:: WAYBEL12 semantic presentation

Lemma1: for b1 being non empty RelStr
for b2 being Function of NAT ,the carrier of b1
for b3 being Nat holds
{ (b2 . b4) where B is Nat : b4 <= b3 } is non empty finite Subset of b1
proof end;

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
redefine attr a2 is closed means :: WAYBEL12:def 1
a2 ` is open;
compatibility
( c2 is closed iff c2 ` is open )
proof end;
end;

:: deftheorem Def1 defines closed WAYBEL12:def 1 :
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is closed iff b2 ` is open );

definition
let c1 be TopStruct ;
let c2 be Subset-Family of c1;
attr a2 is dense means :Def2: :: WAYBEL12:def 2
for b1 being Subset of a1 holds
( b1 in a2 implies b1 is dense );
end;

:: deftheorem Def2 defines dense WAYBEL12:def 2 :
for b1 being TopStruct
for b2 being Subset-Family of b1 holds
( b2 is dense iff for b3 being Subset of b1 holds
( b3 in b2 implies b3 is dense ) );

registration
cluster empty 1-sorted ;
existence
ex b1 being 1-sorted st b1 is empty
proof end;
end;

registration
let c1 be empty 1-sorted ;
cluster the carrier of a1 -> empty ;
coherence
the carrier of c1 is empty
by STRUCT_0:def 1;
end;

registration
let c1 be empty 1-sorted ;
cluster -> empty Element of bool the carrier of a1;
coherence
for b1 being Subset of c1 holds b1 is empty
by XBOOLE_1:3;
end;

registration
cluster finite -> countable set ;
coherence
for b1 being set holds
( b1 is finite implies b1 is countable )
by CARD_4:43;
end;

registration
let c1 be 1-sorted ;
cluster empty countable Element of bool the carrier of a1;
existence
ex b1 being Subset of c1 st b1 is empty
proof end;
end;

registration
cluster non empty finite countable set ;
existence
ex b1 being set st
( not b1 is empty & b1 is finite )
proof end;
end;

registration
let c1 be non empty RelStr ;
cluster non empty finite countable Element of bool the carrier of a1;
existence
ex b1 being Subset of c1 st
( not b1 is empty & b1 is finite )
proof end;
end;

registration
cluster infinite countable set ;
existence
ex b1 being set st
( not b1 is finite & b1 is countable )
by CARD_4:44;
end;

registration
let c1 be 1-sorted ;
cluster empty countable Element of bool (bool the carrier of a1);
existence
ex b1 being Subset-Family of c1 st b1 is empty
proof end;
end;

theorem Th1: :: WAYBEL12:1
canceled;

theorem Th2: :: WAYBEL12:2
for b1, b2 being set holds
( Card b1 <=` Card b2 & b2 is countable implies b1 is countable )
proof end;

theorem Th3: :: WAYBEL12:3
for b1 being infinite countable set holds NAT ,b1 are_equipotent
proof end;

theorem Th4: :: WAYBEL12:4
for b1 being non empty countable set holds
ex b2 being Function of NAT ,b1 st rng b2 = b1
proof end;

Lemma6: for b1 being 1-sorted
for b2, b3 being Subset of b1 holds (b2 \/ b3) ` = (b2 ` ) /\ (b3 ` )
by XBOOLE_1:53;

Lemma7: for b1 being 1-sorted
for b2, b3 being Subset of b1 holds (b2 /\ b3) ` = (b2 ` ) \/ (b3 ` )
by XBOOLE_1:54;

theorem Th5: :: WAYBEL12:5
canceled;

theorem Th6: :: WAYBEL12:6
canceled;

theorem Th7: :: WAYBEL12:7
for b1 being non empty transitive RelStr
for b2, b3 being Subset of b1 holds
( b2 is_finer_than b3 implies downarrow b2 c= downarrow b3 )
proof end;

theorem Th8: :: WAYBEL12:8
for b1 being non empty transitive RelStr
for b2, b3 being Subset of b1 holds
( b2 is_coarser_than b3 implies uparrow b2 c= uparrow b3 )
proof end;

