:: MCART_6 semantic presentation
theorem Th1: :: MCART_6:1
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, b
13, b
14, b
15, b
16 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
14 & b
14 in b
15 & b
15 in b
16 & b
16 in b
2 implies b
3 misses b
1 ) ) ) ) )
theorem Th2: :: MCART_6:2
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, b
14, b
15, b
16, b
17 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
14 & b
14 in b
15 & b
15 in b
16 & b
16 in b
17 & b
17 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, c
8, c
9 be
set ;
func [c1,c2,c3,c4,c5,c6,c7,c8,c9] -> set equals :: MCART_6:def 1
[[a1,a2,a3,a4,a5,a6,a7,a8],a9];
coherence
[[c1,c2,c3,c4,c5,c6,c7,c8],c9] is set
;
end;
:: deftheorem Def1 defines [ MCART_6:def 1 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4,b5,b6,b7,b8],b9];
theorem Th3: :: MCART_6:3
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[[[[[[[b1,b2],b3],b4],b5],b6],b7],b8],b9]
theorem Th4: :: MCART_6:4
canceled;
theorem Th5: :: MCART_6:5
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4,b5,b6,b7],b8,b9]
theorem Th6: :: MCART_6:6
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4,b5,b6],b7,b8,b9]
theorem Th7: :: MCART_6:7
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4,b5],b6,b7,b8,b9]
theorem Th8: :: MCART_6:8
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3,b4],b5,b6,b7,b8,b9]
theorem Th9: :: MCART_6:9
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2,b3],b4,b5,b6,b7,b8,b9]
theorem Th10: :: MCART_6:10
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [[b1,b2],b3,b4,b5,b6,b7,b8,b9]
theorem Th11: :: MCART_6:11
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, b
15, b
16, b
17, b
18 being
set holds
(
[b1,b2,b3,b4,b5,b6,b7,b8,b9] = [b10,b11,b12,b13,b14,b15,b16,b17,b18] implies ( b
1 = b
10 & b
2 = b
11 & b
3 = b
12 & b
4 = b
13 & b
5 = b
14 & b
6 = b
15 & b
7 = b
16 & b
8 = b
17 & b
9 = b
18 ) )
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8, c
9 be
set ;
func [:c1,c2,c3,c4,c5,c6,c7,c8,c9:] -> set equals :: MCART_6:def 2
[:[:a1,a2,a3,a4,a5,a6,a7,a8:],a9:];
coherence
[:[:c1,c2,c3,c4,c5,c6,c7,c8:],c9:] is set
;
end;
:: deftheorem Def2 defines [: MCART_6:def 2 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4,b5,b6,b7,b8:],b9:];
theorem Th12: :: MCART_6:12
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:[:[:[:[:[:[:b1,b2:],b3:],b4:],b5:],b6:],b7:],b8:],b9:]
theorem Th13: :: MCART_6:13
canceled;
theorem Th14: :: MCART_6:14
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4,b5,b6,b7:],b8,b9:]
theorem Th15: :: MCART_6:15
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4,b5,b6:],b7,b8,b9:]
theorem Th16: :: MCART_6:16
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4,b5:],b6,b7,b8,b9:]
theorem Th17: :: MCART_6:17
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3,b4:],b5,b6,b7,b8,b9:]
theorem Th18: :: MCART_6:18
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2,b3:],b4,b5,b6,b7,b8,b9:]
theorem Th19: :: MCART_6:19
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:[:b1,b2:],b3,b4,b5,b6,b7,b8,b9:]
theorem Th20: :: MCART_6:20
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( ( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} ) iff
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] <> {} )
theorem Th21: :: MCART_6:21
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, b
15, b
16, b
17, b
18 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} &
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:b10,b11,b12,b13,b14,b15,b16,b17,b18:] implies ( b
1 = b
10 & b
2 = b
11 & b
3 = b
12 & b
4 = b
13 & b
5 = b
14 & b
6 = b
15 & b
7 = b
16 & b
8 = b
17 & b
9 = b
18 ) )
theorem Th22: :: MCART_6:22
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, b
15, b
16, b
17, b
18 being
set holds
(
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] <> {} &
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] = [:b10,b11,b12,b13,b14,b15,b16,b17,b18:] implies ( b
1 = b
10 & b
2 = b
11 & b
3 = b
12 & b
4 = b
13 & b
5 = b
14 & b
6 = b
15 & b
7 = b
16 & b
8 = b
17 & b
9 = b
18 ) )
theorem Th23: :: MCART_6:23
for b
1, b
2 being
set holds
(
[:b1,b1,b1,b1,b1,b1,b1,b1,b1:] = [:b2,b2,b2,b2,b2,b2,b2,b2,b2:] implies b
1 = b
2 )
theorem Th24: :: MCART_6:24
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
