:: MCART_1 semantic presentation
theorem Th1: :: MCART_1:1
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & b
2 misses b
1 ) ) )
theorem Th2: :: MCART_1:2
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3 being
set holds
( b
3 in b
2 implies b
3 misses b
1 ) ) ) ) )
theorem Th3: :: MCART_1:3
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4 being
set holds
( b
3 in b
4 & b
4 in b
2 implies b
3 misses b
1 ) ) ) ) )
theorem Th4: :: MCART_1:4
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4, b
5 being
set holds
( b
3 in b
4 & b
4 in b
5 & b
5 in b
2 implies b
3 misses b
1 ) ) ) ) )
theorem Th5: :: MCART_1:5
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4, b
5, b
6 being
set holds
( b
3 in b
4 & b
4 in b
5 & b
5 in b
6 & b
6 in b
2 implies b
3 misses b
1 ) ) ) ) )
theorem Th6: :: MCART_1:6
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4, b
5, b
6, b
7 being
set holds
( b
3 in b
4 & b
4 in b
5 & b
5 in b
6 & b
6 in b
7 & b
7 in b
2 implies b
3 misses b
1 ) ) ) ) )
definition
let c
1 be
set ;
given c
2, c
3 being
set such that E5:
c
1 = [c2,c3]
;
func c
1 `1 -> set means :
Def1:
:: MCART_1:def 1
for b
1, b
2 being
set holds
( a
1 = [b1,b2] implies a
2 = b
1 );
existence
ex b1 being set st
for b2, b3 being set holds
( c1 = [b2,b3] implies b1 = b2 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4 being set holds
( c1 = [b3,b4] implies b1 = b3 ) ) & ( for b3, b4 being set holds
( c1 = [b3,b4] implies b2 = b3 ) ) implies b1 = b2 )
func c
1 `2 -> set means :
Def2:
:: MCART_1:def 2
for b
1, b
2 being
set holds
( a
1 = [b1,b2] implies a
2 = b
2 );
existence
ex b1 being set st
for b2, b3 being set holds
( c1 = [b2,b3] implies b1 = b3 )
uniqueness
for b1, b2 being set holds
( ( for b3, b4 being set holds
( c1 = [b3,b4] implies b1 = b4 ) ) & ( for b3, b4 being set holds
( c1 = [b3,b4] implies b2 = b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines `1 MCART_1:def 1 :
for b
1 being
set holds
( ex b
2, b
3 being
set st b
1 = [b2,b3] implies for b
2 being
set holds
( b
2 = b
1 `1 iff for b
3, b
4 being
set holds
( b
1 = [b3,b4] implies b
2 = b
3 ) ) );
:: deftheorem Def2 defines `2 MCART_1:def 2 :
for b
1 being
set holds
( ex b
2, b
3 being
set st b
1 = [b2,b3] implies for b
2 being
set holds
( b
2 = b
1 `2 iff for b
3, b
4 being
set holds
( b
1 = [b3,b4] implies b
2 = b
4 ) ) );
theorem Th7: :: MCART_1:7
theorem Th8: :: MCART_1:8
for b
1 being
set holds
( ex b
2, b
3 being
set st b
1 = [b2,b3] implies
[(b1 `1 ),(b1 `2 )] = b
1 )
theorem Th9: :: MCART_1:9
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4 being
set holds
not ( ( b
3 in b
1 or b
4 in b
1 ) & b
2 = [b3,b4] ) ) ) ) )
theorem Th10: :: MCART_1:10
for b
1, b
2, b
3 being
set holds
( b
1 in [:b2,b3:] implies ( b
1 `1 in b
2 & b
1 `2 in b
3 ) )
theorem Th11: :: MCART_1:11
for b
1, b
2, b
3 being
set holds
( ex b
4, b
5 being
set st b
