:: MCART_4 semantic presentation
theorem Th1: :: MCART_4:1
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, b
8, b
9, b
10, b
11, b
12 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
8 & b
8 in b
9 & b
9 in b
10 & b
10 in b
11 & b
11 in b
12 & b
12 in b
2 implies b
3 misses b
1 ) ) ) ) )
theorem Th2: :: MCART_4:2
for b
1 being
set holds
not ( b
1 <> {} & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13 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
8 & b
8 in b
9 & b
9 in b
10 & b
10 in b
11 & b
11 in b
12 & b
12 in b
13 & b
13 in b
2 implies b
3 misses b
1 ) ) ) ) )
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func [c1,c2,c3,c4,c5,c6,c7] -> set equals :: MCART_4:def 1
[[a1,a2,a3,a4,a5,a6],a7];
correctness
coherence
[[c1,c2,c3,c4,c5,c6],c7] is set ;
;
end;
:: deftheorem Def1 defines [ MCART_4:def 1 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2,b3,b4,b5,b6],b7];
theorem Th3: :: MCART_4:3
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[b1,b2,b3,b4,b5,b6,b7] = [[[[[[b1,b2],b3],b4],b5],b6],b7]
theorem Th4: :: MCART_4:4
canceled;
theorem Th5: :: MCART_4:5
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2,b3,b4,b5],b6,b7]
theorem Th6: :: MCART_4:6
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2,b3,b4],b5,b6,b7]
theorem Th7: :: MCART_4:7
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2,b3],b4,b5,b6,b7]
theorem Th8: :: MCART_4:8
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[b1,b2,b3,b4,b5,b6,b7] = [[b1,b2],b3,b4,b5,b6,b7]
theorem Th9: :: MCART_4:9
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14 being
set holds
(
[b1,b2,b3,b4,b5,b6,b7] = [b8,b9,b10,b11,b12,b13,b14] implies ( b
1 = b
8 & b
2 = b
9 & b
3 = b
10 & b
4 = b
11 & b
5 = b
12 & b
6 = b
13 & b
7 = b
14 ) )
theorem Th10: :: MCART_4:10
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, b
8, b
9 being
set holds
not ( ( b
3 in b
1 or b
4 in b
1 ) & b
2 = [b3,b4,b5,b6,b7,b8,b9] ) ) ) ) )
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func [:c1,c2,c3,c4,c5,c6,c7:] -> set equals :: MCART_4:def 2
[:[:a1,a2,a3,a4,a5,a6:],a7:];
correctness
coherence
[:[:c1,c2,c3,c4,c5,c6:],c7:] is set ;
;
end;
:: deftheorem Def2 defines [: MCART_4:def 2 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2,b3,b4,b5,b6:],b7:];
theorem Th11: :: MCART_4:11
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:[:[:[:[:b1,b2:],b3:],b4:],b5:],b6:],b7:]
theorem Th12: :: MCART_4:12
canceled;
theorem Th13: :: MCART_4:13
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2,b3,b4,b5:],b6,b7:]
theorem Th14: :: MCART_4:14
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2,b3,b4:],b5,b6,b7:]
theorem Th15: :: MCART_4:15
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2,b3:],b4,b5,b6,b7:]
theorem Th16: :: MCART_4:16
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[:b1,b2,b3,b4,b5,b6,b7:] = [:[:b1,b2:],b3,b4,b5,b6,b7:]
theorem Th17: :: MCART_4:17
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( ( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} ) iff
[:b1,b2,b3,b4,b5,b6,b7:] <> {} )
theorem Th18: :: MCART_4:18
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} &
[:b1,b2,b3,b4,b5,b6,b7:] = [:b8,b9,b10,b11,b12,b13,b14:] implies ( b
1 = b
8 & b
2 = b
9 & b
3 = b
10 & b
4 = b
11 & b
5 = b
12 & b
6 = b
13 & b
7 = b
14 ) )
theorem Th19: :: MCART_4:19
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14 being
set holds
(
[:b1,b2,b3,b4,b5,b6,b7:] <> {} &
[:b1,b2,b3,b4,b5,b6,b7:] = [:b8,b9,b10,b11,b12,b13,b14:] implies ( b
1 = b
8 & b
2 = b
9 & b
3 = b
10 & b
