:: WAYBEL30 semantic presentation

theorem Th1: :: WAYBEL30:1
for b1 being set
for b2 being non empty set holds b1 /\ (union b2) = union { (b1 /\ b3) where B is Element of b2 : verum }
proof end;

theorem Th2: :: WAYBEL30:2
for b1 being non empty reflexive transitive RelStr
for b2 being non empty directed Subset of (InclPoset (Ids b1)) holds
union b2 is Ideal of b1
proof end;

E3: now
let c1 be non empty reflexive transitive RelStr ;
let c2 be non empty directed Subset of (InclPoset (Ids c1));
let c3 be Element of (InclPoset (Ids c1));
assume E4: c3 = union c2 ;
thus c2 is_<=_than c3
proof
let c4 be Element of (InclPoset (Ids c1)); :: according to LATTICE3:def 9
assume E5: c4 in c2 ;
c4 c= c3
proof
let c5 be set ; :: according to TARSKI:def 3
assume c5 in c4 ;
hence c5 in c3 by E4, E5, TARSKI:def 4;
end;
hence c4 <= c3 by YELLOW_1:3;
end;
end;

E4: now
let c1 be non empty reflexive transitive RelStr ;
let c2 be non empty directed Subset of (InclPoset (Ids c1));
let c3 be Element of (InclPoset (Ids c1));
assume E5: c3 = union c2 ;
thus for b1 being Element of (InclPoset (Ids c1)) holds
( b1 is_>=_than c2 implies c3 <= b1 )
proof
let c4 be Element of (InclPoset (Ids c1));
assume E6: for b1 being Element of (InclPoset (Ids c1)) holds
( b1 in c2 implies c4 >= b1 ) ; :: according to LATTICE3:def 9
c3 c= c4
proof
let c5 be set ; :: according to TARSKI:def 3
assume c5 in c3 ;
then consider c6 being set such that
E7: ( c5 in c6 & c6 in c2 ) by E5, TARSKI:def 4;
reconsider c7 = c6 as Element of (InclPoset (Ids c1)) by E7;
c4 >= c7 by E6, E7;
then c7 c= c4 by YELLOW_1:3;
hence c5 in c4 by E7;
end;
hence c3 <= c4 by YELLOW_1:3;
end;
end;

registration
let c1 be non empty reflexive transitive RelStr ;
cluster InclPoset (Ids a1) -> up-complete ;
coherence
InclPoset (Ids c1) is up-complete
proof end;
end;

theorem Th3: :: WAYBEL30:3
for b1 being non empty reflexive transitive RelStr
for b2 being non empty directed Subset of (InclPoset (Ids b1)) holds sup b2 = union b2
proof end;

theorem Th4: :: WAYBEL30:4
for b1 being Semilattice
for b2 being non empty directed Subset of (InclPoset (Ids b1))
for b3 being Element of (InclPoset (Ids b1)) holds sup ({b3} "/\" b2) = union { (b3 /\ b4) where B is Element of b2 : verum }
proof end;

registration
let c1 be Semilattice;
cluster InclPoset (Ids a1) -> up-complete satisfying_MC ;
coherence
InclPoset (Ids c1) is satisfying_MC
proof end;
end;

registration
let c1 be non empty trivial RelStr ;
cluster -> trivial TopAugmentation of a1;
coherence
for b1 being TopAugmentation of c1 holds b1 is trivial
proof end;
end;

theorem Th5: :: WAYBEL30:5
for b1 being complete Scott TopLattice
for b2 being complete LATTICE
for b3 being Scott TopAugmentation of b2 holds
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of b2,the InternalRel of b2 #) implies TopRelStr(# the carrier of b3,the InternalRel of b3,the topology of b3 #) = TopRelStr(# the carrier of b1,the InternalRel of b1,the topology of b1 #) )
proof end;

theorem Th6: :: WAYBEL30:6
for b1 being complete Lawson TopLattice
for b2 being complete LATTICE
for b3 being correct Lawson TopAugmentation of b2 holds
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of b2,the InternalRel of b2 #) implies TopRelStr(# the carrier of b3,the InternalRel of b3,the topology of b3 #) = TopRelStr(# the carrier of b1,the InternalRel of b1,the topology of b1 #) )
proof end;

theorem Th7: :: WAYBEL30:7
for b1 being complete Lawson TopLattice
for b2 being Scott TopAugmentation of b1
for b3 being Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 & b4 is closed implies b3 is closed )
proof end;

