:: WAYBEL14 semantic presentation

theorem Th1: :: WAYBEL14:1
for b1 being set
for b2 being finite Subset-Family of b1 holds
ex b3 being finite Subset-Family of b1 st
( b3 c= b2 & union b3 = union b2 & ( for b4 being Subset of b1 holds
not ( b4 in b3 & b4 c= union (b3 \ {b4}) ) ) )
proof end;

Lemma2: for b1 being 1-sorted
for b2, b3 being Subset of b1 holds
( b2 c= b3 ` iff b3 c= b2 ` )
proof end;

theorem Th2: :: WAYBEL14:2
for b1 being 1-sorted
for b2 being Subset of b1 holds
( b2 ` = the carrier of b1 iff b2 is empty )
proof end;

theorem Th3: :: WAYBEL14:3
for b1 being non empty transitive antisymmetric with_infima RelStr
for b2, b3 being Element of b1 holds downarrow (b2 "/\" b3) = (downarrow b2) /\ (downarrow b3)
proof end;

theorem Th4: :: WAYBEL14:4
for b1 being non empty transitive antisymmetric with_suprema RelStr
for b2, b3 being Element of b1 holds uparrow (b2 "\/" b3) = (uparrow b2) /\ (uparrow b3)
proof end;

theorem Th5: :: WAYBEL14:5
for b1 being non empty antisymmetric complete RelStr
for b2 being lower Subset of b1 holds
( sup b2 in b2 implies b2 = downarrow (sup b2) )
proof end;

theorem Th6: :: WAYBEL14:6
for b1 being non empty antisymmetric complete RelStr
for b2 being upper Subset of b1 holds
( inf b2 in b2 implies b2 = uparrow (inf b2) )
proof end;

theorem Th7: :: WAYBEL14:7
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of b1 holds
( b2 << b3 iff uparrow b3 c= wayabove b2 )
proof end;

theorem Th8: :: WAYBEL14:8
for b1 being non empty reflexive transitive RelStr
for b2, b3 being Element of b1 holds
( b2 << b3 iff downarrow b2 c= waybelow b3 )
proof end;

theorem Th9: :: WAYBEL14:9
for b1 being non empty reflexive antisymmetric complete RelStr
for b2 being Element of b1 holds
( sup (waybelow b2) <= b2 & b2 <= inf (wayabove b2) )
proof end;

theorem Th10: :: WAYBEL14:10
for b1 being non empty antisymmetric lower-bounded RelStr holds uparrow (Bottom b1) = the carrier of b1
proof end;

theorem Th11: :: WAYBEL14:11
for b1 being non empty antisymmetric upper-bounded RelStr holds downarrow (Top b1) = the carrier of b1
proof end;

theorem Th12: :: WAYBEL14:12
for b1 being with_suprema Poset
for b2, b3 being Element of b1 holds (wayabove b2) "\/" (wayabove b3) c= uparrow (b2 "\/" b3)
proof end;

theorem Th13: :: WAYBEL14:13
for b1 being with_infima Poset
for b2, b3 being Element of b1 holds (waybelow b2) "/\" (waybelow b3) c= downarrow (b2 "/\" b3)
proof end;

theorem Th14: :: WAYBEL14:14
for b1 being non empty with_suprema Poset
for b2 being Element of b1 holds
( b2 is co-prime iff for b3, b4 being Element of b1 holds
not ( b2 <= b3 "\/" b4 & not b2 <= b3 & not b2 <= b4 ) )
proof end;

theorem Th15: :: WAYBEL14:15
for b1 being non empty complete Poset
for b2 being non empty Subset of b1 holds downarrow (inf b2) = meet { (downarrow b3) where B is Element of b1 : b3 in b2 }
proof end;

theorem Th16: :: WAYBEL14:16
for b1 being non empty complete Poset
for b2 being non empty Subset of b1 holds uparrow (sup b2) = meet { (uparrow b3) where B is Element of b1 : b3 in b2 }
proof end;

registration
let c1 be sup-Semilattice;
let c2 be Element of c1;
cluster compactbelow a2 -> directed ;
coherence
compactbelow c2 is directed
proof end;
end;

theorem Th17: :: WAYBEL14:17
for b1 being non empty TopSpace
for b2 being irreducible Subset of b1
for b3 being Element of (InclPoset the topology of b1) holds
( b3 = b2 ` implies b3 is prime )
proof end;

theorem Th18: :: WAYBEL14:18
for b1 being non empty TopSpace
for b2, b3 being Element of (InclPoset the topology of b1) holds
( b2 "\/" b3 = b2 \/ b3 & b2 "/\" b3 = b2 /\ b3 )
proof end;

theorem Th19: :: WAYBEL14:19
for b1 being non empty TopSpace
for b2 being Element of (InclPoset the topology of b1) holds
( b2 is prime iff for b3, b4 being Element of (InclPoset the topology of b1) holds
not ( b3 /\ b4 c= b2 & not b3 c= b2 & not b4 c= b2 ) )
proof end;

