:: DOMAIN_1 semantic presentation
theorem Th1: :: DOMAIN_1:1
canceled;
theorem Th2: :: DOMAIN_1:2
canceled;
theorem Th3: :: DOMAIN_1:3
canceled;
theorem Th4: :: DOMAIN_1:4
canceled;
theorem Th5: :: DOMAIN_1:5
canceled;
theorem Th6: :: DOMAIN_1:6
canceled;
theorem Th7: :: DOMAIN_1:7
canceled;
theorem Th8: :: DOMAIN_1:8
canceled;
theorem Th9: :: DOMAIN_1:9
theorem Th10: :: DOMAIN_1:10
canceled;
theorem Th11: :: DOMAIN_1:11
canceled;
theorem Th12: :: DOMAIN_1:12
theorem Th13: :: DOMAIN_1:13
canceled;
theorem Th14: :: DOMAIN_1:14
canceled;
theorem Th15: :: DOMAIN_1:15
theorem Th16: :: DOMAIN_1:16
for b
1, b
2, b
3, b
4 being non
empty set holds
( ( for b
5 being
set holds
( b
5 in b
1 iff ex b
6 being
Element of b
2ex b
7 being
Element of b
3ex b
8 being
Element of b
4 st b
5 = [b6,b7,b8] ) ) implies b
1 = [:b2,b3,b4:] )
theorem Th17: :: DOMAIN_1:17
definition
let c
1, c
2, c
3 be non
empty set ;
let c
4 be
Element of c
1;
let c
5 be
Element of c
2;
let c
6 be
Element of c
3;
redefine func [ as
[c4,c5,c6] -> Element of
[:a1,a2,a3:];
coherence
[c4,c5,c6] is Element of [:c1,c2,c3:]
by MCART_1:73;
end;
theorem Th18: :: DOMAIN_1:18
canceled;
theorem Th19: :: DOMAIN_1:19
canceled;
theorem Th20: :: DOMAIN_1:20
canceled;
theorem Th21: :: DOMAIN_1:21
canceled;
theorem Th22: :: DOMAIN_1:22
canceled;
theorem Th23: :: DOMAIN_1:23
canceled;
theorem Th24: :: DOMAIN_1:24
theorem Th25: :: DOMAIN_1:25
theorem Th26: :: DOMAIN_1:26
theorem Th27: :: DOMAIN_1:27
canceled;
theorem Th28: :: DOMAIN_1:28
theorem Th29: :: DOMAIN_1:29
canceled;
theorem Th30: :: DOMAIN_1:30
canceled;
theorem Th31: :: DOMAIN_1:31
theorem Th32: :: DOMAIN_1:32
for b
1, b
2, b
3, b
4, b
5 being non
empty set holds
( ( for b
6 being
set holds
( b
6 in b
1 iff ex b
7 being
Element of b
2ex b
8 being
Element of b
3ex b
9 being
Element of b
4ex b
10 being
Element of b
5 st b
6 = [b7,b8,b9,b10] ) ) implies b
1 = [:b2,b3,b4,b5:] )
theorem Th33: :: DOMAIN_1:33
for b
1, b
2, b
3, b
4, b
5 being non
empty set holds
( b
1 = [:b2,b3,b4,b5:] iff for b
6 being
set holds
( b
6 in b
1 iff ex b
7 being
Element of b
2ex b
8 being
Element of b
3ex b
9 being
Element of b
4ex b
10 being
Element of b
5 st b
6 = [b7,b8,b9,b10] ) )
by Th31, Th32;
definition
let c
1, c
2, c
3, c
4 be non
empty set ;
let c
5 be
Element of c
1;
let c
6 be
Element of c
2;
let c
7 be
Element of c
3;
let c
8 be
Element of c
4;
redefine func [ as
[c5,c6,c7,c8] -> Element of
[:a1,a2,a3,a4:];
coherence
[c5,c6,c7,c8] is Element of [:c1,c2,c3,c4:]
by MCART_1:84;
end;
theorem Th34: :: DOMAIN_1:34
canceled;
theorem Th35: :: DOMAIN_1:35
canceled;
theorem Th36: :: DOMAIN_1:36
canceled;
theorem Th37: :: DOMAIN_1:37
canceled;
theorem Th38: :: DOMAIN_1:38
canceled;
theorem Th39: :: DOMAIN_1:39
canceled;
theorem Th40: :: DOMAIN_1:40
theorem Th41: :: DOMAIN_1:41
theorem Th42: :: DOMAIN_1:42
theorem Th43: :: DOMAIN_1:43
theorem Th44: :: DOMAIN_1:44
canceled;
