:: WAYBEL29 semantic presentation

theorem Th1: :: WAYBEL29:1
for b1, b2 being non empty 1-sorted
for b3 being Function of b1,b2 holds
( b3 is one-to-one & b3 is onto implies ( b3 * (b3 " ) = id b2 & (b3 " ) * b3 = id b1 & b3 " is one-to-one & b3 " is onto ) ) by GRCAT_1:56;

theorem Th2: :: WAYBEL29:2
for b1, b2 being non empty set
for b3 being non empty RelStr
for b4 being non empty SubRelStr of b3 |^ [:b1,b2:]
for b5 being non empty SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function of b4,b5 holds
( b6 is currying & b6 is one-to-one & b6 is onto implies b6 " is uncurrying )
proof end;

theorem Th3: :: WAYBEL29:3
for b1, b2 being non empty set
for b3 being non empty RelStr
for b4 being non empty SubRelStr of b3 |^ [:b1,b2:]
for b5 being non empty SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function of b5,b4 holds
( b6 is uncurrying & b6 is one-to-one & b6 is onto implies b6 " is currying )
proof end;

theorem Th4: :: WAYBEL29:4
for b1, b2 being non empty set
for b3 being non empty Poset
for b4 being non empty full SubRelStr of b3 |^ [:b1,b2:]
for b5 being non empty full SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function of b4,b5 holds
( b6 is currying & b6 is one-to-one & b6 is onto implies b6 is isomorphic )
proof end;

theorem Th5: :: WAYBEL29:5
for b1, b2 being non empty set
for b3 being non empty Poset
for b4 being non empty full SubRelStr of b3 |^ [:b1,b2:]
for b5 being non empty full SubRelStr of (b3 |^ b2) |^ b1
for b6 being Function of b5,b4 holds
( b6 is uncurrying & b6 is one-to-one & b6 is onto implies b6 is isomorphic )
proof end;

theorem Th6: :: WAYBEL29:6
for b1, b2, b3, b4 being RelStr holds
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of b2,the InternalRel of b2 #) & RelStr(# the carrier of b3,the InternalRel of b3 #) = RelStr(# the carrier of b4,the InternalRel of b4 #) implies for b5 being Function of b1,b3 holds
( b5 is isomorphic implies for b6 being Function of b2,b4 holds
( b6 = b5 implies b6 is isomorphic ) ) )
proof end;

theorem Th7: :: WAYBEL29:7
for b1, b2, b3 being RelStr
for b4 being Function of b1,b2 holds
( b4 is isomorphic implies for b5 being Function of b2,b3 holds
( b5 is isomorphic implies for b6 being Function of b1,b3 holds
( b6 = b5 * b4 implies b6 is isomorphic ) ) )
proof end;

theorem Th8: :: WAYBEL29:8
canceled;

theorem Th9: :: WAYBEL29:9
canceled;

theorem Th10: :: WAYBEL29:10
for b1, b2, b3, b4 being TopSpace holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b3,the topology of b3 #) & TopStruct(# the carrier of b2,the topology of b2 #) = TopStruct(# the carrier of b4,the topology of b4 #) implies [:b1,b2:] = [:b3,b4:] )
proof end;

theorem Th11: :: WAYBEL29:11
for b1 being non empty TopSpace
for b2 being non empty up-complete Scott TopPoset
for b3 being non empty directed Subset of (ContMaps b1,b2) holds
"\/" b3,(b2 |^ the carrier of b1) is continuous Function of b1,b2
proof end;

theorem Th12: :: WAYBEL29:12
for b1 being non empty TopSpace
for b2 being non empty up-complete Scott TopPoset holds
ContMaps b1,b2 is directed-sups-inheriting SubRelStr of b2 |^ the carrier of b1
proof end;

theorem Th13: :: WAYBEL29:13
for b1, b2 being TopStruct holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) implies for b3, b4 being non empty TopRelStr holds
( TopRelStr(# the carrier of b3,the InternalRel of b3,the topology of b3 #) = TopRelStr(# the carrier of b4,the InternalRel of b4,the topology of b4 #) implies ContMaps b1,b3 = ContMaps b2,b4 ) )
proof end;