4 = b
11 & b
5 = b
12 & b
6 = b
13 & b
7 = b
14 ) )
theorem Th20: :: MCART_4:20
for b
1, b
2 being
set holds
(
[:b1,b1,b1,b1,b1,b1,b1:] = [:b2,b2,b2,b2,b2,b2,b2:] implies b
1 = b
2 )
theorem Th21: :: MCART_4:21
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:] holds
ex b
9 being
Element of b
1ex b
10 being
Element of b
2ex b
11 being
Element of b
3ex b
12 being
Element of b
4ex b
13 being
Element of b
5ex b
14 being
Element of b
6ex b
15 being
Element of b
7 st b
8 = [b9,b10,b11,b12,b13,b14,b15] )
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
assume E11:
( c
1 <> {} & c
2 <> {} & c
3 <> {} & c
4 <> {} & c
5 <> {} & c
6 <> {} & c
7 <> {} )
;
let c
8 be
Element of
[:c1,c2,c3,c4,c5,c6,c7:];
func c
8 `1 -> Element of a
1 means :
Def3:
:: MCART_4:def 3
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( a
8 = [b1,b2,b3,b4,b5,b6,b7] implies a
9 = b
1 );
existence
ex b1 being Element of c1 st
for b2, b3, b4, b5, b6, b7, b8 being set holds
( c8 = [b2,b3,b4,b5,b6,b7,b8] implies b1 = b2 )
uniqueness
for b1, b2 being Element of c1 holds
( ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b1 = b3 ) ) & ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b2 = b3 ) ) implies b1 = b2 )
func c
8 `2 -> Element of a
2 means :
Def4:
:: MCART_4:def 4
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( a
8 = [b1,b2,b3,b4,b5,b6,b7] implies a
9 = b
2 );
existence
ex b1 being Element of c2 st
for b2, b3, b4, b5, b6, b7, b8 being set holds
( c8 = [b2,b3,b4,b5,b6,b7,b8] implies b1 = b3 )
uniqueness
for b1, b2 being Element of c2 holds
( ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b1 = b4 ) ) & ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b2 = b4 ) ) implies b1 = b2 )
func c
8 `3 -> Element of a
3 means :
Def5:
:: MCART_4:def 5
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( a
8 = [b1,b2,b3,b4,b5,b6,b7] implies a
9 = b
3 );
existence
ex b1 being Element of c3 st
for b2, b3, b4, b5, b6, b7, b8 being set holds
( c8 = [b2,b3,b4,b5,b6,b7,b8] implies b1 = b4 )
uniqueness
for b1, b2 being Element of c3 holds
( ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b1 = b5 ) ) & ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b2 = b5 ) ) implies b1 = b2 )
func c
8 `4 -> Element of a
4 means :
Def6:
:: MCART_4:def 6
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( a
8 = [b1,b2,b3,b4,b5,b6,b7] implies a
9 = b
4 );
existence
ex b1 being Element of c4 st
for b2, b3, b4, b5, b6, b7, b8 being set holds
( c8 = [b2,b3,b4,b5,b6,b7,b8] implies b1 = b5 )
uniqueness
for b1, b2 being Element of c4 holds
( ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b1 = b6 ) ) & ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b2 = b6 ) ) implies b1 = b2 )
func c
8 `5 -> Element of a
5 means :
Def7:
:: MCART_4:def 7
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( a
8 = [b1,b2,b3,b4,b5,b6,b7] implies a
9 = b
5 );
existence
ex b1 being Element of c5 st
for b2, b3, b4, b5, b6, b7, b8 being set holds
( c8 = [b2,b3,b4,b5,b6,b7,b8] implies b1 = b6 )
uniqueness
for b1, b2 being Element of c5 holds
( ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b1 = b7 ) ) & ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b2 = b7 ) ) implies b1 = b2 )
func c
8 `6 -> Element of a
6 means :
Def8:
:: MCART_4:def 8
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( a
8 = [b1,b2,b3,b4,b5,b6,b7] implies a
9 = b
6 );
existence
ex b1 being Element of c6 st
for b2, b3, b4, b5, b6, b7, b8 being set holds
( c8 = [b2,b3,b4,b5,b6,b7,b8] implies b1 = b7 )
uniqueness
for b1, b2 being Element of c6 holds
( ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b1 = b8 ) ) & ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b2 = b8 ) ) implies b1 = b2 )
func c
8 `7 -> Element of a
7 means :
Def9:
:: MCART_4:def 9
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( a
8 = [b1,b2,b3,b4,b5,b6,b7] implies a
9 = b
7 );
existence
ex b1 being Element of c7 st
for b2, b3, b4, b5, b6, b7, b8 being set holds
( c8 = [b2,b3,b4,b5,b6,b7,b8] implies b1 = b8 )
uniqueness
for b1, b2 being Element of c7 holds
( ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b1 = b9 ) ) & ( for b3, b4, b5, b6, b7, b8, b9 being set holds
( c8 = [b3,b4,b5,b6,b7,b8,b9] implies b2 = b9 ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines `1 MCART_4:def 3 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
9 being
Element of b
1 holds
( b
9 = b
8 `1 iff for b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
set holds
( b
8 = [b10,b11,b12,b13,b14,b15,b16] implies b
9 = b
10 ) ) );
:: deftheorem Def4 defines `2 MCART_4:def 4 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
9 being
Element of b
2 holds
( b
9 = b
8 `2 iff for b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
set holds
( b
8 = [b10,b11,b12,b13,b14,b15,b16] implies b
9 = b
11 ) ) );
:: deftheorem Def5 defines `3 MCART_4:def 5 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
9 being
Element of b
3 holds
( b
9 = b
8 `3 iff for b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
set holds
( b
8 = [b10,b11,b12,b13,b14,b15,b16] implies b
9 = b
12 ) ) );
:: deftheorem Def6 defines `4 MCART_4:def 6 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
9 being
Element of b
4 holds
( b
9 = b
8 `4 iff for b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
set holds
( b
8 = [b10,b11,b12,b13,b14,b15,b16] implies b
9 = b
13 ) ) );
:: deftheorem Def7 defines `5 MCART_4:def 7 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
9 being
Element of b
5 holds
( b
9 = b
8 `5 iff for b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
set holds
( b
8 = [b10,b11,b12,b13,b14,b15,b16] implies b
9 = b
14 ) ) );
:: deftheorem Def8 defines `6 MCART_4:def 8 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
9 being
Element of b
6 holds
( b
9 = b
8 `6 iff for b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
set holds
( b
8 = [b10,b11,b12,b13,b14,b15,b16] implies b
9 = b
15 ) ) );
:: deftheorem Def9 defines `7 MCART_4:def 9 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
9 being
Element of b
7 holds
( b
9 = b
8 `7 iff for b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
set holds
( b
8 = [b10,b11,b12,b13,b14,b15,b16] implies b
9 = b
16 ) ) );
theorem Th22: :: MCART_4:22
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
9, b
10, b
11, b
12, b
13, b
14, b
15 being
set holds
( b
8 = [b9,b10,b11,b12,b13,b14,b15] implies ( b
8 `1 = b
9 & b
8 `2 = b
10 & b
8 `3 = b
11 & b
8 `4 = b
12 & b
8 `5 = b
13 & b
8 `6 = b
14 & b
8 `7 = b
15 ) ) )
by Def3, Def4, Def5, Def6, Def7, Def8, Def9;
theorem Th23: :: MCART_4:23
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:] holds b
8 = [(b8 `1 ),(b8 `2 ),(b8 `3 ),(b8 `4 ),(b8 `5 ),(b8 `6 ),(b8 `7 )] )
theorem Th24: :: MCART_4:24
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:] holds
( b
8 `1 = (((((b8 `1 ) `1 ) `1 ) `1 ) `1 ) `1 & b
8 `2 = (((((b8 `1 ) `1 ) `1 ) `1 ) `1 ) `2 & b
8 `3 = ((((b8 `1 ) `1 ) `1 ) `1 ) `2 & b
8 `4 = (((b8 `1 ) `1 ) `1 ) `2 & b
8 `5 = ((b8 `1 ) `1 ) `2 & b
8 `6 = (b8 `1 ) `2 & b
8 `7 = b
8 `2 ) )
theorem Th25: :: MCART_4:25