registration
let c1 be complete LATTICE;
cluster lambda a1 -> non empty ;
coherence
not lambda c1 is empty
proof end;
end;

registration
let c1 be complete Scott TopLattice;
cluster InclPoset (sigma a1) -> non trivial complete ;
coherence
( InclPoset (sigma c1) is complete & not InclPoset (sigma c1) is trivial )
proof end;
end;

registration
let c1 be complete Lawson TopLattice;
cluster InclPoset (sigma a1) -> non trivial complete ;
coherence
( InclPoset (sigma c1) is complete & not InclPoset (sigma c1) is trivial )
proof end;
cluster InclPoset (lambda a1) -> non trivial complete ;
coherence
( InclPoset (lambda c1) is complete & not InclPoset (lambda c1) is trivial )
proof end;
end;

theorem Th8: :: WAYBEL30:8
for b1 being non empty reflexive RelStr holds sigma b1 c= { (b2 \ (uparrow b3)) where B is Subset of b1, B is Subset of b1 : ( b2 in sigma b1 & b3 is finite ) }
proof end;

theorem Th9: :: WAYBEL30:9
for b1 being complete Lawson TopLattice holds lambda b1 = the topology of b1
proof end;

theorem Th10: :: WAYBEL30:10
for b1 being complete Lawson TopLattice holds sigma b1 c= lambda b1
proof end;

theorem Th11: :: WAYBEL30:11
for b1, b2 being complete LATTICE holds
( RelStr(# the carrier of b1,the InternalRel of b1 #) = RelStr(# the carrier of b2,the InternalRel of b2 #) implies lambda b1 = lambda b2 )
proof end;

theorem Th12: :: WAYBEL30:12
for b1 being complete Lawson TopLattice
for b2 being Subset of b1 holds
( b2 in lambda b1 iff b2 is open )
proof end;

registration
cluster non empty TopSpace-like trivial reflexive -> non empty reflexive Scott TopRelStr ;
coherence
for b1 being non empty reflexive TopRelStr holds
( b1 is trivial & b1 is TopSpace-like implies b1 is Scott )
proof end;
end;

registration
cluster trivial complete -> complete Lawson TopRelStr ;
coherence
for b1 being complete TopLattice holds
( b1 is trivial implies b1 is Lawson )
proof end;
end;

registration
cluster lower-bounded complete meet-continuous strict Scott continuous TopRelStr ;
existence
ex b1 being complete TopLattice st
( b1 is strict & b1 is continuous & b1 is lower-bounded & b1 is meet-continuous & b1 is Scott )
proof end;
end;

registration
cluster complete compact Hausdorff strict Lawson continuous TopRelStr ;
existence
ex b1 being complete TopLattice st
( b1 is strict & b1 is continuous & b1 is compact & b1 is being_T2 & b1 is Lawson )
proof end;
end;

scheme :: WAYBEL30:sch 1
s1{ F1() -> Scott TopLattice, F2() -> set , F3( set ) -> set } :
{ F3(b1) where B is Element of F1() : b1 in {} } = {}
proof end;

theorem Th13: :: WAYBEL30:13
for b1 being meet-continuous LATTICE
for b2 being Subset of b1 holds
( b2 has_the_property_(S) implies uparrow b2 has_the_property_(S) )
proof end;

registration
let c1 be meet-continuous LATTICE;
let c2 be property(S) Subset of c1;
cluster uparrow a2 -> property(S) ;
coherence
uparrow c2 is property(S)
by Th13;
end;

theorem Th14: :: WAYBEL30:14
for b1 being complete meet-continuous Lawson TopLattice
for b2 being Scott TopAugmentation of b1
for b3 being Subset of b1 holds
( b3 in lambda b1 implies uparrow b3 in sigma b2 )
proof end;

theorem Th15: :: WAYBEL30:15
for b1 being complete meet-continuous Lawson TopLattice
for b2 being Scott TopAugmentation of b1
for b3 being Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 & b3 is open implies uparrow b4 is open )
proof end;

theorem Th16: :: WAYBEL30:16
for b1 being complete meet-continuous Lawson TopLattice
for b2 being Scott TopAugmentation of b1
for b3 being Point of b2
for b4 being Point of b1
for b5 being Basis of b4 holds
( b3 = b4 implies { (uparrow b6) where B is Subset of b1 : b6 in b5 } is Basis of b3 )
proof end;

