:: MCART_1 semantic presentation
theorem Th1: :: MCART_1:1
theorem Th2: :: MCART_1:2
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 ) )
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 ) )
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 ) )
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 ) )
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
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
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
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
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
theorem Th8: :: MCART_1:8
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] ) ) )
theorem Th10: :: MCART_1:10
theorem Th11: :: MCART_1:11
theorem Th12: :: MCART_1:12
theorem Th13: :: MCART_1:13
theorem Th14: :: MCART_1:14
theorem Th15: :: MCART_1:15
theorem Th16: :: MCART_1:16
theorem Th17: :: MCART_1:17
theorem Th18: :: MCART_1:18
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 ) )
theorem Th20: :: MCART_1:20
theorem Th21: :: MCART_1:21
canceled;
theorem Th22: :: MCART_1:22
canceled;
theorem Th23: :: MCART_1:23
theorem Th24: :: MCART_1:24
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]
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]}
theorem Th26: :: MCART_1:26
:: 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 )
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] ) ) )
:: 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 )
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] ) ) )
theorem Th35: :: MCART_1:35
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 )
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 )
theorem Th38: :: MCART_1:38
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]
theorem Th39: :: MCART_1:39
theorem Th40: :: MCART_1:40
for
x1,
y1,
x2,
x3 being
set holds
[:{x1,y1},{x2},{x3}:] = {[x1,x2,x3],[y1,x2,x3]}
theorem Th41: :: MCART_1:41
for
x1,
x2,
y2,
x3 being
set holds
[:{x1},{x2,y2},{x3}:] = {[x1,x2,x3],[x1,y2,x3]}
theorem Th42: :: MCART_1:42
for
x1,
x2,
x3,
y3 being
set holds
[:{x1},{x2},{x3,y3}:] = {[x1,x2,x3],[x1,x2,y3]}
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]}
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]}
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]}
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]}
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
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
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
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
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
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
end;
:: deftheorem Def5 defines `1 MCART_1:def 5 :
:: deftheorem Def6 defines `2 MCART_1:def 6 :
:: deftheorem Def7 defines `3 MCART_1:def 7 :
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
theorem Th49: :: MCART_1:49
theorem Th50: :: MCART_1:50
theorem Th51: :: MCART_1:51
theorem Th52: :: MCART_1:52
theorem Th53: :: MCART_1:53
theorem Th54: :: MCART_1:54
theorem Th55: :: MCART_1:55
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 )
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 )
theorem Th58: :: MCART_1:58
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]
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
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
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
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
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
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
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
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
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
theorem Th61: :: MCART_1:61
theorem Th62: :: MCART_1:62
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 = {}
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 )
theorem Th65: :: MCART_1:65
theorem Th66: :: MCART_1:66
theorem Th67: :: MCART_1:67
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
theorem Th70: :: MCART_1:70
theorem Th71: :: MCART_1:71
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] )
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 ) )
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:]
theorem Th75: :: MCART_1:75
theorem Th76: :: MCART_1:76
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:]
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
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
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
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
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] )
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 ) )
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:]
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 )
theorem Th87: :: MCART_1:87
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:]
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;
:: deftheorem Def12 defines pr1 MCART_1:def 12 :
:: deftheorem Def13 defines pr2 MCART_1:def 13 :
:: deftheorem Def14 defines `11 MCART_1:def 14 :
:: deftheorem Def15 defines `12 MCART_1:def 15 :
:: deftheorem Def16 defines `21 MCART_1:def 16 :
:: deftheorem Def17 defines `22 MCART_1:def 17 :
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 )