ex b
11 being
Element of b
1ex b
12 being
Element of b
2ex b
13 being
Element of b
3ex b
14 being
Element of b
4ex b
15 being
Element of b
5ex b
16 being
Element of b
6ex b
17 being
Element of b
7ex b
18 being
Element of b
8ex b
19 being
Element of b
9 st b
10 = [b11,b12,b13,b14,b15,b16,b17,b18,b19] )
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8, c
9 be
set ;
assume E9:
( c
1 <> {} & c
2 <> {} & c
3 <> {} & c
4 <> {} & c
5 <> {} & c
6 <> {} & c
7 <> {} & c
8 <> {} & c
9 <> {} )
;
let c
10 be
Element of
[:c1,c2,c3,c4,c5,c6,c7,c8,c9:];
func c
10 `1 -> Element of a
1 means :
Def3:
:: MCART_6:def 3
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
1 );
existence
ex b1 being Element of c1 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b2 )
uniqueness
for b1, b2 being Element of c1 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b3 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b3 ) ) implies b1 = b2 )
func c
10 `2 -> Element of a
2 means :
Def4:
:: MCART_6:def 4
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
2 );
existence
ex b1 being Element of c2 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b3 )
uniqueness
for b1, b2 being Element of c2 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b4 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b4 ) ) implies b1 = b2 )
func c
10 `3 -> Element of a
3 means :
Def5:
:: MCART_6:def 5
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
3 );
existence
ex b1 being Element of c3 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b4 )
uniqueness
for b1, b2 being Element of c3 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b5 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b5 ) ) implies b1 = b2 )
func c
10 `4 -> Element of a
4 means :
Def6:
:: MCART_6:def 6
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
4 );
existence
ex b1 being Element of c4 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b5 )
uniqueness
for b1, b2 being Element of c4 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b6 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b6 ) ) implies b1 = b2 )
func c
10 `5 -> Element of a
5 means :
Def7:
:: MCART_6:def 7
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
5 );
existence
ex b1 being Element of c5 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b6 )
uniqueness
for b1, b2 being Element of c5 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b7 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b7 ) ) implies b1 = b2 )
func c
10 `6 -> Element of a
6 means :
Def8:
:: MCART_6:def 8
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
6 );
existence
ex b1 being Element of c6 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b7 )
uniqueness
for b1, b2 being Element of c6 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b8 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b8 ) ) implies b1 = b2 )
func c
10 `7 -> Element of a
7 means :
Def9:
:: MCART_6:def 9
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
7 );
existence
ex b1 being Element of c7 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b8 )
uniqueness
for b1, b2 being Element of c7 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b9 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b9 ) ) implies b1 = b2 )
func c
10 `8 -> Element of a
8 means :
Def10:
:: MCART_6:def 10
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
8 );
existence
ex b1 being Element of c8 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b9 )
uniqueness
for b1, b2 being Element of c8 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b10 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b10 ) ) implies b1 = b2 )
func c
10 `9 -> Element of a
9 means :
Def11:
:: MCART_6:def 11
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( a
10 = [b1,b2,b3,b4,b5,b6,b7,b8,b9] implies a
11 = b
9 );
existence
ex b1 being Element of c9 st
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being set holds
( c10 = [b2,b3,b4,b5,b6,b7,b8,b9,b10] implies b1 = b10 )
uniqueness
for b1, b2 being Element of c9 holds
( ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b1 = b11 ) ) & ( for b3, b4, b5, b6, b7, b8, b9, b10, b11 being set holds