theorem Th9: :: WAYBEL12:9
for b1 being non empty Poset
for b2 being non empty finite filtered Subset of b1 holds
( ex_inf_of b2,b1 implies inf b2 in b2 )
proof end;

theorem Th10: :: WAYBEL12:10
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being non empty lower Subset of b1 holds Bottom b1 in b2
proof end;

theorem Th11: :: WAYBEL12:11
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being non empty Subset of b1 holds Bottom b1 in downarrow b2
proof end;

theorem Th12: :: WAYBEL12:12
for b1 being non empty antisymmetric upper-bounded RelStr
for b2 being non empty upper Subset of b1 holds Top b1 in b2
proof end;

theorem Th13: :: WAYBEL12:13
for b1 being non empty antisymmetric upper-bounded RelStr
for b2 being non empty Subset of b1 holds Top b1 in uparrow b2
proof end;

theorem Th14: :: WAYBEL12:14
for b1 being antisymmetric lower-bounded with_infima RelStr
for b2 being Subset of b1 holds b2 "/\" {(Bottom b1)} c= {(Bottom b1)}
proof end;

theorem Th15: :: WAYBEL12:15
for b1 being antisymmetric lower-bounded with_infima RelStr
for b2 being non empty Subset of b1 holds b2 "/\" {(Bottom b1)} = {(Bottom b1)}
proof end;

theorem Th16: :: WAYBEL12:16
for b1 being antisymmetric upper-bounded with_suprema RelStr
for b2 being Subset of b1 holds b2 "\/" {(Top b1)} c= {(Top b1)}
proof end;

theorem Th17: :: WAYBEL12:17
for b1 being antisymmetric upper-bounded with_suprema RelStr
for b2 being non empty Subset of b1 holds b2 "\/" {(Top b1)} = {(Top b1)}
proof end;

theorem Th18: :: WAYBEL12:18
for b1 being upper-bounded Semilattice
for b2 being Subset of b1 holds {(Top b1)} "/\" b2 = b2
proof end;

theorem Th19: :: WAYBEL12:19
for b1 being lower-bounded with_suprema Poset
for b2 being Subset of b1 holds {(Bottom b1)} "\/" b2 = b2
proof end;

theorem Th20: :: WAYBEL12:20
for b1 being non empty reflexive RelStr
for b2, b3 being Subset of b1 holds
( b2 c= b3 implies ( b2 is_finer_than b3 & b2 is_coarser_than b3 ) )
proof end;

theorem Th21: :: WAYBEL12:21
for b1 being transitive antisymmetric with_infima RelStr
for b2 being Subset of b1
for b3, b4 being Element of b1 holds
( b3 <= b4 implies {b4} "/\" b2 is_coarser_than {b3} "/\" b2 )
proof end;

theorem Th22: :: WAYBEL12:22
for b1 being transitive antisymmetric with_suprema RelStr
for b2 being Subset of b1
for b3, b4 being Element of b1 holds
( b3 <= b4 implies {b3} "\/" b2 is_finer_than {b4} "\/" b2 )
proof end;

theorem Th23: :: WAYBEL12:23
for b1 being non empty RelStr
for b2, b3, b4 being Subset of b1 holds
( b3 is_coarser_than b4 & b2 is upper & b4 c= b2 implies b3 c= b2 )
proof end;

theorem Th24: :: WAYBEL12:24
for b1 being non empty RelStr
for b2, b3, b4 being Subset of b1 holds
( b3 is_finer_than b4 & b2 is lower & b4 c= b2 implies b3 c= b2 )
proof end;

theorem Th25: :: WAYBEL12:25
for b1 being Semilattice
for b2 being filtered upper Subset of b1 holds b2 "/\" b2 = b2
proof end;

theorem Th26: :: WAYBEL12:26
for b1 being sup-Semilattice
for b2 being directed lower Subset of b1 holds b2 "\/" b2 = b2
proof end;