1 = [b4,b5] & b
1 `1 in b
2 & b
1 `2 in b
3 implies b
1 in [:b2,b3:] )
theorem Th12: :: MCART_1:12
theorem Th13: :: MCART_1:13
theorem Th14: :: MCART_1:14
theorem Th15: :: MCART_1:15
for b
1, b
2, b
3, b
4 being
set holds
( b
1 in [:{b2,b3},b4:] implies ( ( b
1 `1 = b
2 or b
1 `1 = b
3 ) & b
1 `2 in b
4 ) )
theorem Th16: :: MCART_1:16
for b
1, b
2, b
3, b
4 being
set holds
( b
1 in [:b2,{b3,b4}:] implies ( b
1 `1 in b
2 & ( b
1 `2 = b
3 or b
1 `2 = b
4 ) ) )
theorem Th17: :: MCART_1:17
for b
1, b
2, b
3, b
4 being
set holds
( b
1 in [:{b2,b3},{b4}:] implies ( ( b
1 `1 = b
2 or b
1 `1 = b
3 ) & b
1 `2 = b
4 ) )
theorem Th18: :: MCART_1:18
for b
1, b
2, b
3, b
4 being
set holds
( b
1 in [:{b2},{b3,b4}:] implies ( b
1 `1 = b
2 & ( b
1 `2 = b
3 or b
1 `2 = b
4 ) ) )
theorem Th19: :: MCART_1:19
for b
1, b
2, b
3, b
4, b
5 being
set holds
( b
1 in [:{b2,b3},{b4,b5}:] implies ( ( b
1 `1 = b
2 or b
1 `1 = b
3 ) & ( b
1 `2 = b
4 or b
1 `2 = b
5 ) ) )
theorem Th20: :: MCART_1:20
for b
1 being
set holds
( ex b
2, b
3 being
set st b
1 = [b2,b3] implies ( b
1 <> b
1 `1 & b
1 <> b
1 `2 ) )
theorem Th21: :: MCART_1:21
canceled;
theorem Th22: :: MCART_1:22
canceled;
theorem Th23: :: MCART_1:23
for b
1, b
2, b
3 being
set holds
( b
1 in [:b2,b3:] implies b
1 = [(b1 `1 ),(b1 `2 )] )
theorem Th24: :: MCART_1:24
Lemma13:
for b1, b2 being set holds
( b1 <> {} & b2 <> {} implies for b3 being Element of [:b1,b2:] holds
ex b4 being Element of b1ex b5 being Element of b2 st b3 = [b4,b5] )
theorem Th25: :: MCART_1:25
for b
1, b
2, b
3, b
4 being
set holds
[:{b1,b2},{b3,b4}:] = {[b1,b3],[b1,b4],[b2,b3],[b2,b4]}
theorem Th26: :: MCART_1:26
:: deftheorem Def3 defines [ MCART_1:def 3 :
for b
1, b
2, b
3 being
set holds
[b1,b2,b3] = [[b1,b2],b3];
theorem Th27: :: MCART_1:27
canceled;
theorem Th28: :: MCART_1:28
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
(
[b1,b2,b3] = [b4,b5,b6] implies ( b
1 = b
4 & b
2 = b
5 & b
3 = b
6 ) )
theorem Th29: :: MCART_1:29
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4, b
5 being
set holds
not ( ( b
3 in b
1 or b
4 in b
1 ) & b
2 = [b3,b4,b5] ) ) ) ) )
definition
let c
1, c
2, c
3, c
4 be
set ;
func [c1,c2,c3,c4] -> set equals :: MCART_1:def 4
[[a1,a2,a3],a4];
coherence
[[c1,c2,c3],c4] is set
;
end;
:: deftheorem Def4 defines [ MCART_1:def 4 :
for b
1, b
2, b
3, b
4 being
set holds
[b1,b2,b3,b4] = [[b1,b2,b3],b4];
theorem Th30: :: MCART_1:30
canceled;
theorem Th31: :: MCART_1:31
for b
1, b
2, b
3, b
4 being
set holds
[b1,b2,b3,b4] = [[[b1,b2],b3],b4] ;
theorem Th32: :: MCART_1:32
for b
1, b
2, b
3, b
4 being
set holds
[b1,b2,b3,b4] = [[b1,b2],b3,b4] ;
theorem Th33: :: MCART_1:33
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
(
[b1,b2,b3,b4] = [b5,b6,b7,b8] implies ( b
1 = b
5 & b
2 = b
6 & b
3 = b
7 & b
4 = b
8 ) )
theorem Th34: :: MCART_1:34
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4, b