registration
cluster complete Scott continuous -> T_0 complete continuous injective TopRelStr ;
coherence
for b1 being complete continuous TopLattice holds
( b1 is Scott implies ( b1 is injective & b1 is T_0 ) )
proof end;
end;

registration
cluster T_0 complete Scott continuous injective TopRelStr ;
existence
ex b1 being TopLattice st
( b1 is Scott & b1 is continuous & b1 is complete )
proof end;
end;

registration
let c1 be non empty TopSpace;
let c2 be non empty up-complete Scott TopPoset;
cluster ContMaps a1,a2 -> up-complete ;
coherence
ContMaps c1,c2 is up-complete
proof end;
end;

theorem Th14: :: WAYBEL29:14
for b1 being non empty set
for b2 being non-Empty Poset-yielding ManySortedSet of b1 holds
( ( for b3 being Element of b1 holds b2 . b3 is up-complete ) implies b1 -POS_prod b2 is up-complete )
proof end;

theorem Th15: :: WAYBEL29:15
for b1 being non empty set
for b2 being non-Empty reflexive-yielding Poset-yielding ManySortedSet of b1 holds
( ( for b3 being Element of b1 holds
( b2 . b3 is up-complete & b2 . b3 is lower-bounded ) ) implies for b3, b4 being Element of (product b2) holds
( b3 << b4 iff ( ( for b5 being Element of b1 holds b3 . b5 << b4 . b5 ) & ex b5 being finite Subset of b1 st
for b6 being Element of b1 holds
( not b6 in b5 implies b3 . b6 = Bottom (b2 . b6) ) ) ) )
proof end;

registration
let c1 be set ;
let c2 be non empty reflexive antisymmetric lower-bounded RelStr ;
cluster a2 |^ a1 -> lower-bounded ;
coherence
c2 |^ c1 is lower-bounded
proof end;
end;

registration
let c1 be non empty TopSpace;
let c2 be non empty lower-bounded TopPoset;
cluster ContMaps a1,a2 -> lower-bounded ;
coherence
ContMaps c1,c2 is lower-bounded
proof end;
end;

registration
let c1 be non empty up-complete Poset;
cluster -> up-complete TopAugmentation of a1;
coherence
for b1 being TopAugmentation of c1 holds b1 is up-complete
proof end;
cluster Scott -> correct TopAugmentation of a1;
coherence
for b1 being TopAugmentation of c1 holds
( b1 is Scott implies b1 is TopSpace-like )
proof end;
end;

registration
let c1 be non empty up-complete Poset;
cluster correct up-complete strict Scott TopAugmentation of a1;
existence
ex b1 being TopAugmentation of c1 st
( b1 is strict & b1 is Scott )
proof end;
end;

theorem Th16: :: WAYBEL29:16
canceled;

theorem Th17: :: WAYBEL29:17
for b1 being non empty up-complete Poset
for b2, b3 being Scott TopAugmentation of b1 holds the topology of b2 = the topology of b3
proof end;

theorem Th18: :: WAYBEL29:18
for b1, b2 being non empty reflexive antisymmetric up-complete TopRelStr holds
( TopRelStr(# the carrier of b1,the InternalRel of b1,the topology of b1 #) = TopRelStr(# the carrier of b2,the InternalRel of b2,the topology of b2 #) & b1 is Scott implies b2 is Scott )
proof end;

definition
let c1 be non empty up-complete Poset;
func Sigma c1 -> strict Scott TopAugmentation of a1 means :Def1: :: WAYBEL29:def 1
verum;
uniqueness
for b1, b2 being strict Scott TopAugmentation of c1 holds b1 = b2
proof end;
existence
ex b1 being strict Scott TopAugmentation of c1 st
verum
;
end;