theorem Th17: :: WAYBEL30:17
for b1 being complete meet-continuous Lawson TopLattice
for b2 being Scott TopAugmentation of b1
for b3 being upper Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 implies Int b3 = Int b4 )
proof end;

theorem Th18: :: WAYBEL30:18
for b1 being complete meet-continuous Lawson TopLattice
for b2 being Scott TopAugmentation of b1
for b3 being lower Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 implies Cl b3 = Cl b4 )
proof end;

theorem Th19: :: WAYBEL30:19
for b1, b2 being complete LATTICE
for b3 being correct Lawson TopAugmentation of b1
for b4 being correct Lawson TopAugmentation of b2 holds
( InclPoset (sigma b2) is continuous implies the topology of [:b3,b4:] = lambda [:b1,b2:] )
proof end;

theorem Th20: :: WAYBEL30:20
for b1, b2 being complete LATTICE
for b3 being correct Lawson TopAugmentation of [:b1,b2:]
for b4 being correct Lawson TopAugmentation of b1
for b5 being correct Lawson TopAugmentation of b2 holds
( InclPoset (sigma b2) is continuous implies TopStruct(# the carrier of b3,the topology of b3 #) = [:b4,b5:] )
proof end;

theorem Th21: :: WAYBEL30:21
for b1 being complete meet-continuous Lawson TopLattice
for b2 being Element of b1 holds b2 "/\" is continuous
proof end;

registration
let c1 be complete meet-continuous Lawson TopLattice;
let c2 be Element of c1;
cluster a2 "/\" -> continuous ;
coherence
c2 "/\" is continuous
by Th21;
end;

theorem Th22: :: WAYBEL30:22
for b1 being complete meet-continuous Lawson TopLattice holds
( InclPoset (sigma b1) is continuous implies b1 is topological_semilattice )
proof end;

E20: now
let c1, c2, c3, c4 be set ;
assume E21: ( c3 in c1 & c4 in c2 ) ;
E22: dom <:(pr2 c1,c2),(pr1 c1,c2):> = (dom (pr2 c1,c2)) /\ (dom (pr1 c1,c2)) by FUNCT_3:def 8
.= (dom (pr2 c1,c2)) /\ [:c1,c2:] by FUNCT_3:def 5
.= [:c1,c2:] /\ [:c1,c2:] by FUNCT_3:def 6
.= [:c1,c2:] ;
[c3,c4] in [:c1,c2:] by E21, ZFMISC_1:106;
hence <:(pr2 c1,c2),(pr1 c1,c2):> . [c3,c4] = [((pr2 c1,c2) . [c3,c4]),((pr1 c1,c2) . [c3,c4])] by E22, FUNCT_3:def 8
.= [c4,((pr1 c1,c2) . [c3,c4])] by E21, FUNCT_3:def 6
.= [c4,c3] by E21, FUNCT_3:def 5 ;

end;

theorem Th23: :: WAYBEL30:23
for b1 being complete meet-continuous Lawson TopLattice holds
( InclPoset (sigma b1) is continuous implies ( b1 is being_T2 iff for b2 being Subset of [:b1,b1:] holds
( b2 = the InternalRel of b1 implies b2 is closed ) ) )
proof end;

definition
let c1 be non empty reflexive RelStr ;
let c2 be Subset of c1;
func c2 ^0 -> Subset of a1 equals :: WAYBEL30:def 1
{ b1 where B is Element of a1 : for b1 being non empty directed Subset of a1 holds
not ( b1 <= sup b2 & not a2 meets b2 )
}
;
coherence
{ b1 where B is Element of c1 : for b1 being non empty directed Subset of c1 holds
not ( b1 <= sup b2 & not c2 meets b2 )
}
is Subset of c1
proof end;
end;

:: deftheorem Def1 defines ^0 WAYBEL30:def 1 :
for b1 being non empty reflexive RelStr
for b2 being Subset of b1 holds b2 ^0 = { b3 where B is Element of b1 : for b1 being non empty directed Subset of b1 holds
not ( b3 <= sup b4 & not b2 meets b4 )
}
;

registration
let c1 be non empty reflexive antisymmetric RelStr ;
let c2 be empty Subset of c1;
cluster a2 ^0 -> empty ;
coherence
c2 ^0 is empty
proof end;
end;

theorem Th24: :: WAYBEL30:24
for b1 being non empty reflexive RelStr
for b2, b3 being Subset of b1 holds
( b2 c= b3 implies b2 ^0 c= b3 ^0 )
proof end;

