:: 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
theorem Th17: :: DOMAIN_1:17
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
D,
X1,
X2,
X3,
X4 being non
empty set st ( for
a being
set holds
(
a in D iff ex
x1 being
Element of
X1 ex
x2 being
Element of
X2 ex
x3 being
Element of
X3 ex
x4 being
Element of
X4 st
a = [x1,x2,x3,x4] ) ) holds
D = [:X1,X2,X3,X4:]
theorem Th33: :: DOMAIN_1:33
for
D,
X1,
X2,
X3,
X4 being non
empty set holds
(
D = [:X1,X2,X3,X4:] iff for
a being
set holds
(
a in D iff ex
x1 being
Element of
X1 ex
x2 being
Element of
X2 ex
x3 being
Element of
X3 ex
x4 being
Element of
X4 st
a = [x1,x2,x3,x4] ) )
by , ;
definition
let X1 be non
empty set ;
let X2 be non
empty set ;
let X3 be non
empty set ;
let X4 be non
empty set ;
let x1 be
Element of
X1;
let x2 be
Element of
X2;
let x3 be
Element of
X3;
let x4 be
Element of
X4;
redefine func [ as
[c5,c6,c7,c8] -> Element of
[:a1,a2,a3,a4:];
coherence
[x1,x2,x3,x4] is Element of [:X1,X2,X3,X4:]
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 40
s40{
P1[
set ,
set ,
set ] } :
for
X1,
X2,
X3 being non
empty set holds
{ [x1,x2,x3] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3 : P1[x1,x2,x3] } is
Subset of
[:X1,X2,X3:]
scheme :: DOMAIN_1:sch 41
s41{
P1[
set ,
set ,
set ,
set ] } :
for
X1,
X2,
X3,
X4 being non
empty set holds
{ [x1,x2,x3,x4] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3, x4 is Element of X4 : P1[x1,x2,x3,x4] } is
Subset of
[:X1,X2,X3,X4:]
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
X1,
X2,
X3,
X4 being non
empty set holds
[:X1,X2,X3,X4:] = { [x1,x2,x3,x4] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3, x4 is Element of X4 : verum }
theorem Th52: :: DOMAIN_1:52
theorem Th53: :: DOMAIN_1:53
theorem Th54: :: DOMAIN_1:54
theorem Th55: :: DOMAIN_1:55
for
X1,
X2,
X3,
X4 being non
empty set for
A1 being
Subset of
X1 for
A2 being
Subset of
X2 for
A3 being
Subset of
X3 for
A4 being
Subset of
X4 holds
[:A1,A2,A3,A4:] = { [x1,x2,x3,x4] where x1 is Element of X1, x2 is Element of X2, x3 is Element of X3, x4 is Element of X4 : ( x1 in A1 & x2 in A2 & x3 in A3 & x4 in A4 ) }
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
for
X1 being non
empty set for
A1,
B1 being
Subset of
X1 holds
A1 \ B1 = { x1 where x1 is Element of X1 : ( x1 in A1 & not x1 in B1 ) }
theorem Th61: :: DOMAIN_1:61
for
X1 being non
empty set for
A1,
B1 being
Subset of
X1 holds
A1 \+\ B1 = { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) }
theorem Th62: :: DOMAIN_1:62
theorem Th63: :: DOMAIN_1:63
theorem Th64: :: DOMAIN_1:64
for
X1 being non
empty set for
A1,
B1 being
Subset of
X1 holds
A1 \+\ B1 = { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( x1 in B1 & not x1 in A1 ) ) }
definition
let D be non
empty set ;
let x1 be
Element of
D;
redefine func { as
{c2} -> Subset of
a1;
coherence
{x1} is Subset of D
by SUBSET_1:55;
let x2 be
Element of
D;
redefine func { as
{c2,c3} -> Subset of
a1;
coherence
{x1,x2} is Subset of D
by SUBSET_1:56;
let x3 be
Element of
D;
redefine func { as
{c2,c3,c4} -> Subset of
a1;
coherence
{x1,x2,x3} is Subset of D
by SUBSET_1:57;
let x4 be
Element of
D;
redefine func { as
{c2,c3,c4,c5} -> Subset of
a1;
coherence
{x1,x2,x3,x4} is Subset of D
by SUBSET_1:58;
let x5 be
Element of
D;
redefine func { as
{c2,c3,c4,c5,c6} -> Subset of
a1;
coherence
{x1,x2,x3,x4,x5} is Subset of D
by SUBSET_1:59;
let x6 be
Element of
D;
redefine func { as
{c2,c3,c4,c5,c6,c7} -> Subset of
a1;
coherence
{x1,x2,x3,x4,x5,x6} is Subset of D
by SUBSET_1:60;
let x7 be
Element of
D;
redefine func { as
{c2,c3,c4,c5,c6,c7,c8} -> Subset of
a1;
coherence
{x1,x2,x3,x4,x5,x6,x7} is Subset of D
by SUBSET_1:61;
let x8 be
Element of
D;
redefine func { as
{c2,c3,c4,c5,c6,c7,c8,c9} -> Subset of
a1;
coherence
{x1,x2,x3,x4,x5,x6,x7,x8} is Subset of D
by SUBSET_1:62;
end;