5, b
6 being
set holds
not ( ( b
3 in b
1 or b
4 in b
1 ) & b
2 = [b3,b4,b5,b6] ) ) ) ) )
theorem Th35: :: MCART_1:35
theorem Th36: :: MCART_1:36
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} &
[:b1,b2,b3:] = [:b4,b5,b6:] implies ( b
1 = b
4 & b
2 = b
5 & b
3 = b
6 ) )
theorem Th37: :: MCART_1:37
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
(
[:b1,b2,b3:] <> {} &
[:b1,b2,b3:] = [:b4,b5,b6:] implies ( b
1 = b
4 & b
2 = b
5 & b
3 = b
6 ) )
theorem Th38: :: MCART_1:38
for b
1, b
2 being
set holds
(
[:b1,b1,b1:] = [:b2,b2,b2:] implies b
1 = b
2 )
Lemma22:
for b1, b2, b3 being set holds
( b1 <> {} & b2 <> {} & b3 <> {} implies for b4 being Element of [:b1,b2,b3:] holds
ex b5 being Element of b1ex b6 being Element of b2ex b7 being Element of b3 st b4 = [b5,b6,b7] )
theorem Th39: :: MCART_1:39
theorem Th40: :: MCART_1:40
for b
1, b
2, b
3, b
4 being
set holds
[:{b1,b2},{b3},{b4}:] = {[b1,b3,b4],[b2,b3,b4]}
theorem Th41: :: MCART_1:41
for b
1, b
2, b
3, b
4 being
set holds
[:{b1},{b2,b3},{b4}:] = {[b1,b2,b4],[b1,b3,b4]}
theorem Th42: :: MCART_1:42
for b
1, b
2, b
3, b
4 being
set holds
[:{b1},{b2},{b3,b4}:] = {[b1,b2,b3],[b1,b2,b4]}
theorem Th43: :: MCART_1:43
for b
1, b
2, b
3, b
4, b
5 being
set holds
[:{b1,b2},{b3,b4},{b5}:] = {[b1,b3,b5],[b2,b3,b5],[b1,b4,b5],[b2,b4,b5]}
theorem Th44: :: MCART_1:44
for b
1, b
2, b
3, b
4, b
5 being
set holds
[:{b1,b2},{b3},{b4,b5}:] = {[b1,b3,b4],[b2,b3,b4],[b1,b3,b5],[b2,b3,b5]}
theorem Th45: :: MCART_1:45
for b
1, b
2, b
3, b
4, b
5 being
set holds
[:{b1},{b2,b3},{b4,b5}:] = {[b1,b2,b4],[b1,b3,b4],[b1,b2,b5],[b1,b3,b5]}
theorem Th46: :: MCART_1:46
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
[:{b1,b2},{b3,b4},{b5,b6}:] = {[b1,b3,b5],[b1,b4,b5],[b1,b3,b6],[b1,b4,b6],[b2,b3,b5],[b2,b4,b5],[b2,b3,b6],[b2,b4,b6]}
definition
let c
1, c
2, c
3 be
set ;
assume E24:
( c
1 <> {} & c
2 <> {} & c
3 <> {} )
;
let c
4 be
Element of
[:c1,c2,c3:];
func c
4 `1 -> Element of a
1 means :
Def5:
:: MCART_1:def 5
for b
1, b
2, b
3 being
set holds
( a
4 = [b1,b2,b3] implies a
5 = b
1 );
existence
ex b1 being Element of c1 st
for b2, b3, b4 being set holds
( c4 = [b2,b3,b4] implies b1 = b2 )
uniqueness
for b1, b2 being Element of c1 holds
( ( for b3, b4, b5 being set holds
( c4 = [b3,b4,b5] implies b1 = b3 ) ) & ( for b3, b4, b5 being set holds
( c4 = [b3,b4,b5] implies b2 = b3 ) ) implies b1 = b2 )
func c
4 `2 -> Element of a
2 means :
Def6:
:: MCART_1:def 6
for b
1, b
2, b
3 being
set holds
( a
4 = [b1,b2,b3] implies a
5 = b
2 );
existence
ex b1 being Element of c2 st
for b2, b3, b4 being set holds
( c4 = [b2,b3,b4] implies b1 = b3 )
uniqueness
for b1, b2 being Element of c2 holds
( ( for b3, b4, b5 being set holds
( c4 = [b3,b4,b5] implies b1 = b4 ) ) & ( for b3, b4, b5 being set holds
( c4 = [b3,b4,b5] implies b2 = b4 ) ) implies b1 = b2 )
func c
4 `3 -> Element of a
3 means :