( c10 = [b3,b4,b5,b6,b7,b8,b9,b10,b11] implies b2 = b11 ) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines `1 MCART_6:def 3 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
1 holds
( b
11 = b
10 `1 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
12 ) ) );
:: deftheorem Def4 defines `2 MCART_6:def 4 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
2 holds
( b
11 = b
10 `2 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
13 ) ) );
:: deftheorem Def5 defines `3 MCART_6:def 5 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
3 holds
( b
11 = b
10 `3 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
14 ) ) );
:: deftheorem Def6 defines `4 MCART_6:def 6 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
4 holds
( b
11 = b
10 `4 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
15 ) ) );
:: deftheorem Def7 defines `5 MCART_6:def 7 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
5 holds
( b
11 = b
10 `5 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
16 ) ) );
:: deftheorem Def8 defines `6 MCART_6:def 8 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
6 holds
( b
11 = b
10 `6 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
17 ) ) );
:: deftheorem Def9 defines `7 MCART_6:def 9 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
7 holds
( b
11 = b
10 `7 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
18 ) ) );
:: deftheorem Def10 defines `8 MCART_6:def 10 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
8 holds
( b
11 = b
10 `8 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
19 ) ) );
:: deftheorem Def11 defines `9 MCART_6:def 11 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11 being
Element of b
9 holds
( b
11 = b
10 `9 iff for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set holds
( b
10 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
11 = b
20 ) ) );
theorem Th25: :: MCART_6:25
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
set holds
( b
10 = [b11,b12,b13,b14,b15,b16,b17,b18,b19] implies ( b
10 `1 = b
11 & b
10 `2 = b
12 & b
10 `3 = b
13 & b
10 `4 = b
14 & b
10 `5 = b
15 & b
10 `6 = b
16 & b
10 `7 = b
17 & b
10 `8 = b
18 & b
10 `9 = b
19 ) ) )
by Def3, Def4, Def5, Def6, Def7, Def8, Def9, Def10, Def11;
theorem Th26: :: MCART_6:26
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds b
10 = [(b10 `1 ),(b10 `2 ),(b10 `3 ),(b10 `4 ),(b10 `5 ),(b10 `6 ),(b10 `7 ),(b10 `8 ),(b10 `9 )] )
theorem Th27: :: MCART_6:27
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
10 `1 = (((((((b10 `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 & b
10 `2 = (((((((b10 `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `2 & b
10 `3 = ((((((b10 `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `2 & b
10 `4 = (((((b10 `1 ) `1 ) `1 ) `1 ) `1 ) `2 & b
10 `5 = ((((b10 `1 ) `1 ) `1 ) `1 ) `2 & b
10 `6 = (((b10 `1 ) `1 ) `1 ) `2 & b
10 `7 = ((b10 `1 ) `1 ) `2 & b
10 `8 = (b10 `1 ) `2 & b
10 `9 = b
10 `2 ) )
theorem Th28: :: MCART_6:28
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, b
15, b
16, b
17, b
18 being
set holds
(
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] meets [:b10,b11,b12,b13,b14,b15,b16,b17,b18:] implies ( b
1 meets b
10 & b
2 meets b
11 & b
3 meets b
12 & b
4 meets b
13 & b
5 meets b
14 & b
6 meets b
15 & b
7 meets b
16 & b
8 meets b
17 & b
9 meets b
18 ) )
theorem Th29: :: MCART_6:29
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
[:{b1},{b2},{b3},{b4},{b5},{b6},{b7},{b8},{b9}:] = {[b1,b2,b3,b4,b5,b6,b7,b8,b9]}
theorem Th30: :: MCART_6:30
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set for b
10 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} implies for b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
set holds
( b
10 = [b11,b12,b13,b14,b15,b16,b17,b18,b19] implies ( b
10 `1 = b
11 & b
10 `2 = b
12 & b
10 `3 = b
13 & b
10 `4 = b
14 & b
10 `5 = b
15 & b
10 `6 = b
16 & b
10 `7 = b
17 & b
10 `8 = b
18 & b
10 `9 = b
19 ) ) )
by Def3, Def4, Def5, Def6, Def7, Def8, Def9, Def10, Def11;
theorem Th31: :: MCART_6:31
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
12 ) ) implies b
10 = b
11 `1 )
theorem Th32: :: MCART_6:32
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
13 ) ) implies b
10 = b
11 `2 )
theorem Th33: :: MCART_6:33
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
14 ) ) implies b
10 = b
11 `3 )
theorem Th34: :: MCART_6:34