theorem Th45: :: DOMAIN_1:45
scheme :: DOMAIN_1:sch 3
s3{ P
1[
set ,
set ,
set ] } :
for b
1, b
2, b
3 being non
empty set holds
{ [b4,b5,b6] where B is Element of b1, B is Element of b2, B is Element of b3 : P1[b4,b5,b6] } is
Subset of
[:b1,b2,b3:]
scheme :: DOMAIN_1:sch 4
s4{ P
1[
set ,
set ,
set ,
set ] } :
for b
1, b
2, b
3, b
4 being non
empty set holds
{ [b5,b6,b7,b8] where B is Element of b1, B is Element of b2, B is Element of b3, B is Element of b4 : P1[b5,b6,b7,b8] } is
Subset of
[:b1,b2,b3,b4:]
theorem Th46: :: DOMAIN_1:46
canceled;
theorem Th47: :: DOMAIN_1:47
canceled;
theorem Th48: :: DOMAIN_1:48
theorem Th49: :: DOMAIN_1:49
theorem Th50: :: DOMAIN_1:50
theorem Th51: :: DOMAIN_1:51
for b
1, b
2, b
3, b
4 being non
empty set holds
[:b1,b2,b3,b4:] = { [b5,b6,b7,b8] where B is Element of b1, B is Element of b2, B is Element of b3, B is Element of b4 : verum }
theorem Th52: :: DOMAIN_1:52
theorem Th53: :: DOMAIN_1:53
theorem Th54: :: DOMAIN_1:54
theorem Th55: :: DOMAIN_1:55
for b
1, b
2, b
3, b
4 being non
empty set for b
5 being
Subset of b
1for b
6 being
Subset of b
2for b
7 being
Subset of b
3for b
8 being
Subset of b
4 holds
[:b5,b6,b7,b8:] = { [b9,b10,b11,b12] where B is Element of b1, B is Element of b2, B is Element of b3, B is Element of b4 : ( b9 in b5 & b10 in b6 & b11 in b7 & b12 in b8 ) }
theorem Th56: :: DOMAIN_1:56
theorem Th57: :: DOMAIN_1:57
theorem Th58: :: DOMAIN_1:58
theorem Th59: :: DOMAIN_1:59
theorem Th60: :: DOMAIN_1:60
theorem Th61: :: DOMAIN_1:61
for b
1 being non
empty set for b
2, b
3 being
Subset of b
1 holds b
2 \+\ b
3 = { b4 where B is Element of b1 : ( ( b4 in b2 & not b4 in b3 ) or ( not b4 in b2 & b4 in b3 ) ) }
theorem Th62: :: DOMAIN_1:62
theorem Th63: :: DOMAIN_1:63
theorem Th64: :: DOMAIN_1:64
definition
let c
1 be non
empty set ;
let c
2 be
Element of c
1;
redefine func { as
{c2} -> Subset of a
1;
coherence
{c2} is Subset of c1
by SUBSET_1:55;
let c
3 be
Element of c
1;
redefine func { as
{c2,c3} -> Subset of a
1;
coherence
{c2,c3} is Subset of c1
by SUBSET_1:56;
let c
4 be
Element of c
1;
redefine func { as
{c2,c3,c4} -> Subset of a
1;
coherence
{c2,c3,c4} is Subset of c1
by SUBSET_1:57;
let c
5 be
Element of c
1;
redefine func { as
{c2,c3,c4,c5} -> Subset of a
1;
coherence
{c2,c3,c4,c5} is Subset of c1
by SUBSET_1:58;
let c
6 be
Element of c
1;
redefine func { as
{c2,c3,c4,c5,c6} -> Subset of a
1;
coherence
{c2,c3,c4,c5,c6} is Subset of c1
by SUBSET_1:59;
let c
7 be
Element of c
1;
redefine func { as
{c2,c3,c4,c5,c6,c7} -> Subset of a
1;
coherence
{c2,c3,c4,c5,c6,c7} is Subset of c1
by SUBSET_1:60;
let c
8 be
Element of c
1;
redefine func { as
{c2,c3,c4,c5,c6,c7,c8} -> Subset of a
1;
coherence
{c2,c3,c4,c5,c6,c7,c8} is Subset of c1
by SUBSET_1:61;
let c
9 be
Element of c
1;
redefine func { as
{c2,c3,c4,c5,c6,c7,c8,c9} -> Subset of a
1;
coherence
{c2,c3,c4,c5,c6,c7,c8,c9} is Subset of c1
by SUBSET_1:62;
end;