Lemma18: for b1 being non empty RelStr
for b2 being Subset of b1 holds
{ b3 where B is Element of b1 : b2 "/\" {b3} c= b2 } is Subset of b1
proof end;

theorem Th27: :: WAYBEL12:27
for b1 being upper-bounded Semilattice
for b2 being Subset of b1 holds
not { b3 where B is Element of b1 : b2 "/\" {b3} c= b2 } is empty
proof end;

theorem Th28: :: WAYBEL12:28
for b1 being transitive antisymmetric with_infima RelStr
for b2 being Subset of b1 holds
{ b3 where B is Element of b1 : b2 "/\" {b3} c= b2 } is filtered Subset of b1
proof end;

theorem Th29: :: WAYBEL12:29
for b1 being transitive antisymmetric with_infima RelStr
for b2 being upper Subset of b1 holds
{ b3 where B is Element of b1 : b2 "/\" {b3} c= b2 } is upper Subset of b1
proof end;

theorem Th30: :: WAYBEL12:30
for b1 being with_infima Poset
for b2 being Subset of b1 holds
( b2 is Open & b2 is lower implies b2 is filtered )
proof end;

registration
let c1 be with_infima Poset;
cluster lower Open -> filtered Element of bool the carrier of a1;
coherence
for b1 being Subset of c1 holds
( b1 is Open & b1 is lower implies b1 is filtered )
by Th30;
end;

registration
let c1 be non empty reflexive antisymmetric continuous RelStr ;
cluster lower -> Open Element of bool the carrier of a1;
coherence
for b1 being Subset of c1 holds
( b1 is lower implies b1 is Open )
proof end;
end;

registration
let c1 be continuous Semilattice;
let c2 be Element of c1;
cluster (downarrow a2) ` -> Open ;
coherence
(downarrow c2) ` is Open
proof end;
end;

theorem Th31: :: WAYBEL12:31
for b1 being Semilattice
for b2 being non empty Subset of b1 holds
( ( for b3, b4 being Element of b1 holds
not ( b3 in b2 & b4 in b2 & not b3 <= b4 & not b4 <= b3 ) ) implies for b3 being non empty finite Subset of b2 holds "/\" b3,b1 in b3 )
proof end;

theorem Th32: :: WAYBEL12:32
for b1 being sup-Semilattice
for b2 being non empty Subset of b1 holds
( ( for b3, b4 being Element of b1 holds
not ( b3 in b2 & b4 in b2 & not b3 <= b4 & not b4 <= b3 ) ) implies for b3 being non empty finite Subset of b2 holds "\/" b3,b1 in b3 )
proof end;

Lemma24: for b1 being Semilattice
for b2 being Filter of b1 holds b2 = uparrow (fininfs b2)
proof end;

definition
let c1 be Semilattice;
let c2 be Filter of c1;
mode GeneratorSet of c2 -> Subset of a1 means :Def3: :: WAYBEL12:def 3
a2 = uparrow (fininfs a3);
existence
ex b1 being Subset of c1 st c2 = uparrow (fininfs b1)
proof end;
end;

:: deftheorem Def3 defines GeneratorSet WAYBEL12:def 3 :
for b1 being Semilattice
for b2 being Filter of b1
for b3 being Subset of b1 holds
( b3 is GeneratorSet of b2 iff b2 = uparrow (fininfs b3) );

registration
let c1 be Semilattice;
let c2 be Filter of c1;
cluster non empty GeneratorSet of a2;
existence
not for b1 being GeneratorSet of c2 holds b1 is empty
proof end;
end;

Lemma26: for b1 being Semilattice
for b2 being Filter of b1
for b3 being GeneratorSet of b2 holds b3 c= b2
proof end;

theorem Th33: :: WAYBEL12:33
for b1 being Semilattice
for b2 being Subset of b1
for b3 being non empty Subset of b1 holds
( b2 is_coarser_than b3 implies fininfs b2 is_coarser_than fininfs b3 )
proof end;

theorem Th34: :: WAYBEL12:34
for b1 being Semilattice
for b2 being Filter of b1
for b3 being GeneratorSet of b2
for b4 being non empty Subset of b1 holds
( b3 is_coarser_than b4 & b4 is_coarser_than b2 implies b4 is GeneratorSet of b2 )
proof end;