:: deftheorem Def1 defines Sigma WAYBEL29:def 1 :
for b1 being non empty up-complete Poset
for b2 being strict Scott TopAugmentation of b1 holds
( b2 = Sigma b1 iff verum );

theorem Th19: :: WAYBEL29:19
for b1 being non empty up-complete Scott TopPoset holds Sigma b1 = TopRelStr(# the carrier of b1,the InternalRel of b1,the topology of b1 #)
proof end;

theorem Th20: :: WAYBEL29:20
for b1, b2 being non empty up-complete Poset holds
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of b2,the InternalRel of b2 #) implies Sigma b1 = Sigma b2 )
proof end;

definition
let c1, c2 be non empty up-complete Poset;
let c3 be Function of c1,c2;
func Sigma c3 -> Function of (Sigma a1),(Sigma a2) equals :: WAYBEL29:def 2
a3;
coherence
c3 is Function of (Sigma c1),(Sigma c2)
proof end;
end;

:: deftheorem Def2 defines Sigma WAYBEL29:def 2 :
for b1, b2 being non empty up-complete Poset
for b3 being Function of b1,b2 holds Sigma b3 = b3;

registration
let c1, c2 be non empty up-complete Poset;
let c3 be directed-sups-preserving Function of c1,c2;
cluster Sigma a3 -> continuous ;
coherence
Sigma c3 is continuous
proof end;
end;

theorem Th21: :: WAYBEL29:21
for b1, b2 being non empty up-complete Poset
for b3 being Function of b1,b2 holds
( b3 is isomorphic iff Sigma b3 is isomorphic )
proof end;

theorem Th22: :: WAYBEL29:22
for b1 being non empty TopSpace
for b2 being complete Scott TopLattice holds oContMaps b1,b2 = ContMaps b1,b2
proof end;

definition
let c1, c2 be non empty TopSpace;
func Theta c1,c2 -> Function of (InclPoset the topology of [:a1,a2:]),(ContMaps a1,(Sigma (InclPoset the topology of a2))) means :Def3: :: WAYBEL29:def 3
for b1 being open Subset of [:a1,a2:] holds a3 . b1 = b1,the carrier of a1 *graph ;
existence
ex b1 being Function of (InclPoset the topology of [:c1,c2:]),(ContMaps c1,(Sigma (InclPoset the topology of c2))) st
for b2 being open Subset of [:c1,c2:] holds b1 . b2 = b2,the carrier of c1 *graph
proof end;
correctness
uniqueness
for b1, b2 being Function of (InclPoset the topology of [:c1,c2:]),(ContMaps c1,(Sigma (InclPoset the topology of c2))) holds
( ( for b3 being open Subset of [:c1,c2:] holds b1 . b3 = b3,the carrier of c1 *graph ) & ( for b3 being open Subset of [:c1,c2:] holds b2 . b3 = b3,the carrier of c1 *graph ) implies b1 = b2 )
;
proof end;
end;

:: deftheorem Def3 defines Theta WAYBEL29:def 3 :
for b1, b2 being non empty TopSpace
for b3 being Function of (InclPoset the topology of [:b1,b2:]),(ContMaps b1,(Sigma (InclPoset the topology of b2))) holds
( b3 = Theta b1,b2 iff for b4 being open Subset of [:b1,b2:] holds b3 . b4 = b4,the carrier of b1 *graph );

defpred S1[ T_0-TopSpace] means for b1 being non empty TopSpace
for b2 being complete Scott continuous TopLattice
for b3 being Scott TopAugmentation of ContMaps a1,b2 holds
ex b4 being Function of (ContMaps b1,b3),(ContMaps [:b1,a1:],b2)ex b5 being Function of (ContMaps [:b1,a1:],b2),(ContMaps b1,b3) st
( b4 is uncurrying & b4 is one-to-one & b4 is onto & b5 is currying & b5 is one-to-one & b5 is onto );

defpred S2[ T_0-TopSpace] means for b1 being non empty TopSpace
for b2 being complete Scott continuous TopLattice
for b3 being Scott TopAugmentation of ContMaps a1,b2 holds
ex b4 being Function of (ContMaps b1,b3),(ContMaps [:b1,a1:],b2)ex b5 being Function of (ContMaps [:b1,a1:],b2),(ContMaps b1,b3) st
( b4 is uncurrying & b4 is isomorphic & b5 is currying & b5 is isomorphic );

defpred S3[ T_0-TopSpace] means for b1 being non empty TopSpace holds Theta b1,a1 is isomorphic;

