:: Bases of Continuous Lattices
:: by Robert Milewski
::
:: Copyright (c) 1998-2021 Association of Mizar Users

theorem Th1: :: WAYBEL23:1
for L being non empty Poset
for x being Element of L holds compactbelow x = () /\ the carrier of ()
proof end;

definition
let L be non empty reflexive transitive RelStr ;
let X be Subset of (InclPoset (Ids L));
:: original: union
redefine func union X -> Subset of L;
coherence
union X is Subset of L
proof end;
end;

Lm1: for X being non empty set
for Y being Subset of () st ex_sup_of Y, InclPoset X holds
union Y c= sup Y

proof end;

theorem Th2: :: WAYBEL23:2
for L being non empty RelStr
for X, Y being Subset of L st X c= Y holds
finsups X c= finsups Y
proof end;

theorem Th3: :: WAYBEL23:3
for L being non empty transitive RelStr
for S being non empty full sups-inheriting SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
finsups X c= finsups Y
proof end;

theorem :: WAYBEL23:4
for L being non empty transitive antisymmetric complete RelStr
for S being non empty full sups-inheriting SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
finsups X = finsups Y
proof end;

theorem Th5: :: WAYBEL23:5
for L being complete sup-Semilattice
for S being non empty full join-inheriting SubRelStr of L st Bottom L in the carrier of S holds
for X being Subset of L
for Y being Subset of S st X = Y holds
finsups Y c= finsups X
proof end;

theorem Th6: :: WAYBEL23:6
for L being lower-bounded sup-Semilattice
for X being Subset of (InclPoset (Ids L)) holds sup X = downarrow (finsups ())
proof end;

theorem Th7: :: WAYBEL23:7
for L being reflexive transitive RelStr
for X being Subset of L holds downarrow () = downarrow X
proof end;

theorem :: WAYBEL23:8
for L being reflexive transitive RelStr
for X being Subset of L holds uparrow () = uparrow X
proof end;

theorem :: WAYBEL23:9
for L being non empty reflexive transitive RelStr
for x being Element of L holds downarrow () = downarrow x
proof end;

theorem :: WAYBEL23:10
for L being non empty reflexive transitive RelStr
for x being Element of L holds uparrow () = uparrow x
proof end;

theorem Th11: :: WAYBEL23:11
for L being non empty RelStr
for S being non empty SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
downarrow Y c= downarrow X
proof end;

theorem Th12: :: WAYBEL23:12
for L being non empty RelStr
for S being non empty SubRelStr of L
for X being Subset of L
for Y being Subset of S st X = Y holds
uparrow Y c= uparrow X
proof end;

theorem :: WAYBEL23:13
for L being non empty RelStr
for S being non empty SubRelStr of L
for x being Element of L
for y being Element of S st x = y holds
downarrow y c= downarrow x
proof end;

theorem :: WAYBEL23:14
for L being non empty RelStr
for S being non empty SubRelStr of L
for x being Element of L
for y being Element of S st x = y holds
uparrow y c= uparrow x
proof end;

definition
let L be non empty RelStr ;
let S be Subset of L;
attr S is meet-closed means :Def1: :: WAYBEL23:def 1
subrelstr S is meet-inheriting ;
end;

:: deftheorem Def1 defines meet-closed WAYBEL23:def 1 :
for L being non empty RelStr
for S being Subset of L holds
( S is meet-closed iff subrelstr S is meet-inheriting );

definition
let L be non empty RelStr ;
let S be Subset of L;
attr S is join-closed means :Def2: :: WAYBEL23:def 2
subrelstr S is join-inheriting ;
end;

:: deftheorem Def2 defines join-closed WAYBEL23:def 2 :
for L being non empty RelStr
for S being Subset of L holds
( S is join-closed iff subrelstr S is join-inheriting );

definition
let L be non empty RelStr ;
let S be Subset of L;
attr S is infs-closed means :Def3: :: WAYBEL23:def 3
subrelstr S is infs-inheriting ;
end;

:: deftheorem Def3 defines infs-closed WAYBEL23:def 3 :
for L being non empty RelStr
for S being Subset of L holds
( S is infs-closed iff subrelstr S is infs-inheriting );

definition
let L be non empty RelStr ;
let S be Subset of L;
attr S is sups-closed means :Def4: :: WAYBEL23:def 4
subrelstr S is sups-inheriting ;
end;

