:: WAYBEL25 semantic presentation
theorem Th1: :: WAYBEL25:1
theorem Th2: :: WAYBEL25:2
theorem Th3: :: WAYBEL25:3
theorem Th4: :: WAYBEL25:4
theorem Th5: :: WAYBEL25:5
theorem Th6: :: WAYBEL25:6
theorem Th7: :: WAYBEL25:7
:: deftheorem Def1 defines is_Retract_of WAYBEL25:def 1 :
theorem Th8: :: WAYBEL25:8
theorem Th9: :: WAYBEL25:9
theorem Th10: :: WAYBEL25:10
theorem Th11: :: WAYBEL25:11
theorem Th12: :: WAYBEL25:12
definition
let c
1 be
TopStruct ;
func Omega c
1 -> strict TopRelStr means :
Def2:
:: WAYBEL25:def 2
(
TopStruct(# the
carrier of a
2,the
topology of a
2 #)
= TopStruct(# the
carrier of a
1,the
topology of a
1 #) & ( for b
1, b
2 being
Element of a
2 holds
( b
1 <= b
2 iff ex b
3 being
Subset of a
1 st
( b
3 = {b2} & b
1 in Cl b
3 ) ) ) );
existence
ex b1 being strict TopRelStr st
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of c1,the topology of c1 #) & ( for b2, b3 being Element of b1 holds
( b2 <= b3 iff ex b4 being Subset of c1 st
( b4 = {b3} & b2 in Cl b4 ) ) ) )
uniqueness
for b1, b2 being strict TopRelStr holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of c1,the topology of c1 #) & ( for b3, b4 being Element of b1 holds
( b3 <= b4 iff ex b5 being Subset of c1 st
( b5 = {b4} & b3 in Cl b5 ) ) ) & TopStruct(# the carrier of b2,the topology of b2 #) = TopStruct(# the carrier of c1,the topology of c1 #) & ( for b3, b4 being Element of b2 holds
( b3 <= b4 iff ex b5 being Subset of c1 st
( b5 = {b4} & b3 in Cl b5 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Omega WAYBEL25:def 2 :
Lemma10:
for b1 being TopStruct holds the carrier of b1 = the carrier of (Omega b1)
then Lemma11:
for b1 being TopStruct
for b2 being set holds
( b2 is Subset of b1 iff b2 is Subset of (Omega b1) )
;
Lemma12:
for b1 being TopStruct
for b2, b3 being Element of b1
for b4, b5 being Subset of b1 holds
( b4 = {b2} & b5 = {b3} implies ( b2 in Cl b5 iff Cl b4 c= Cl b5 ) )
theorem Th13: :: WAYBEL25:13
theorem Th14: :: WAYBEL25:14
theorem Th15: :: WAYBEL25:15
theorem Th16: :: WAYBEL25:16
theorem Th17: :: WAYBEL25:17
theorem Th18: :: WAYBEL25:18
theorem Th19: :: WAYBEL25:19
Lemma19:
for b1, b2 being non empty RelStr
for b3 being Function of b1,b1
for b4 being Function of b2,b2 holds
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of b2,the InternalRel of b2 #) & b3 = b4 & b3 is projection implies b4 is projection )
theorem Th20: :: WAYBEL25:20
theorem Th21: :: WAYBEL25:21
theorem Th22: :: WAYBEL25:22
theorem Th23: :: WAYBEL25:23
E24:
now
let c
1, c
2 be non
empty TopSpace;
let c
3 be
net of
ContMaps c
1,
(Omega c2);
E25:
the
mapping of c
3 in Funcs the
carrier of c
3,the
carrier of
(ContMaps c1,(Omega c2))
by FUNCT_2:11;
E26:
the
carrier of c
2 = the
carrier of
(Omega c2)
by Lemma10;
the
carrier of
(ContMaps c1,(Omega c2)) c= Funcs the
carrier of c
1,the
carrier of c
2
then
Funcs the
carrier of c
3,the
carrier of
(ContMaps c1,(Omega c2)) c= Funcs the
carrier of c
3,
(Funcs the carrier of c1,the carrier of c2)
by FUNCT_5:63;
hence
the
mapping of c
3 in Funcs the
carrier of c
3,
(Funcs the carrier of c1,the carrier of c2)
by E25;
end;
definition
let c
1 be non
empty set ;
let c
2, c
3 be non
empty RelStr ;
let c
4 be
net of c
3;
let c
5 be
Element of c
1;
assume E25:
the
carrier of c
3 c= the
carrier of
(c2 |^ c1)
;
func commute c
4,c
5,c
2 -> strict net of a
2 means :
Def3:
:: WAYBEL25:def 3
(
RelStr(# the
carrier of a
6,the
InternalRel of a
6 #)
= RelStr(# the
carrier of a
4,the
InternalRel of a
4 #) & the
mapping of a
6 = (commute the mapping of a4) . a
5 );
existence
ex b1 being strict net of c2 st
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of c4,the InternalRel of c4 #) & the mapping of b1 = (commute the mapping of c4) . c5 )
uniqueness
for b1, b2 being strict net of c2 holds
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of c4,the InternalRel of c4 #) & the mapping of b1 = (commute the mapping of c4) . c5 & RelStr(# the carrier of b2,the InternalRel of b2 #) = RelStr(# the carrier of c4,the InternalRel of c4 #) & the mapping of b2 = (commute the mapping of c4) . c5 implies b1 = b2 )
;
end;
:: deftheorem Def3 defines commute WAYBEL25:def 3 :
theorem Th24: :: WAYBEL25:24
theorem Th25: :: WAYBEL25:25
E29:
now
let c
1, c
2 be non
empty TopSpace;
let c
3 be
net of
ContMaps c
1,
(Omega c2);
let c
4 be
Point of c
1;
ContMaps c
1,
(Omega c2) is
SubRelStr of
(Omega c2) |^ the
carrier of c
1
by WAYBEL24:def 3;
then
the
carrier of
(ContMaps c1,(Omega c2)) c= the
carrier of
((Omega c2) |^ the carrier of c1)
by YELLOW_0:def 13;
then E30:
RelStr(# the
carrier of c
3,the
InternalRel of c
3 #)
= RelStr(# the
carrier of
(commute c3,c4,(Omega c2)),the
InternalRel of
(commute c3,c4,(Omega c2)) #)
by Def3;
thus dom the
mapping of c
3 =
the
carrier of c
3
by FUNCT_2:def 1
.=
dom the
mapping of
(commute c3,c4,(Omega c2))
by E30, FUNCT_2:def 1
;
end;
theorem Th26: :: WAYBEL25:26
:: deftheorem Def4 defines monotone-convergence WAYBEL25:def 4 :
theorem Th27: :: WAYBEL25:27
theorem Th28: :: WAYBEL25:28
theorem Th29: :: WAYBEL25:29
theorem Th30: :: WAYBEL25:30
theorem Th31: :: WAYBEL25:31
theorem Th32: :: WAYBEL25:32
theorem Th33: :: WAYBEL25:33
theorem Th34: :: WAYBEL25:34
theorem Th35: :: WAYBEL25:35
theorem Th36: :: WAYBEL25:36
theorem Th37: :: WAYBEL25:37
theorem Th38: :: WAYBEL25:38
theorem Th39: :: WAYBEL25:39
theorem Th40: :: WAYBEL25:40
theorem Th41: :: WAYBEL25:41
theorem Th42: :: WAYBEL25:42
theorem Th43: :: WAYBEL25:43