:: YELLOW_3 semantic presentation
theorem Th1: :: YELLOW_3:1
Lemma37:
for x, a1, a2, b1, b2 being set st x = [[a1,a2],[b1,b2]] holds
( (x `1 ) `1 = a1 & (x `1 ) `2 = a2 & (x `2 ) `1 = b1 & (x `2 ) `2 = b2 )
theorem Th2: :: YELLOW_3:2
theorem Th3: :: YELLOW_3:3
theorem Th4: :: YELLOW_3:4
theorem Th5: :: YELLOW_3:5
theorem Th6: :: YELLOW_3:6
theorem Th7: :: YELLOW_3:7
theorem Th8: :: YELLOW_3:8
theorem Th9: :: YELLOW_3:9
definition
let P be
Relation,
R be
Relation;
func ["c1,c2"] -> Relation means :
Def1:
:: YELLOW_3:def 1
for
x,
y being
set holds
(
[x,y] in it iff ex
p,
q,
s,
t being
set st
(
x = [p,q] &
y = [s,t] &
[p,s] in P &
[q,t] in R ) );
existence
ex b1 being Relation st
for x, y being set holds
( [x,y] in b1 iff ex p, q, s, t being set st
( x = [p,q] & y = [s,t] & [p,s] in P & [q,t] in R ) )
uniqueness
for b1, b2 being Relation st ( for x, y being set holds
( [x,y] in b1 iff ex p, q, s, t being set st
( x = [p,q] & y = [s,t] & [p,s] in P & [q,t] in R ) ) ) & ( for x, y being set holds
( [x,y] in b2 iff ex p, q, s, t being set st
( x = [p,q] & y = [s,t] & [p,s] in P & [q,t] in R ) ) ) holds
b1 = b2
end;
:: deftheorem Def1 defines [" YELLOW_3:def 1 :
for
P,
R,
b3 being
Relation holds
(
b3 = ["P,R"] iff for
x,
y being
set holds
(
[x,y] in b3 iff ex
p,
q,
s,
t being
set st
(
x = [p,q] &
y = [s,t] &
[p,s] in P &
[q,t] in R ) ) );
theorem Th10: :: YELLOW_3:10
for
P,
R being
Relation for
x being
set holds
(
x in ["P,R"] iff (
[((x `1 ) `1 ),((x `2 ) `1 )] in P &
[((x `1 ) `2 ),((x `2 ) `2 )] in R & ex
a,
b being
set st
x = [a,b] & ex
c,
d being
set st
x `1 = [c,d] & ex
e,
f being
set st
x `2 = [e,f] ) )
definition
let A be
set ,
B be
set ,
X be
set ,
Y be
set ;
let P be
Relation of
A,
B;
let R be
Relation of
X,
Y;
redefine func [" as
["c5,c6"] -> Relation of
[:a1,a3:],
[:a2,a4:];
coherence
["P,R"] is Relation of [:A,X:],[:B,Y:]
end;
definition
let X be
RelStr ,
Y be
RelStr ;
func [:c1,c2:] -> strict RelStr means :
Def2:
:: YELLOW_3:def 2
( the
carrier of
it = [:the carrier of X,the carrier of Y:] & the
InternalRel of
it = ["the InternalRel of X,the InternalRel of Y"] );
existence
ex b1 being strict RelStr st
( the carrier of b1 = [:the carrier of X,the carrier of Y:] & the InternalRel of b1 = ["the InternalRel of X,the InternalRel of Y"] )
uniqueness
for b1, b2 being strict RelStr st the carrier of b1 = [:the carrier of X,the carrier of Y:] & the InternalRel of b1 = ["the InternalRel of X,the InternalRel of Y"] & the carrier of b2 = [:the carrier of X,the carrier of Y:] & the InternalRel of b2 = ["the InternalRel of X,the InternalRel of Y"] holds
b1 = b2
;
end;
:: deftheorem Def2 defines [: YELLOW_3:def 2 :
theorem Th11: :: YELLOW_3:11
theorem Th12: :: YELLOW_3:12
theorem Th13: :: YELLOW_3:13
theorem Th14: :: YELLOW_3:14
theorem Th15: :: YELLOW_3:15
theorem Th16: :: YELLOW_3:16
theorem Th17: :: YELLOW_3:17
theorem Th18: :: YELLOW_3:18
theorem Th19: :: YELLOW_3:19
theorem Th20: :: YELLOW_3:20
theorem Th21: :: YELLOW_3:21
theorem Th22: :: YELLOW_3:22
theorem Th23: :: YELLOW_3:23
theorem Th24: :: YELLOW_3:24
theorem Th25: :: YELLOW_3:25
theorem Th26: :: YELLOW_3:26
theorem Th27: :: YELLOW_3:27
theorem Th28: :: YELLOW_3:28
:: deftheorem Def3 defines void YELLOW_3:def 3 :
theorem Th29: :: YELLOW_3:29
theorem Th30: :: YELLOW_3:30
theorem Th31: :: YELLOW_3:31
theorem Th32: :: YELLOW_3:32
theorem Th33: :: YELLOW_3:33
theorem Th34: :: YELLOW_3:34
theorem Th35: :: YELLOW_3:35
theorem Th36: :: YELLOW_3:36
theorem Th37: :: YELLOW_3:37
theorem Th38: :: YELLOW_3:38
theorem Th39: :: YELLOW_3:39
theorem Th40: :: YELLOW_3:40
theorem Th41: :: YELLOW_3:41
theorem Th42: :: YELLOW_3:42
theorem Th43: :: YELLOW_3:43
theorem Th44: :: YELLOW_3:44
theorem Th45: :: YELLOW_3:45
theorem Th46: :: YELLOW_3:46
theorem Th47: :: YELLOW_3:47
theorem Th48: :: YELLOW_3:48
theorem Th49: :: YELLOW_3:49