theorem Th20: :: WAYBEL14:20
for b1 being non empty TopSpace
for b2 being Element of (InclPoset the topology of b1) holds
( b2 is co-prime iff for b3, b4 being Element of (InclPoset the topology of b1) holds
not ( b2 c= b3 \/ b4 & not b2 c= b3 & not b2 c= b4 ) )
proof end;

registration
let c1 be non empty TopSpace;
cluster InclPoset the topology of a1 -> distributive ;
coherence
InclPoset the topology of c1 is distributive
proof end;
end;

theorem Th21: :: WAYBEL14:21
for b1 being non empty TopSpace
for b2 being TopLattice
for b3 being Point of b1
for b4 being Point of b2
for b5 being Subset-Family of b2 holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b3 = b4 & b5 is Basis of b4 implies b5 is Basis of b3 )
proof end;

theorem Th22: :: WAYBEL14:22
for b1 being TopLattice
for b2 being Element of b1 holds
( ( for b3 being Subset of b1 holds
( b3 is open implies b3 is upper ) ) implies uparrow b2 is compact )
proof end;

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

theorem Th23: :: WAYBEL14:23
for b1 being complete Scott TopLattice holds sigma b1 = the topology of b1
proof end;

theorem Th24: :: WAYBEL14:24
for b1 being complete Scott TopLattice
for b2 being Subset of b1 holds
( b2 in sigma b1 iff b2 is open )
proof end;

Lemma19: for b1 being complete Scott TopLattice
for b2 being filtered Subset of b1
for b3 being Subset-Family of b1
for b4 being Subset of (InclPoset (sigma b1)) holds
( b3 = { (downarrow b5) where B is Element of b1 : b5 in b2 } & b4 = COMPLEMENT b3 implies b4 is directed )
proof end;

theorem Th25: :: WAYBEL14:25
for b1 being complete Scott TopLattice
for b2 being Subset of (InclPoset (sigma b1))
for b3 being filtered Subset of b1 holds
( b2 = { ((downarrow b4) ` ) where B is Element of b1 : b4 in b3 } implies b2 is directed )
proof end;

theorem Th26: :: WAYBEL14:26
for b1 being complete Scott TopLattice
for b2 being Element of b1
for b3 being Subset of b1 holds
( b3 is open & b2 in b3 implies inf b3 << b2 )
proof end;

definition
let c1 be non empty reflexive RelStr ;
let c2 be Function of [:c1,c1:],c1;
attr a2 is jointly_Scott-continuous means :Def1: :: WAYBEL14:def 1
for b1 being non empty TopSpace holds
not ( TopStruct(# the carrier of b1,the topology of b1 #) = ConvergenceSpace (Scott-Convergence a1) & ( for b2 being Function of [:b1,b1:],b1 holds
not ( b2 = a2 & b2 is continuous ) ) );
end;

:: deftheorem Def1 defines jointly_Scott-continuous WAYBEL14:def 1 :
for b1 being non empty reflexive RelStr
for b2 being Function of [:b1,b1:],b1 holds
( b2 is jointly_Scott-continuous iff for b3 being non empty TopSpace holds
not ( TopStruct(# the carrier of b3,the topology of b3 #) = ConvergenceSpace (Scott-Convergence b1) & ( for b4 being Function of [:b3,b3:],b3 holds
not ( b4 = b2 & b4 is continuous ) ) ) );

theorem Th27: :: WAYBEL14:27
for b1 being complete Scott TopLattice
for b2 being Subset of b1
for b3 being Element of (InclPoset (sigma b1)) holds
( b3 = b2 implies ( b3 is co-prime iff ( b2 is filtered & b2 is upper ) ) )
proof end;

theorem Th28: :: WAYBEL14:28
for b1 being complete Scott TopLattice
for b2 being Subset of b1
for b3 being Element of (InclPoset (sigma b1)) holds
( b3 = b2 & ex b4 being Element of b1 st b2 = (downarrow b4) ` implies ( b3 is prime & b3 <> the carrier of b1 ) )
proof end;

theorem Th29: :: WAYBEL14:29
for b1 being complete Scott TopLattice
for b2 being Subset of b1
for b3 being Element of (InclPoset (sigma b1)) holds
not ( b3 = b2 & sup_op b1 is jointly_Scott-continuous & b3 is prime & b3 <> the carrier of b1 & ( for b4 being Element of b1 holds
not b2 = (downarrow b4) ` ) )
proof end;

theorem Th30: :: WAYBEL14:30
for b1 being complete Scott TopLattice holds
( b1 is continuous implies sup_op b1 is jointly_Scott-continuous )
proof end;

theorem Th31: :: WAYBEL14:31
for b1 being complete Scott TopLattice holds
( sup_op b1 is jointly_Scott-continuous implies b1 is sober )
proof end;

theorem Th32: :: WAYBEL14:32
for b1 being complete Scott TopLattice holds
( b1 is continuous implies ( b1 is compact & b1 is locally-compact & b1 is sober & b1 is Baire ) )
proof end;