theorem Th25: :: WAYBEL30:25
for b1 being non empty reflexive RelStr
for b2 being Element of b1 holds (uparrow b2) ^0 = wayabove b2
proof end;

theorem Th26: :: WAYBEL30:26
for b1 being Scott TopLattice
for b2 being upper Subset of b1 holds Int b2 c= b2 ^0
proof end;

theorem Th27: :: WAYBEL30:27
for b1 being non empty reflexive RelStr
for b2, b3 being Subset of b1 holds (b2 ^0 ) \/ (b3 ^0 ) c= (b2 \/ b3) ^0
proof end;

theorem Th28: :: WAYBEL30:28
for b1 being meet-continuous LATTICE
for b2, b3 being upper Subset of b1 holds (b2 ^0 ) \/ (b3 ^0 ) = (b2 \/ b3) ^0
proof end;

theorem Th29: :: WAYBEL30:29
for b1 being meet-continuous Scott TopLattice
for b2 being finite Subset of b1 holds Int (uparrow b2) c= union { (wayabove b3) where B is Element of b1 : b3 in b2 }
proof end;

theorem Th30: :: WAYBEL30:30
for b1 being complete Lawson TopLattice holds
( b1 is continuous iff ( b1 is meet-continuous & b1 is being_T2 ) )
proof end;

registration
cluster complete Lawson continuous -> complete Hausdorff TopRelStr ;
coherence
for b1 being complete TopLattice holds
( b1 is continuous & b1 is Lawson implies b1 is being_T2 )
by Th30;
cluster complete meet-continuous Hausdorff Lawson -> complete continuous TopRelStr ;
coherence
for b1 being complete TopLattice holds
( b1 is meet-continuous & b1 is Lawson & b1 is being_T2 implies b1 is continuous )
by Th30;
end;

definition
let c1 be non empty TopRelStr ;
attr a1 is with_small_semilattices means :: WAYBEL30:def 2
for b1 being Point of a1 holds
ex b2 being basis of b1 st
for b3 being Subset of a1 holds
( b3 in b2 implies subrelstr b3 is meet-inheriting );
attr a1 is with_compact_semilattices means :: WAYBEL30:def 3
for b1 being Point of a1 holds
ex b2 being basis of b1 st
for b3 being Subset of a1 holds
( b3 in b2 implies ( subrelstr b3 is meet-inheriting & b3 is compact ) );
attr a1 is with_open_semilattices means :Def4: :: WAYBEL30:def 4
for b1 being Point of a1 holds
ex b2 being Basis of b1 st
for b3 being Subset of a1 holds
( b3 in b2 implies subrelstr b3 is meet-inheriting );
end;

:: deftheorem Def2 defines with_small_semilattices WAYBEL30:def 2 :
for b1 being non empty TopRelStr holds
( b1 is with_small_semilattices iff for b2 being Point of b1 holds
ex b3 being basis of b2 st
for b4 being Subset of b1 holds
( b4 in b3 implies subrelstr b4 is meet-inheriting ) );

:: deftheorem Def3 defines with_compact_semilattices WAYBEL30:def 3 :
for b1 being non empty TopRelStr holds
( b1 is with_compact_semilattices iff for b2 being Point of b1 holds
ex b3 being basis of b2 st
for b4 being Subset of b1 holds
( b4 in b3 implies ( subrelstr b4 is meet-inheriting & b4 is compact ) ) );

:: deftheorem Def4 defines with_open_semilattices WAYBEL30:def 4 :
for b1 being non empty TopRelStr holds
( b1 is with_open_semilattices iff for b2 being Point of b1 holds
ex b3 being Basis of b2 st
for b4 being Subset of b1 holds
( b4 in b3 implies subrelstr b4 is meet-inheriting ) );

registration
cluster non empty TopSpace-like with_open_semilattices -> non empty TopSpace-like with_small_semilattices TopRelStr ;
coherence
for b1 being non empty TopSpace-like TopRelStr holds
( b1 is with_open_semilattices implies b1 is with_small_semilattices )
proof end;
cluster non empty TopSpace-like with_compact_semilattices -> non empty TopSpace-like with_small_semilattices TopRelStr ;
coherence
for b1 being non empty TopSpace-like TopRelStr holds
( b1 is with_compact_semilattices implies b1 is with_small_semilattices )
proof end;
cluster non empty anti-discrete -> non empty with_small_semilattices with_open_semilattices TopRelStr ;
coherence
for b1 being non empty TopRelStr holds
( b1 is anti-discrete implies ( b1 is with_small_semilattices & b1 is with_open_semilattices ) )
proof end;
cluster non empty TopSpace-like trivial reflexive -> non empty with_compact_semilattices TopRelStr ;
coherence
for b1 being non empty TopRelStr holds
( b1 is reflexive & b1 is trivial & b1 is TopSpace-like implies b1 is with_compact_semilattices )
proof end;
end;