theorem Th35: :: WAYBEL12:35
for b1 being Semilattice
for b2 being Subset of b1
for b3, b4 being Function of NAT ,the carrier of b1 holds
( rng b3 = b2 & ( for b5 being Element of NAT holds b4 . b5 = "/\" { (b3 . b6) where B is Nat : b6 <= b5 } ,b1 ) implies b2 is_coarser_than rng b4 )
proof end;

theorem Th36: :: WAYBEL12:36
for b1 being Semilattice
for b2 being Filter of b1
for b3 being GeneratorSet of b2
for b4, b5 being Function of NAT ,the carrier of b1 holds
( rng b4 = b3 & ( for b6 being Element of NAT holds b5 . b6 = "/\" { (b4 . b7) where B is Nat : b7 <= b6 } ,b1 ) implies rng b5 is GeneratorSet of b2 )
proof end;

theorem Th37: :: WAYBEL12:37
for b1 being lower-bounded continuous LATTICE
for b2 being upper Open Subset of b1
for b3 being Filter of b1
for b4 being Element of b1 holds
not ( b2 "/\" b3 c= b2 & b4 in b2 & ex b5 being non empty GeneratorSet of b3 st b5 is countable & ( for b5 being Open Filter of b1 holds
not ( b5 c= b2 & b4 in b5 & b3 c= b5 ) ) )
proof end;

theorem Th38: :: WAYBEL12:38
for b1 being lower-bounded continuous LATTICE
for b2 being upper Open Subset of b1
for b3 being non empty countable Subset of b1
for b4 being Element of b1 holds
not ( b2 "/\" b3 c= b2 & b4 in b2 & ( for b5 being Open Filter of b1 holds
not ( {b4} "/\" b3 c= b5 & b5 c= b2 & b4 in b5 ) ) )
proof end;

theorem Th39: :: WAYBEL12:39
for b1 being lower-bounded continuous LATTICE
for b2 being upper Open Subset of b1
for b3 being non empty countable Subset of b1
for b4, b5 being Element of b1 holds
not ( b2 "/\" b3 c= b2 & b5 in b2 & not b4 in b2 & ( for b6 being irreducible Element of b1 holds
not ( b4 <= b6 & not b6 in uparrow ({b5} "/\" b3) ) ) )
proof end;

theorem Th40: :: WAYBEL12:40
for b1 being lower-bounded continuous LATTICE
for b2 being Element of b1
for b3 being non empty countable Subset of b1 holds
( ( for b4, b5 being Element of b1 holds
not ( not b5 <= b2 & b4 in b3 & b5 "/\" b4 <= b2 ) ) implies for b4 being Element of b1 holds
not ( not b4 <= b2 & ( for b5 being irreducible Element of b1 holds
not ( b2 <= b5 & not b5 in uparrow ({b4} "/\" b3) ) ) ) )
proof end;

definition
let c1 be non empty RelStr ;
let c2 be Element of c1;
attr a2 is dense means :Def4: :: WAYBEL12:def 4
for b1 being Element of a1 holds
not ( b1 <> Bottom a1 & not a2 "/\" b1 <> Bottom a1 );
end;

:: deftheorem Def4 defines dense WAYBEL12:def 4 :
for b1 being non empty RelStr
for b2 being Element of b1 holds
( b2 is dense iff for b3 being Element of b1 holds
not ( b3 <> Bottom b1 & not b2 "/\" b3 <> Bottom b1 ) );

registration
let c1 be upper-bounded Semilattice;
cluster Top a1 -> dense ;
coherence
Top c1 is dense
proof end;
end;

registration
let c1 be upper-bounded Semilattice;
cluster dense Element of the carrier of a1;
existence
ex b1 being Element of c1 st b1 is dense
proof end;
end;

theorem Th41: :: WAYBEL12:41
for b1 being non trivial bounded Semilattice
for b2 being Element of b1 holds
not ( b2 is dense & not b2 <> Bottom b1 )
proof end;

definition
let c1 be non empty RelStr ;
let c2 be Subset of c1;
attr a2 is dense means :Def5: :: WAYBEL12:def 5
for b1 being Element of a1 holds
( b1 in a2 implies b1 is dense );
end;