theorem Th33: :: WAYBEL14:33
for b1 being complete Scott TopLattice
for b2 being Subset of b1 holds
( b1 is continuous & b2 in sigma b1 implies b2 = union { (wayabove b3) where B is Element of b1 : b3 in b2 } )
proof end;

theorem Th34: :: WAYBEL14:34
for b1 being complete Scott TopLattice holds
( ( for b2 being Subset of b1 holds
( b2 in sigma b1 implies b2 = union { (wayabove b3) where B is Element of b1 : b3 in b2 } ) ) implies b1 is continuous )
proof end;

theorem Th35: :: WAYBEL14:35
for b1 being complete Scott TopLattice
for b2 being Element of b1 holds
not ( b1 is continuous & ( for b3 being Basis of b2 holds
ex b4 being Subset of b1 st
( b4 in b3 & not ( b4 is open & b4 is filtered ) ) ) )
proof end;

theorem Th36: :: WAYBEL14:36
for b1 being complete Scott TopLattice holds
( b1 is continuous implies InclPoset (sigma b1) is continuous )
proof end;

theorem Th37: :: WAYBEL14:37
for b1 being complete Scott TopLattice
for b2 being Element of b1 holds
( ( for b3 being Element of b1 holds
ex b4 being Basis of b3 st
for b5 being Subset of b1 holds
( b5 in b4 implies ( b5 is open & b5 is filtered ) ) ) & InclPoset (sigma b1) is continuous implies b2 = "\/" { (inf b3) where B is Subset of b1 : ( b2 in b3 & b3 in sigma b1 ) } ,b1 )
proof end;

theorem Th38: :: WAYBEL14:38
for b1 being complete Scott TopLattice holds
( ( for b2 being Element of b1 holds b2 = "\/" { (inf b3) where B is Subset of b1 : ( b2 in b3 & b3 in sigma b1 ) } ,b1 ) implies b1 is continuous )
proof end;

theorem Th39: :: WAYBEL14:39
for b1 being complete Scott TopLattice holds
( ( for b2 being Element of b1 holds
ex b3 being Basis of b2 st
for b4 being Subset of b1 holds
( b4 in b3 implies ( b4 is open & b4 is filtered ) ) ) iff for b2 being Element of (InclPoset (sigma b1)) holds
ex b3 being Subset of (InclPoset (sigma b1)) st
( b2 = sup b3 & ( for b4 being Element of (InclPoset (sigma b1)) holds
( b4 in b3 implies b4 is co-prime ) ) ) )
proof end;

theorem Th40: :: WAYBEL14:40
for b1 being complete Scott TopLattice holds
( ( ( for b2 being Element of (InclPoset (sigma b1)) holds
ex b3 being Subset of (InclPoset (sigma b1)) st
( b2 = sup b3 & ( for b4 being Element of (InclPoset (sigma b1)) holds
( b4 in b3 implies b4 is co-prime ) ) ) ) & InclPoset (sigma b1) is continuous ) iff InclPoset (sigma b1) is completely-distributive )
proof end;

theorem Th41: :: WAYBEL14:41
for b1 being complete Scott TopLattice holds
( InclPoset (sigma b1) is completely-distributive iff ( InclPoset (sigma b1) is continuous & (InclPoset (sigma b1)) opp is continuous ) )
proof end;

theorem Th42: :: WAYBEL14:42
for b1 being complete Scott TopLattice holds
not ( b1 is algebraic & ( for b2 being Basis of b1 holds
not b2 = { (uparrow b3) where B is Element of b1 : b3 in the carrier of (CompactSublatt b1) } ) )
proof end;

theorem Th43: :: WAYBEL14:43
for b1 being complete Scott TopLattice holds
( ex b2 being Basis of b1 st b2 = { (uparrow b3) where B is Element of b1 : b3 in the carrier of (CompactSublatt b1) } implies ( InclPoset (sigma b1) is algebraic & ( for b2 being Element of (InclPoset (sigma b1)) holds
ex b3 being Subset of (InclPoset (sigma b1)) st
( b2 = sup b3 & ( for b4 being Element of (InclPoset (sigma b1)) holds
( b4 in b3 implies b4 is co-prime ) ) ) ) ) )
proof end;

theorem Th44: :: WAYBEL14:44
for b1 being complete Scott TopLattice holds
not ( InclPoset (sigma b1) is algebraic & ( for b2 being Element of (InclPoset (sigma b1)) holds
ex b3 being Subset of (InclPoset (sigma b1)) st
( b2 = sup b3 & ( for b4 being Element of (InclPoset (sigma b1)) holds
( b4 in b3 implies b4 is co-prime ) ) ) ) & ( for b2 being Basis of b1 holds
not b2 = { (uparrow b3) where B is Element of b1 : b3 in the carrier of (CompactSublatt b1) } ) )
proof end;

theorem Th45: :: WAYBEL14:45
for b1 being complete Scott TopLattice holds
( ex b2 being Basis of b1 st b2 = { (uparrow b3) where B is Element of b1 : b3 in the carrier of (CompactSublatt b1) } implies b1 is algebraic )
proof end;