 being set holds proj1 (proj1 {[x1,x2,x3],[y1,y2,y3]}) = {x1,y1}

proof end;

theorem :: MCART_1:97

for x1, x2, x3 being set holds proj1 (proj1 {[x1,x2,x3]}) = {x1}

proof end;