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( not ( not b
1 c= [:b1,b2,b3,b4,b5,b6,b7:] & not b
1 c= [:b2,b3,b4,b5,b6,b7,b1:] & not b
1 c= [:b3,b4,b5,b6,b7,b1,b2:] & not b
1 c= [:b4,b5,b6,b7,b1,b2,b3:] & not b
1 c= [:b5,b6,b7,b1,b2,b3,b4:] & not b
1 c= [:b6,b7,b1,b2,b3,b4,b5:] & not b
1 c= [:b7,b1,b2,b3,b4,b5,b6:] ) implies b
1 = {} )
theorem Th26: :: MCART_4:26
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14 being
set holds
(
[:b1,b2,b3,b4,b5,b6,b7:] meets [:b8,b9,b10,b11,b12,b13,b14:] implies ( b
1 meets b
8 & b
2 meets b
9 & b
3 meets b
10 & b
4 meets b
11 & b
5 meets b
12 & b
6 meets b
13 & b
7 meets b
14 ) )
theorem Th27: :: MCART_4:27
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
[:{b1},{b2},{b3},{b4},{b5},{b6},{b7}:] = {[b1,b2,b3,b4,b5,b6,b7]}
theorem Th28: :: MCART_4:28
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set for b
8 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} implies for b
9, b
10, b
11, b
12, b
13, b
14, b
15 being
set holds
( b
8 = [b9,b10,b11,b12,b13,b14,b15] implies ( b
8 `1 = b
9 & b
8 `2 = b
10 & b
8 `3 = b
11 & b
8 `4 = b
12 & b
8 `5 = b
13 & b
8 `6 = b
14 & b
8 `7 = b
15 ) ) )
by Def3, Def4, Def5, Def6, Def7, Def8, Def9;
theorem Th29: :: MCART_4:29
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Element of
[:b2,b3,b4,b5,b6,b7,b8:] holds
( b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & ( for b
10 being
Element of b
2for b
11 being
Element of b
3for b
12 being
Element of b
4for b
13 being
Element of b
5for b
14 being
Element of b
6for b
15 being
Element of b
7for b
16 being
Element of b
8 holds
( b
9 = [b10,b11,b12,b13,b14,b15,b16] implies b
1 = b
10 ) ) implies b
1 = b
9 `1 )
theorem Th30: :: MCART_4:30
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Element of
[:b2,b3,b4,b5,b6,b7,b8:] holds
( b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & ( for b
10 being
Element of b
2for b
11 being
Element of b
3for b
12 being
Element of b
4for b
13 being
Element of b
5for b
14 being
Element of b
6for b
15 being
Element of b
7for b
16 being
Element of b
8 holds
( b
9 = [b10,b11,b12,b13,b14,b15,b16] implies b
1 = b
11 ) ) implies b
1 = b
9 `2 )
theorem Th31: :: MCART_4:31
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Element of
[:b2,b3,b4,b5,b6,b7,b8:] holds
( b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & ( for b
10 being
Element of b
2for b
11 being
Element of b
3for b
12 being
Element of b
4for b
13 being
Element of b
5for b
14 being
Element of b
6for b
15 being
Element of b
7for b
16 being
Element of b
8 holds
( b
9 = [b10,b11,b12,b13,b14,b15,b16] implies b
1 = b
12 ) ) implies b
1 = b
9 `3 )
theorem Th32: :: MCART_4:32
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Element of
[:b2,b3,b4,b5,b6,b7,b8:] holds
( b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & ( for b
10 being
Element of b
2for b
11 being
Element of b
3for b
12 being
Element of b
4for b
13 being
Element of b
5for b
14 being
Element of b
6for b
15 being
Element of b
7for b
16 being
Element of b
8 holds
( b
9 = [b10,b11,b12,b13,b14,b15,b16] implies b
1 = b
13 ) ) implies b
1 = b
9 `4 )
theorem Th33: :: MCART_4:33
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Element of
[:b2,b3,b4,b5,b6,b7,b8:] holds
( b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & ( for b
10 being
Element of b
2for b
11 being
Element of b
3for b
12 being
Element of b
4for b
13 being
Element of b
5for b
14 being
Element of b
6for b
15 being
Element of b
7for b
16 being
Element of b
8 holds
( b
9 = [b10,b11,b12,b13,b14,b15,b16] implies b