Def7:
:: MCART_1:def 7
for b
1, b
2, b
3 being
set holds
( a
4 = [b1,b2,b3] implies a
5 = b
3 );
existence
ex b1 being Element of c3 st
for b2, b3, b4 being set holds
( c4 = [b2,b3,b4] implies b1 = b4 )
uniqueness
for b1, b2 being Element of c3 holds
( ( for b3, b4, b5 being set holds
( c4 = [b3,b4,b5] implies b1 = b5 ) ) & ( for b3, b4, b5 being set holds
( c4 = [b3,b4,b5] implies b2 = b5 ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines `1 MCART_1:def 5 :
for b
1, b
2, b
3 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} implies for b
4 being
Element of
[:b1,b2,b3:]for b
5 being
Element of b
1 holds
( b
5 = b
4 `1 iff for b
6, b
7, b
8 being
set holds
( b
4 = [b6,b7,b8] implies b
5 = b
6 ) ) );
:: deftheorem Def6 defines `2 MCART_1:def 6 :
for b
1, b
2, b
3 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} implies for b
4 being
Element of
[:b1,b2,b3:]for b
5 being
Element of b
2 holds
( b
5 = b
4 `2 iff for b
6, b
7, b
8 being
set holds
( b
4 = [b6,b7,b8] implies b
5 = b
7 ) ) );
:: deftheorem Def7 defines `3 MCART_1:def 7 :
for b
1, b
2, b
3 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} implies for b
4 being
Element of
[:b1,b2,b3:]for b
5 being
Element of b
3 holds
( b
5 = b
4 `3 iff for b
6, b
7, b
8 being
set holds
( b
4 = [b6,b7,b8] implies b
5 = b
8 ) ) );
theorem Th47: :: MCART_1:47
for b
1, b
2, b
3 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} implies for b
4 being
Element of
[:b1,b2,b3:]for b
5, b
6, b
7 being
set holds
( b
4 = [b5,b6,b7] implies ( b
4 `1 = b
5 & b
4 `2 = b
6 & b
4 `3 = b
7 ) ) )
by Def5, Def6, Def7;
theorem Th48: :: MCART_1:48
theorem Th49: :: MCART_1:49
for b
1, b
2, b
3 being
set holds
( not ( not b
1 c= [:b1,b2,b3:] & not b
1 c= [:b2,b3,b1:] & not b
1 c= [:b3,b1,b2:] ) implies b
1 = {} )
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 b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} &
[:b1,b2,b3,b4:] = [:b5,b6,b7,b8:] implies ( b
1 = b
5 & b
2 = b
6 & b
3 = b
7 & b
4 = b
8 ) )
theorem Th57: :: MCART_1:57
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
(
[:b1,b2,b3,b4:] <> {} &
[:b1,b2,b3,b4:] = [:b5,b6,b7,b8:] implies ( b
1 = b
5 & b
2 = b
6 & b
3 = b
7 & b
4 = b
8 ) )
theorem Th58: :: MCART_1:58
for b
1, b
2 being
set holds
(
[:b1,b1,b1,b1:] = [:b2,b2,b2,b2:] implies b
1 = b
2 )
Lemma35:
for b1, b2, b3, b4 being set holds
( b1 <> {} & b2 <> {} & b3 <> {} & b4 <> {} implies for b5 being Element of [:b1,b2,b3,b4:] holds
ex b6 being Element of b1ex b7 being Element of b2ex b8 being Element of b3ex b9 being Element of b4 st b5 = [b6,b7,b8,b9] )
definition
let c
1, c
2, c
3, c
4 be
set ;
assume E36:
( c
1 <> {} & c
2 <> {} & c
3 <> {} & c
4 <> {} )
;
let c
5 be
Element of
[:c1,c2,c3,c4:];
func c
5 `1 -> Element of a
1 means :
Def8:
:: MCART_1:def 8
for b
1, b
2, b
3, b
4 being
set holds
( a
5 = [b1,b2,b3,b4] implies a
6 = b
1 );
existence