defpred S4[ T_0-TopSpace] means for b1 being non empty TopSpace
for b2 being Scott TopAugmentation of InclPoset the topology of a1
for b3 being continuous Function of b1,b2 holds
*graph b3 is open Subset of [:b1,a1:];

defpred S5[ T_0-TopSpace] means for b1 being Scott TopAugmentation of InclPoset the topology of a1 holds
{ [b2,b3] where B is open Subset of a1, B is Element of a1 : b3 in b2 } is open Subset of [:b1,a1:];

defpred S6[ T_0-TopSpace] means for b1 being Scott TopAugmentation of InclPoset the topology of a1
for b2 being Element of a1
for b3 being open a_neighborhood of b2 holds
ex b4 being open Subset of b1 st
( b3 in b4 & meet b4 is a_neighborhood of b2 );

Lemma18: for b1 being T_0-TopSpace holds
( S1[b1] iff S2[b1] )
proof end;

definition
let c1 be non empty TopSpace;
func alpha c1 -> Function of (oContMaps a1,Sierpinski_Space ),(InclPoset the topology of a1) means :Def4: :: WAYBEL29:def 4
for b1 being continuous Function of a1,Sierpinski_Space holds a2 . b1 = b1 " {1};
existence
ex b1 being Function of (oContMaps c1,Sierpinski_Space ),(InclPoset the topology of c1) st
for b2 being continuous Function of c1,Sierpinski_Space holds b1 . b2 = b2 " {1}
proof end;
uniqueness
for b1, b2 being Function of (oContMaps c1,Sierpinski_Space ),(InclPoset the topology of c1) holds
( ( for b3 being continuous Function of c1,Sierpinski_Space holds b1 . b3 = b3 " {1} ) & ( for b3 being continuous Function of c1,Sierpinski_Space holds b2 . b3 = b3 " {1} ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def4 defines alpha WAYBEL29:def 4 :
for b1 being non empty TopSpace
for b2 being Function of (oContMaps b1,Sierpinski_Space ),(InclPoset the topology of b1) holds
( b2 = alpha b1 iff for b3 being continuous Function of b1,Sierpinski_Space holds b2 . b3 = b3 " {1} );

theorem Th23: :: WAYBEL29:23
for b1 being non empty TopSpace
for b2 being open Subset of b1 holds ((alpha b1) " ) . b2 = chi b2,the carrier of b1
proof end;

registration
let c1 be non empty TopSpace;
cluster alpha a1 -> isomorphic ;
coherence
alpha c1 is isomorphic
proof end;
end;

registration
let c1 be non empty TopSpace;
cluster (alpha a1) " -> isomorphic ;
coherence
(alpha c1) " is isomorphic
by YELLOW14:11;
end;

registration
let c1 be injective T_0-TopSpace;
cluster Omega a1 -> T_0 Scott injective ;
coherence
Omega c1 is Scott
proof end;
end;

registration
let c1 be non empty TopSpace;
cluster oContMaps a1,Sierpinski_Space -> complete ;
coherence
oContMaps c1,Sierpinski_Space is complete
proof end;
end;

theorem Th24: :: WAYBEL29:24
Omega Sierpinski_Space = Sigma (BoolePoset 1)
proof end;

registration
let c1 be non empty set ;
let c2 be injective T_0-TopSpace;
cluster product (a1 => a2) -> injective ;
coherence
c1 -TOP_prod (c1 => c2) is injective
proof end;
end;

theorem Th25: :: WAYBEL29:25
for b1 being non empty set
for b2 being complete continuous LATTICE holds Omega (b1 -TOP_prod (b1 => (Sigma b2))) = Sigma (b1 -POS_prod (b1 => b2))
proof end;

theorem Th26: :: WAYBEL29:26
for b1 being non empty set
for b2 being injective T_0-TopSpace holds Omega (b1 -TOP_prod (b1 => b2)) = Sigma (b1 -POS_prod (b1 => (Omega b2)))
proof end;