registration
cluster trivial Hausdorff strict lower Lawson Scott with_small_semilattices with_compact_semilattices with_open_semilattices TopRelStr ;
existence
ex b1 being TopLattice st
( b1 is strict & b1 is trivial & b1 is lower )
proof end;
end;

theorem Th31: :: WAYBEL30:31
for b1 being with_infima topological_semilattice TopPoset
for b2 being Subset of b1 holds
( subrelstr b2 is meet-inheriting implies subrelstr (Cl b2) is meet-inheriting )
proof end;

theorem Th32: :: WAYBEL30:32
for b1 being complete meet-continuous Lawson TopLattice
for b2 being Scott TopAugmentation of b1 holds
( ( for b3 being Point of b2 holds
ex b4 being Basis of b3 st
for b5 being Subset of b2 holds
( b5 in b4 implies b5 is Filter of b2 ) ) iff b1 is with_open_semilattices )
proof end;

theorem Th33: :: WAYBEL30:33
for b1 being complete Lawson TopLattice
for b2 being Scott TopAugmentation of b1
for b3 being Element of b1 holds { (inf b4) where B is Subset of b2 : ( b3 in b4 & b4 in sigma b2 ) } c= { (inf b4) where B is Subset of b1 : ( b3 in b4 & b4 in lambda b1 ) }
proof end;

theorem Th34: :: WAYBEL30:34
for b1 being complete meet-continuous Lawson TopLattice
for b2 being Scott TopAugmentation of b1
for b3 being Element of b1 holds { (inf b4) where B is Subset of b2 : ( b3 in b4 & b4 in sigma b2 ) } = { (inf b4) where B is Subset of b1 : ( b3 in b4 & b4 in lambda b1 ) }
proof end;

theorem Th35: :: WAYBEL30:35
for b1 being complete meet-continuous Lawson TopLattice holds
( b1 is continuous iff ( b1 is with_open_semilattices & InclPoset (sigma b1) is continuous ) )
proof end;

registration
cluster complete Lawson continuous -> complete Lawson with_small_semilattices with_open_semilattices TopRelStr ;
coherence
for b1 being complete Lawson TopLattice holds
( b1 is continuous implies b1 is with_open_semilattices )
proof end;
end;

registration
let c1 be complete Lawson continuous TopLattice;
cluster InclPoset (sigma a1) -> non trivial complete continuous ;
coherence
InclPoset (sigma c1) is continuous
by Th35;
end;

theorem Th36: :: WAYBEL30:36
for b1 being complete Lawson continuous TopLattice holds
( b1 is compact & b1 is being_T2 & b1 is topological_semilattice & b1 is with_open_semilattices )
proof end;

theorem Th37: :: WAYBEL30:37
for b1 being complete Hausdorff Lawson topological_semilattice with_open_semilattices TopLattice holds b1 is with_compact_semilattices
proof end;

theorem Th38: :: WAYBEL30:38
for b1 being complete meet-continuous Hausdorff Lawson TopLattice
for b2 being Element of b1 holds b2 = "\/" { (inf b3) where B is Subset of b1 : ( b2 in b3 & b3 in lambda b1 ) } ,b1
proof end;

theorem Th39: :: WAYBEL30:39
for b1 being complete meet-continuous Lawson TopLattice holds
( b1 is continuous iff for b2 being Element of b1 holds b2 = "\/" { (inf b3) where B is Subset of b1 : ( b2 in b3 & b3 in lambda b1 ) } ,b1 )
proof end;

theorem Th40: :: WAYBEL30:40
for b1 being complete meet-continuous Lawson TopLattice holds
( b1 is algebraic iff ( b1 is with_open_semilattices & InclPoset (sigma b1) is algebraic ) )
proof end;

registration
let c1 be complete algebraic meet-continuous Lawson TopLattice;
cluster InclPoset (sigma a1) -> non trivial complete algebraic continuous ;
coherence
InclPoset (sigma c1) is algebraic
by Th40;
end;