:: deftheorem Def4 defines sups-closed WAYBEL23:def 4 :
for L being non empty RelStr
for S being Subset of L holds
( S is sups-closed iff subrelstr S is sups-inheriting );

registration
let L be non empty RelStr ;
cluster infs-closed -> meet-closed for Element of K32( the carrier of L);
coherence
for b1 being Subset of L st b1 is infs-closed holds
b1 is meet-closed
;
cluster sups-closed -> join-closed for Element of K32( the carrier of L);
coherence
for b1 being Subset of L st b1 is sups-closed holds
b1 is join-closed
;
end;

registration
let L be non empty RelStr ;
cluster non empty infs-closed sups-closed for Element of K32( the carrier of L);
existence
ex b1 being Subset of L st
( b1 is infs-closed & b1 is sups-closed & not b1 is empty )
proof end;
end;

theorem Th15: :: WAYBEL23:15
for L being non empty RelStr
for S being Subset of L holds
( S is meet-closed iff for x, y being Element of L st x in S & y in S & ex_inf_of {x,y},L holds
inf {x,y} in S )
proof end;

theorem Th16: :: WAYBEL23:16
for L being non empty RelStr
for S being Subset of L holds
( S is join-closed iff for x, y being Element of L st x in S & y in S & ex_sup_of {x,y},L holds
sup {x,y} in S )
proof end;

theorem :: WAYBEL23:17
for L being antisymmetric with_infima RelStr
for S being Subset of L holds
( S is meet-closed iff for x, y being Element of L st x in S & y in S holds
inf {x,y} in S )
proof end;

theorem Th18: :: WAYBEL23:18
for L being antisymmetric with_suprema RelStr
for S being Subset of L holds
( S is join-closed iff for x, y being Element of L st x in S & y in S holds
sup {x,y} in S )
proof end;

theorem :: WAYBEL23:19
for L being non empty RelStr
for S being Subset of L holds
( S is infs-closed iff for X being Subset of S st ex_inf_of X,L holds
"/\" (X,L) in S )
proof end;

theorem :: WAYBEL23:20
for L being non empty RelStr
for S being Subset of L holds
( S is sups-closed iff for X being Subset of S st ex_sup_of X,L holds
"\/" (X,L) in S )
proof end;

theorem Th21: :: WAYBEL23:21
for L being non empty transitive RelStr
for S being non empty infs-closed Subset of L
for X being Subset of S st ex_inf_of X,L holds
( ex_inf_of X, subrelstr S & "/\" (X,()) = "/\" (X,L) )
proof end;

theorem Th22: :: WAYBEL23:22
for L being non empty transitive RelStr
for S being non empty sups-closed Subset of L
for X being Subset of S st ex_sup_of X,L holds
( ex_sup_of X, subrelstr S & "\/" (X,()) = "\/" (X,L) )
proof end;

theorem :: WAYBEL23:23
for L being non empty transitive RelStr
for S being non empty meet-closed Subset of L
for x, y being Element of S st ex_inf_of {x,y},L holds
( ex_inf_of {x,y}, subrelstr S & "/\" ({x,y},()) = "/\" ({x,y},L) )
proof end;

theorem :: WAYBEL23:24
for L being non empty transitive RelStr
for S being non empty join-closed Subset of L
for x, y being Element of S st ex_sup_of {x,y},L holds
( ex_sup_of {x,y}, subrelstr S & "\/" ({x,y},()) = "\/" ({x,y},L) )
proof end;

theorem Th25: :: WAYBEL23:25
for L being transitive antisymmetric with_infima RelStr
for S being non empty meet-closed Subset of L holds subrelstr S is with_infima
proof end;

theorem Th26: :: WAYBEL23:26
for L being transitive antisymmetric with_suprema RelStr
for S being non empty join-closed Subset of L holds subrelstr S is with_suprema
proof end;

registration
let L be transitive antisymmetric with_infima RelStr ;
let S be non empty meet-closed Subset of L;
coherence by Th25;
end;

registration
let L be transitive antisymmetric with_suprema RelStr ;
let S be non empty join-closed Subset of L;
coherence by Th26;
end;

theorem :: WAYBEL23:27
for L being non empty transitive antisymmetric complete RelStr
for S being non empty infs-closed Subset of L
for X being Subset of S holds "/\" (X,()) = "/\" (X,L)
proof end;

theorem :: WAYBEL23:28
for L being non empty transitive antisymmetric complete RelStr
for S being non empty sups-closed Subset of L
for X being Subset of S holds "\/" (X,()) = "\/" (X,L)
proof end;

theorem Th29: :: WAYBEL23:29
for L being Semilattice
for S being meet-closed Subset of L holds S is filtered
proof end;

theorem Th30: :: WAYBEL23:30
for L being sup-Semilattice
for S being join-closed Subset of L holds S is directed
proof end;

registration
let L be Semilattice;
cluster meet-closed -> filtered for Element of K32( the carrier of L);
coherence
for b1 being Subset of L st b1 is meet-closed holds
b1 is filtered
by Th29;
end;