ex b1 being Element of c1 st
for b2, b3, b4, b5 being set holds
( c5 = [b2,b3,b4,b5] implies b1 = b2 )
uniqueness
for b1, b2 being Element of c1 holds
( ( for b3, b4, b5, b6 being set holds
( c5 = [b3,b4,b5,b6] implies b1 = b3 ) ) & ( for b3, b4, b5, b6 being set holds
( c5 = [b3,b4,b5,b6] implies b2 = b3 ) ) implies b1 = b2 )
func c
5 `2 -> Element of a
2 means :
Def9:
:: MCART_1:def 9
for b
1, b
2, b
3, b
4 being
set holds
( a
5 = [b1,b2,b3,b4] implies a
6 = b
2 );
existence
ex b1 being Element of c2 st
for b2, b3, b4, b5 being set holds
( c5 = [b2,b3,b4,b5] implies b1 = b3 )
uniqueness
for b1, b2 being Element of c2 holds
( ( for b3, b4, b5, b6 being set holds
( c5 = [b3,b4,b5,b6] implies b1 = b4 ) ) & ( for b3, b4, b5, b6 being set holds
( c5 = [b3,b4,b5,b6] implies b2 = b4 ) ) implies b1 = b2 )
func c
5 `3 -> Element of a
3 means :
Def10:
:: MCART_1:def 10
for b
1, b
2, b
3, b
4 being
set holds
( a
5 = [b1,b2,b3,b4] implies a
6 = b
3 );
existence
ex b1 being Element of c3 st
for b2, b3, b4, b5 being set holds
( c5 = [b2,b3,b4,b5] implies b1 = b4 )
uniqueness
for b1, b2 being Element of c3 holds
( ( for b3, b4, b5, b6 being set holds
( c5 = [b3,b4,b5,b6] implies b1 = b5 ) ) & ( for b3, b4, b5, b6 being set holds
( c5 = [b3,b4,b5,b6] implies b2 = b5 ) ) implies b1 = b2 )
func c
5 `4 -> Element of a
4 means :
Def11:
:: MCART_1:def 11
for b
1, b
2, b
3, b
4 being
set holds
( a
5 = [b1,b2,b3,b4] implies a
6 = b
4 );
existence
ex b1 being Element of c4 st
for b2, b3, b4, b5 being set holds
( c5 = [b2,b3,b4,b5] implies b1 = b5 )
uniqueness
for b1, b2 being Element of c4 holds
( ( for b3, b4, b5, b6 being set holds
( c5 = [b3,b4,b5,b6] implies b1 = b6 ) ) & ( for b3, b4, b5, b6 being set holds
( c5 = [b3,b4,b5,b6] implies b2 = b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines `1 MCART_1:def 8 :
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} implies for b
5 being
Element of
[:b1,b2,b3,b4:]for b
6 being
Element of b
1 holds
( b
6 = b
5 `1 iff for b
7, b
8, b
9, b
10 being
set holds
( b
5 = [b7,b8,b9,b10] implies b
6 = b
7 ) ) );
:: deftheorem Def9 defines `2 MCART_1:def 9 :
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} implies for b
5 being
Element of
[:b1,b2,b3,b4:]for b
6 being
Element of b
2 holds
( b
6 = b
5 `2 iff for b
7, b
8, b
9, b
10 being
set holds
( b
5 = [b7,b8,b9,b10] implies b
6 = b
8 ) ) );
:: deftheorem Def10 defines `3 MCART_1:def 10 :
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} implies for b
5 being
Element of
[:b1,b2,b3,b4:]for b
6 being
Element of b
3 holds
( b
6 = b
5 `3 iff for b
7, b
8, b
9, b
10 being
set holds
( b
5 = [b7,b8,b9,b10] implies b
6 = b
9 ) ) );
:: deftheorem Def11 defines `4 MCART_1:def 11 :
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} implies for b
5 being
Element of
[:b1,b2,b3,b4:]for b
6 being
Element of b
4 holds
( b
6 = b
5 `4 iff for b