1 = b
14 ) ) implies b
1 = b
9 `5 )
theorem Th34: :: MCART_4:34
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Element of
[:b2,b3,b4,b5,b6,b7,b8:] holds
( b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & ( for b
10 being
Element of b
2for b
11 being
Element of b
3for b
12 being
Element of b
4for b
13 being
Element of b
5for b
14 being
Element of b
6for b
15 being
Element of b
7for b
16 being
Element of b
8 holds
( b
9 = [b10,b11,b12,b13,b14,b15,b16] implies b
1 = b
15 ) ) implies b
1 = b
9 `6 )
theorem Th35: :: MCART_4:35
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Element of
[:b2,b3,b4,b5,b6,b7,b8:] holds
( b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & ( for b
10 being
Element of b
2for b
11 being
Element of b
3for b
12 being
Element of b
4for b
13 being
Element of b
5for b
14 being
Element of b
6for b
15 being
Element of b
7for b
16 being
Element of b
8 holds
( b
9 = [b10,b11,b12,b13,b14,b15,b16] implies b
1 = b
16 ) ) implies b
1 = b
9 `7 )
theorem Th36: :: MCART_4:36
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
not ( b
1 in [:b2,b3,b4,b5,b6,b7,b8:] & ( for b
9, b
10, b
11, b
12, b
13, b
14, b
15 being
set holds
not ( b
9 in b
2 & b
10 in b
3 & b
11 in b
4 & b
12 in b
5 & b
13 in b
6 & b
14 in b
7 & b
15 in b
8 & b
1 = [b9,b10,b11,b12,b13,b14,b15] ) ) )
theorem Th37: :: MCART_4:37
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14 being
set holds
(
[b1,b2,b3,b4,b5,b6,b7] in [:b8,b9,b10,b11,b12,b13,b14:] iff ( b
1 in b
8 & b
2 in b
9 & b
3 in b
10 & b
4 in b
11 & b
5 in b
12 & b
6 in b
13 & b
7 in b
14 ) )
theorem Th38: :: MCART_4:38
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( ( for b
9 being
set holds
( b
9 in b
8 iff ex b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
set st
( b
10 in b
1 & b
11 in b
2 & b
12 in b
3 & b
13 in b
4 & b
14 in b
5 & b
15 in b
6 & b
16 in b
7 & b
9 = [b10,b11,b12,b13,b14,b15,b16] ) ) ) implies b
8 = [:b1,b2,b3,b4,b5,b6,b7:] )
theorem Th39: :: MCART_4:39
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & b
10 <> {} & b
11 <> {} & b
12 <> {} & b
13 <> {} & b
14 <> {} implies for b
15 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:]for b
16 being
Element of
[:b8,b9,b10,b11,b12,b13,b14:] holds
( b
15 = b
16 implies ( b
15 `1 = b
16 `1 & b
15 `2 = b
16 `2 & b
15 `3 = b
16 `3 & b
15 `4 = b
16 `4 & b
15 `5 = b
16 `5 & b
15 `6 = b
16 `6 & b
15 `7 = b
16 `7 ) ) )
theorem Th40: :: MCART_4:40
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set for b
8 being
Subset of b
1for b
9 being
Subset of b
2for b
10 being
Subset of b
3for b
11 being
Subset of b
4for b
12 being
Subset of b
5for b
13 being
Subset of b
6for b
14 being
Subset of b
7for b
15 being
Element of
[:b1,b2,b3,b4,b5,b6,b7:] holds
( b
15 in [:b8,b9,b10,b11,b12,b13,b14:] implies ( b
15 `1 in b
8 & b
15 `2 in b
9 & b
15 `3 in b
10 & b
15 `4 in b
11 & b
15 `5 in b
12 & b
15 `6 in b
13 & b
15 `7 in b
14 ) )
theorem Th41: :: MCART_4:41
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14 being
set holds
( b
1 c= b
8 & b
2 c= b
9 & b
3 c= b
10 & b
4 c= b
11 & b
5 c= b
12 & b
6 c= b
13 & b
7 c= b
14 implies
[:b1,b2,b3,b4,b5,b6,b7:] c= [:b8,b9,b10,b11,b12,b13,b14:] )
theorem Th42: :: MCART_4:42
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set for b
8 being
Subset of b
1for b
9 being
Subset of b
2for b
10 being
Subset of b
3for b
11 being
Subset of b
4for b
12 being
Subset of b
5for b
13 being
Subset of b
6for b
14 being
Subset of b
7 holds
[:b8,b9,b10,b11,b12,b13,b14:] is
Subset of
[:b1,b2,b3,b4,b5,b6,b7:] by Th41;