definition
let c1 be non empty set ;
let c2, c3 be non empty TopSpace;
func commute c2,c1,c3 -> Function of (oContMaps a2,(a1 -TOP_prod (a1 => a3))),((oContMaps a2,a3) |^ a1) means :Def5: :: WAYBEL29:def 5
for b1 being continuous Function of a2,(a1 -TOP_prod (a1 => a3)) holds a4 . b1 = commute b1;
uniqueness
for b1, b2 being Function of (oContMaps c2,(c1 -TOP_prod (c1 => c3))),((oContMaps c2,c3) |^ c1) holds
( ( for b3 being continuous Function of c2,(c1 -TOP_prod (c1 => c3)) holds b1 . b3 = commute b3 ) & ( for b3 being continuous Function of c2,(c1 -TOP_prod (c1 => c3)) holds b2 . b3 = commute b3 ) implies b1 = b2 )
proof end;
existence
ex b1 being Function of (oContMaps c2,(c1 -TOP_prod (c1 => c3))),((oContMaps c2,c3) |^ c1) st
for b2 being continuous Function of c2,(c1 -TOP_prod (c1 => c3)) holds b1 . b2 = commute b2
proof end;
end;

:: deftheorem Def5 defines commute WAYBEL29:def 5 :
for b1 being non empty set
for b2, b3 being non empty TopSpace
for b4 being Function of (oContMaps b2,(b1 -TOP_prod (b1 => b3))),((oContMaps b2,b3) |^ b1) holds
( b4 = commute b2,b1,b3 iff for b5 being continuous Function of b2,(b1 -TOP_prod (b1 => b3)) holds b4 . b5 = commute b5 );

registration
let c1 be non empty set ;
let c2, c3 be non empty TopSpace;
cluster commute a2,a1,a3 -> V5 onto ;
correctness
coherence
( commute c2,c1,c3 is one-to-one & commute c2,c1,c3 is onto )
;
proof end;
end;

registration
let c1 be non empty set ;
let c2 be non empty TopSpace;
cluster commute a2,a1,Sierpinski_Space -> V5 onto isomorphic ;
correctness
coherence
commute c2,c1,Sierpinski_Space is isomorphic
;
proof end;
end;

Lemma21: for b1 being T_0-TopSpace holds
( S3[b1] implies S4[b1] )
proof end;

theorem Th27: :: WAYBEL29:27
for b1, b2 being non empty TopSpace
for b3 being Scott TopAugmentation of InclPoset the topology of b2
for b4, b5 being Element of (ContMaps b1,b3) holds
( b4 <= b5 implies *graph b4 c= *graph b5 )
proof end;

Lemma23: for b1 being T_0-TopSpace holds
( S4[b1] implies S3[b1] )
proof end;

Lemma24: for b1 being T_0-TopSpace holds
( S4[b1] implies S5[b1] )
proof end;

Lemma25: for b1 being T_0-TopSpace holds
( S5[b1] implies S6[b1] )
proof end;

Lemma26: for b1 being T_0-TopSpace holds
( S6[b1] implies S4[b1] )
proof end;

Lemma27: for b1 being T_0-TopSpace holds
( S6[b1] implies InclPoset the topology of b1 is continuous )
proof end;

Lemma28: for b1 being T_0-TopSpace holds
( InclPoset the topology of b1 is continuous implies S6[b1] )
proof end;

theorem Th28: :: WAYBEL29:28
for b1 being T_0-TopSpace holds
( ( for b2 being non empty TopSpace
for b3 being complete Scott continuous TopLattice
for b4 being Scott TopAugmentation of ContMaps b1,b3 holds
ex b5 being Function of (ContMaps b2,b4),(ContMaps [:b2,b1:],b3)ex b6 being Function of (ContMaps [:b2,b1:],b3),(ContMaps b2,b4) st
( b5 is uncurrying & b5 is one-to-one & b5 is onto & b6 is currying & b6 is one-to-one & b6 is onto ) ) iff for b2 being non empty TopSpace
for b3 being complete Scott continuous TopLattice
for b4 being Scott TopAugmentation of ContMaps b1,b3 holds
ex b5 being Function of (ContMaps b2,b4),(ContMaps [:b2,b1:],b3)ex b6 being Function of (ContMaps [:b2,b1:],b3),(ContMaps b2,b4) st
( b5 is uncurrying & b5 is isomorphic & b6 is currying & b6 is isomorphic ) ) by Lemma18;