7, b
8, b
9, b
10 being
set holds
( b
5 = [b7,b8,b9,b10] implies b
6 = b
10 ) ) );
theorem Th59: :: MCART_1:59
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} implies for b
5 being
Element of
[:b1,b2,b3,b4:]for b
6, b
7, b
8, b
9 being
set holds
( b
5 = [b6,b7,b8,b9] implies ( b
5 `1 = b
6 & b
5 `2 = b
7 & b
5 `3 = b
8 & b
5 `4 = b
9 ) ) )
by Def8, Def9, Def10, Def11;
theorem Th60: :: MCART_1:60
theorem Th61: :: MCART_1:61
theorem Th62: :: MCART_1:62
theorem Th63: :: MCART_1:63
for b
1, b
2, b
3, b
4 being
set holds
( not ( not b
1 c= [:b1,b2,b3,b4:] & not b
1 c= [:b2,b3,b4,b1:] & not b
1 c= [:b3,b4,b1,b2:] & not b
1 c= [:b4,b1,b2,b3:] ) implies b
1 = {} )
theorem Th64: :: MCART_1:64
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
(
[:b1,b2,b3,b4:] meets [:b5,b6,b7,b8:] implies ( b
1 meets b
5 & b
2 meets b
6 & b
3 meets b
7 & b
4 meets b
8 ) )
theorem Th65: :: MCART_1:65
theorem Th66: :: MCART_1:66
theorem Th67: :: MCART_1:67
theorem Th68: :: MCART_1:68
for b
1, b
2, b
3 being
set for b
4 being
Element of
[:b1,b2,b3:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} implies for b
5, b
6, b
7 being
set holds
( b
4 = [b5,b6,b7] implies ( b
4 `1 = b
5 & b
4 `2 = b
6 & b
4 `3 = b
7 ) ) )
by Def5, Def6, Def7;
theorem Th69: :: MCART_1:69
theorem Th70: :: MCART_1:70
theorem Th71: :: MCART_1:71
theorem Th72: :: MCART_1:72
for b
1, b
2, b
3, b
4 being
set holds
not ( b
1 in [:b2,b3,b4:] & ( for b
5, b
6, b
7 being
set holds
not ( b
5 in b
2 & b
6 in b
3 & b
7 in b
4 & b
1 = [b5,b6,b7] ) ) )
theorem Th73: :: MCART_1:73
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
(
[b1,b2,b3] in [:b4,b5,b6:] iff ( b
1 in b
4 & b
2 in b
5 & b
3 in b
6 ) )
theorem Th74: :: MCART_1:74
for b
1, b
2, b
3, b
4 being
set holds
( ( for b
5 being
set holds
( b
5 in b
1 iff ex b
6, b
7, b
8 being
set st
( b
6 in b
2 & b
7 in b
3 & b
8 in b
4 & b
5 = [b6,b7,b8] ) ) ) implies b
1 = [:b2,b3,b4:] )
theorem Th75: :: MCART_1:75
theorem Th76: :: MCART_1:76
theorem Th77: :: MCART_1:77
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
1 c= b
2 & b
3 c= b
4 & b
5 c= b
6 implies
[:b1,b3,b5:] c= [:b2,b4,b6:] )
theorem Th78: :: MCART_1:78
for b
1, b
2, b
3, b
4 being
set for b
5 being
Element of
[:b1,b2,b3,b4:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} implies for b
6, b
7, b
8, b
9 being
set holds
( b
5 = [b6,b7,b8,b9] implies ( b
5 `1 = b
6 & b
5 `2 = b
7 & b
5 `3 = b
8 & b
5 `4 = b
9 ) ) )
by Def8, Def9, Def10, Def11;
theorem Th79: :: MCART_1:79
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Element of
[:b1,b2,b3,b4:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & ( for b
7 being
Element of b
1for b
8 being
Element of b
2for b
9 being
Element of b
3for b
10 being
Element of b
4 holds
( b
6 = [b7,b8,b9,b10] implies b
5 = b
7 ) ) implies b
5 = b
6 `1 )
theorem Th80: :: MCART_1:80
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Element of