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
15 ) ) implies b
10 = b
11 `4 )
theorem Th35: :: MCART_6:35
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
16 ) ) implies b
10 = b
11 `5 )
theorem Th36: :: MCART_6:36
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
17 ) ) implies b
10 = b
11 `6 )
theorem Th37: :: MCART_6:37
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
18 ) ) implies b
10 = b
11 `7 )
theorem Th38: :: MCART_6:38
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
19 ) ) implies b
10 = b
11 `8 )
theorem Th39: :: MCART_6:39
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set for b
11 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
1 <> {} & b
2 <> {} & b
3 <> {} & b
4 <> {} & b
5 <> {} & b
6 <> {} & b
7 <> {} & b
8 <> {} & b
9 <> {} & ( for b
12 being
Element of b
1for b
13 being
Element of b
2for b
14 being
Element of b
3for b
15 being
Element of b
4for b
16 being
Element of b
5for b
17 being
Element of b
6for b
18 being
Element of b
7for b
19 being
Element of b
8for b
20 being
Element of b
9 holds
( b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] implies b
10 = b
20 ) ) implies b
10 = b
11 `9 )
theorem Th40: :: MCART_6:40
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set holds
not ( b
1 in [:b2,b3,b4,b5,b6,b7,b8,b9,b10:] & ( for b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
set holds
not ( 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
17 in b
8 & b
18 in b
9 & b
19 in b
10 & b
1 = [b11,b12,b13,b14,b15,b16,b17,b18,b19] ) ) )
theorem Th41: :: MCART_6:41
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, b
15, b
16, b
17, b
18 being
set holds
(
[b1,b2,b3,b4,b5,b6,b7,b8,b9] in [:b10,b11,b12,b13,b14,b15,b16,b17,b18:] iff ( b
1 in b
10 & b
2 in b
11 & b
3 in b
12 & b
4 in b
13 & b
5 in b
14 & b
6 in b
15 & b
7 in b
16 & b
8 in b
17 & b
9 in b
18 ) )
theorem Th42: :: MCART_6:42
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
set holds
( ( for b
11 being
set holds
( b
11 in b
1 iff ex b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
set st
( b
12 in b
2 & b
13 in b
3 & b
14 in b
4 & b
15 in b
5 & b
16 in b
6 & b
17 in b
7 & b
18 in b
8 & b
19 in b
9 & b
20 in b
10 & b
11 = [b12,b13,b14,b15,b16,b17,b18,b19,b20] ) ) ) implies b
1 = [:b2,b3,b4,b5,b6,b7,b8,b9,b10:] )
theorem Th43: :: MCART_6:43
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, b
15, b
16, b
17, b
18 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 <> {} & b
15 <> {} & b
16 <> {} & b
17 <> {} & b
18 <> {} implies for b
19 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:]for b
20 being
Element of
[:b10,b11,b12,b13,b14,b15,b16,b17,b18:] holds
( b
19 = b
20 implies ( b
19 `1 = b
20 `1 & b
19 `2 = b
20 `2 & b
19 `3 = b
20 `3 & b
19 `4 = b
20 `4 & b
19 `5 = b
20 `5 & b
19 `6 = b
20 `6 & b
19 `7 = b
20 `7 & b
19 `8 = b
20 `8 & b
19 `9 = b
20 `9 ) ) )
theorem Th44: :: MCART_6:44
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set for b
10 being
Subset of b
1for b
11 being
Subset of b
2for b
12 being
Subset of b
3for b
13 being
Subset of b
4for b
14 being
Subset of b
5for b
15 being
Subset of b
6for b
16 being
Subset of b
7for b
17 being
Subset of b
8for b
18 being
Subset of b
9for b
19 being
Element of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] holds
( b
19 in [:b10,b11,b12,b13,b14,b15,b16,b17,b18:] implies ( b
19 `1 in b
10 & b
19 `2 in b
11 & b
19 `3 in b
12 & b
19 `4 in b
13 & b
19 `5 in b
14 & b
19 `6 in b
15 & b
19 `7 in b
16 & b
19 `8 in b
17 & b
19 `9 in b
18 ) )
theorem Th45: :: MCART_6:45
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, b
15, b
16, b
17, b
18 being
set holds
( b
1 c= b
2 & b
3 c= b
4 & b
5 c= b
6 & b
7 c= b
8 & b
9 c= b
10 & b
11 c= b
12 & b
13 c= b
14 & b
15 c= b
16 & b
17 c= b
18 implies
[:b1,b3,b5,b7,b9,b11,b13,b15,b17:] c= [:b2,b4,b6,b8,b10,b12,b14,b16,b18:] )
theorem Th46: :: MCART_6:46
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set for b
10 being
Subset of b
1for b
11 being
Subset of b
2for b
12 being
Subset of b
3for b
13 being
Subset of b
4for b
14 being
Subset of b
5for b
15 being
Subset of b
6for b
16 being
Subset of b
7for b
17 being
Subset of b
8for b
18 being
Subset of b
9 holds
[:b10,b11,b12,b13,b14,b15,b16,b17,b18:] is
Subset of
[:b1,b2,b3,b4,b5,b6,b7,b8,b9:] by Th45;