theorem Th29: :: WAYBEL29:29
for b1 being T_0-TopSpace holds
( InclPoset the topology of b1 is continuous iff for b2 being non empty TopSpace holds Theta b2,b1 is isomorphic )
proof end;

theorem Th30: :: WAYBEL29:30
for b1 being T_0-TopSpace holds
( InclPoset the topology of b1 is continuous iff for b2 being non empty TopSpace
for b3 being continuous Function of b2,(Sigma (InclPoset the topology of b1)) holds
*graph b3 is open Subset of [:b2,b1:] )
proof end;

theorem Th31: :: WAYBEL29:31
for b1 being T_0-TopSpace holds
( InclPoset the topology of b1 is continuous iff { [b2,b3] where B is open Subset of b1, B is Element of b1 : b3 in b2 } is open Subset of [:(Sigma (InclPoset the topology of b1)),b1:] )
proof end;

theorem Th32: :: WAYBEL29:32
for b1 being T_0-TopSpace holds
( InclPoset the topology of b1 is continuous iff for b2 being Element of b1
for b3 being open a_neighborhood of b2 holds
ex b4 being open Subset of (Sigma (InclPoset the topology of b1)) st
( b3 in b4 & meet b4 is a_neighborhood of b2 ) )
proof end;

defpred S7[ complete LATTICE] means InclPoset (sigma a1) is continuous;

defpred S8[ complete LATTICE] means for b1 being Scott TopAugmentation of a1
for b2 being complete LATTICE
for b3 being Scott TopAugmentation of b2 holds sigma [:b2,a1:] = the topology of [:b3,b1:];

defpred S9[ complete LATTICE] means for b1 being Scott TopAugmentation of a1
for b2 being complete LATTICE
for b3 being Scott TopAugmentation of b2
for b4 being Scott TopAugmentation of [:b2,a1:] holds TopStruct(# the carrier of b4,the topology of b4 #) = [:b3,b1:];

Lemma29: for b1 being complete LATTICE holds
( S8[b1] iff S9[b1] )
proof end;

theorem Th33: :: WAYBEL29:33
for b1, b2, b3 being non empty RelStr
for b4 being Function of b1,b3 holds
( b4 is isomorphic implies for b5 being Function of b2,b3 holds
( b5 = b4 & b5 is isomorphic implies RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of b2,the InternalRel of b2 #) ) )
proof end;

Lemma30: for b1 being complete LATTICE holds
( S7[b1] implies S8[b1] )
proof end;

Lemma31: for b1 being complete LATTICE holds
( S8[b1] implies S7[b1] )
proof end;

theorem Th34: :: WAYBEL29:34
for b1 being complete LATTICE holds
( InclPoset (sigma b1) is continuous iff for b2 being complete LATTICE holds sigma [:b2,b1:] = the topology of [:(Sigma b2),(Sigma b1):] )
proof end;

theorem Th35: :: WAYBEL29:35
for b1 being complete LATTICE holds
( ( for b2 being complete LATTICE holds sigma [:b2,b1:] = the topology of [:(Sigma b2),(Sigma b1):] ) iff for b2 being complete LATTICE holds TopStruct(# the carrier of (Sigma [:b2,b1:]),the topology of (Sigma [:b2,b1:]) #) = [:(Sigma b2),(Sigma b1):] )
proof end;

theorem Th36: :: WAYBEL29:36
for b1 being complete LATTICE holds
( ( for b2 being complete LATTICE holds sigma [:b2,b1:] = the topology of [:(Sigma b2),(Sigma b1):] ) iff for b2 being complete LATTICE holds Sigma [:b2,b1:] = Omega [:(Sigma b2),(Sigma b1):] )
proof end;

theorem Th37: :: WAYBEL29:37
for b1 being complete LATTICE holds
( InclPoset (sigma b1) is continuous iff for b2 being complete LATTICE holds Sigma [:b2,b1:] = Omega [:(Sigma b2),(Sigma b1):] )
proof end;