[:b1,b2,b3,b4:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & ( for b
7 being
Element of b
1for b
8 being
Element of b
2for b
9 being
Element of b
3for b
10 being
Element of b
4 holds
( b
6 = [b7,b8,b9,b10] implies b
5 = b
8 ) ) implies b
5 = b
6 `2 )
theorem Th81: :: MCART_1:81
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Element of
[:b1,b2,b3,b4:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & ( for b
7 being
Element of b
1for b
8 being
Element of b
2for b
9 being
Element of b
3for b
10 being
Element of b
4 holds
( b
6 = [b7,b8,b9,b10] implies b
5 = b
9 ) ) implies b
5 = b
6 `3 )
theorem Th82: :: MCART_1:82
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Element of
[:b1,b2,b3,b4:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & ( for b
7 being
Element of b
1for b
8 being
Element of b
2for b
9 being
Element of b
3for b
10 being
Element of b
4 holds
( b
6 = [b7,b8,b9,b10] implies b
5 = b
10 ) ) implies b
5 = b
6 `4 )
theorem Th83: :: MCART_1:83
for b
1, b
2, b
3, b
4, b
5 being
set holds
not ( b
1 in [:b2,b3,b4,b5:] & ( for b
6, b
7, b
8, b
9 being
set holds
not ( b
6 in b
2 & b
7 in b
3 & b
8 in b
4 & b
9 in b
5 & b
1 = [b6,b7,b8,b9] ) ) )
theorem Th84: :: MCART_1:84
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
(
[b1,b2,b3,b4] in [:b5,b6,b7,b8:] iff ( b
1 in b
5 & b
2 in b
6 & b
3 in b
7 & b
4 in b
8 ) )
theorem Th85: :: MCART_1:85
for b
1, b
2, b
3, b
4, b
5 being
set holds
( ( for b
6 being
set holds
( b
6 in b
1 iff ex b
7, b
8, b
9, b
10 being
set st
( b
7 in b
2 & b
8 in b
3 & b
9 in b
4 & b
10 in b
5 & b
6 = [b7,b8,b9,b10] ) ) ) implies b
1 = [:b2,b3,b4,b5:] )
theorem Th86: :: MCART_1:86
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} implies for b
9 being
Element of
[:b1,b2,b3,b4:]for b
10 being
Element of
[:b5,b6,b7,b8:] holds
( b
9 = b
10 implies ( b
9 `1 = b
10 `1 & b
9 `2 = b
10 `2 & b
9 `3 = b
10 `3 & b
9 `4 = b
10 `4 ) ) )
theorem Th87: :: MCART_1:87
theorem Th88: :: MCART_1:88
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( b
1 c= b
2 & b
3 c= b
4 & b
5 c= b
6 & b
7 c= b
8 implies
[:b1,b3,b5,b7:] c= [:b2,b4,b6,b8:] )
definition
let c
1, c
2, c
3 be
set ;
let c
4 be
Subset of c
1;
let c
5 be
Subset of c
2;
let c
6 be
Subset of c
3;
redefine func [: as
[:c4,c5,c6:] -> Subset of
[:a1,a2,a3:];
coherence
[:c4,c5,c6:] is Subset of [:c1,c2,c3:]
by Th77;
end;
definition
let c
1, c
2, c
3, c
4 be
set ;
let c
5 be
Subset of c
1;
let c
6 be
Subset of c
2;
let c
7 be
Subset of c
3;
let c
8 be
Subset of c
4;
redefine func [: as
[:c5,c6,c7,c8:] -> Subset of
[:a1,a2,a3,a4:];
coherence
[:c5,c6,c7,c8:] is Subset of [:c1,c2,c3,c4:]
by Th88;
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 b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
(
[[b1,b2],b3] `11 = b
1 &
[[b1,b2],b3] `12 = b
2 &
[b6,[b4,b5]] `21 = b
4 &
[b6,[b4,b5]] `22 = b
5 )