:: deftheorem Def5 defines dense WAYBEL12:def 5 :
for b1 being non empty RelStr
for b2 being Subset of b1 holds
( b2 is dense iff for b3 being Element of b1 holds
( b3 in b2 implies b3 is dense ) );

theorem Th42: :: WAYBEL12:42
for b1 being upper-bounded Semilattice holds {(Top b1)} is dense
proof end;

registration
let c1 be upper-bounded Semilattice;
cluster non empty finite countable dense Element of bool the carrier of a1;
existence
ex b1 being Subset of c1 st
( not b1 is empty & b1 is finite & b1 is countable & b1 is dense )
proof end;
end;

theorem Th43: :: WAYBEL12:43
for b1 being lower-bounded continuous LATTICE
for b2 being non empty countable dense Subset of b1
for b3 being Element of b1 holds
not ( b3 <> Bottom b1 & ( for b4 being irreducible Element of b1 holds
not ( b4 <> Top b1 & not b4 in uparrow ({b3} "/\" b2) ) ) )
proof end;

theorem Th44: :: WAYBEL12:44
for b1 being non empty TopSpace
for b2 being Element of (InclPoset the topology of b1)
for b3 being Subset of b1 holds
( b2 = b3 & b3 ` is irreducible implies b2 is meet-irreducible )
proof end;

theorem Th45: :: WAYBEL12:45
for b1 being non empty TopSpace
for b2 being Element of (InclPoset the topology of b1)
for b3 being Subset of b1 holds
( b2 = b3 & b2 <> Top (InclPoset the topology of b1) implies ( b2 is meet-irreducible iff b3 ` is irreducible ) )
proof end;

theorem Th46: :: WAYBEL12:46
for b1 being non empty TopSpace
for b2 being Element of (InclPoset the topology of b1)
for b3 being Subset of b1 holds
( b2 = b3 implies ( b2 is dense iff b3 is everywhere_dense ) )
proof end;

theorem Th47: :: WAYBEL12:47
for b1 being non empty TopSpace holds
( b1 is locally-compact implies for b2 being countable Subset-Family of b1 holds
( not b2 is empty & b2 is dense & b2 is open implies for b3 being non empty Subset of b1 holds
not ( b3 is open & ( for b4 being irreducible Subset of b1 holds
ex b5 being Subset of b1 st
( b5 in b2 & not b4 /\ b3 meets b5 ) ) ) ) )
proof end;

definition
let c1 be non empty TopSpace;
redefine attr a1 is Baire means :: WAYBEL12:def 6
for b1 being Subset-Family of a1 holds
not ( b1 is countable & ( for b2 being Subset of a1 holds
( b2 in b1 implies ( b2 is open & b2 is dense ) ) ) & ( for b2 being Subset of a1 holds
not ( b2 = Intersect b1 & b2 is dense ) ) );
compatibility
( c1 is Baire iff for b1 being Subset-Family of c1 holds
not ( b1 is countable & ( for b2 being Subset of c1 holds
( b2 in b1 implies ( b2 is open & b2 is dense ) ) ) & ( for b2 being Subset of c1 holds
not ( b2 = Intersect b1 & b2 is dense ) ) ) )
proof end;
end;

:: deftheorem Def6 defines Baire WAYBEL12:def 6 :
for b1 being non empty TopSpace holds
( b1 is Baire iff for b2 being Subset-Family of b1 holds
not ( b2 is countable & ( for b3 being Subset of b1 holds
( b3 in b2 implies ( b3 is open & b3 is dense ) ) ) & ( for b3 being Subset of b1 holds
not ( b3 = Intersect b2 & b3 is dense ) ) ) );

theorem Th48: :: WAYBEL12:48
for b1 being non empty TopSpace holds
( b1 is sober & b1 is locally-compact implies b1 is Baire )
proof end;