registration
let L be sup-Semilattice;
cluster join-closed -> directed for Element of K32( the carrier of L);
coherence
for b1 being Subset of L st b1 is join-closed holds
b1 is directed
by Th30;
end;

theorem :: WAYBEL23:31
for L being Semilattice
for S being non empty upper Subset of L holds
( S is Filter of L iff S is meet-closed ) by WAYBEL_0:63;

theorem :: WAYBEL23:32
for L being sup-Semilattice
for S being non empty lower Subset of L holds
( S is Ideal of L iff S is join-closed ) by WAYBEL_0:64;

theorem Th33: :: WAYBEL23:33
for L being non empty RelStr
for S1, S2 being join-closed Subset of L holds S1 /\ S2 is join-closed
proof end;

theorem :: WAYBEL23:34
for L being non empty RelStr
for S1, S2 being meet-closed Subset of L holds S1 /\ S2 is meet-closed
proof end;

theorem Th35: :: WAYBEL23:35
for L being sup-Semilattice
for x being Element of L holds downarrow x is join-closed
proof end;

theorem Th36: :: WAYBEL23:36
for L being Semilattice
for x being Element of L holds downarrow x is meet-closed
proof end;

theorem Th37: :: WAYBEL23:37
for L being sup-Semilattice
for x being Element of L holds uparrow x is join-closed
proof end;

theorem Th38: :: WAYBEL23:38
for L being Semilattice
for x being Element of L holds uparrow x is meet-closed
proof end;

registration
let L be sup-Semilattice;
let x be Element of L;
coherence by Th35;
coherence by Th37;
end;

registration
let L be Semilattice;
let x be Element of L;
coherence by Th36;
coherence by Th38;
end;

theorem Th39: :: WAYBEL23:39
for L being sup-Semilattice
for x being Element of L holds waybelow x is join-closed
proof end;

theorem Th40: :: WAYBEL23:40
for L being Semilattice
for x being Element of L holds waybelow x is meet-closed
proof end;

theorem Th41: :: WAYBEL23:41
for L being sup-Semilattice
for x being Element of L holds wayabove x is join-closed
proof end;

registration
let L be sup-Semilattice;
let x be Element of L;
coherence by Th39;
coherence by Th41;
end;

registration
let L be Semilattice;
let x be Element of L;
coherence by Th40;
end;

definition
let T be TopStruct ;
func weight T -> Cardinal equals :: WAYBEL23:def 5
meet { (card B) where B is Basis of T : verum } ;
coherence
meet { (card B) where B is Basis of T : verum } is Cardinal
proof end;
end;

:: deftheorem defines weight WAYBEL23:def 5 :
for T being TopStruct holds weight T = meet { (card B) where B is Basis of T : verum } ;

definition
let T be TopStruct ;
end;

:: deftheorem defines second-countable WAYBEL23:def 6 :
for T being TopStruct holds
( T is second-countable iff weight T c= omega );

definition
let L be continuous sup-Semilattice;
mode CLbasis of L -> Subset of L means :Def7: :: WAYBEL23:def 7
( it is join-closed & ( for x being Element of L holds x = sup (() /\ it) ) );
existence
ex b1 being Subset of L st
( b1 is join-closed & ( for x being Element of L holds x = sup (() /\ b1) ) )
proof end;
end;

:: deftheorem Def7 defines CLbasis WAYBEL23:def 7 :
for L being continuous sup-Semilattice
for b2 being Subset of L holds
( b2 is CLbasis of L iff ( b2 is join-closed & ( for x being Element of L holds x = sup (() /\ b2) ) ) );

definition
let L be non empty RelStr ;
let S be Subset of L;
attr S is with_bottom means :Def8: :: WAYBEL23:def 8
Bottom L in S;
end;

:: deftheorem Def8 defines with_bottom WAYBEL23:def 8 :
for L being non empty RelStr
for S being Subset of L holds
( S is with_bottom iff Bottom L in S );

definition
let L be non empty RelStr ;
let S be Subset of L;
attr S is with_top means :: WAYBEL23:def 9
Top L in S;
end;

:: deftheorem defines with_top WAYBEL23:def 9 :
for L being non empty RelStr
for S being Subset of L holds
( S is with_top iff Top L in S );

registration
let L be non empty RelStr ;
cluster with_bottom -> non empty for Element of K32( the carrier of L);
coherence
for b1 being Subset of L st b1 is with_bottom holds
not b1 is empty
;
end;

registration
let L be non empty RelStr ;
cluster with_top -> non empty for Element of K32( the carrier of L);
coherence
for b1 being Subset of L st b1 is with_top holds
not b1 is empty
;
end;

registration
let L be non empty RelStr ;
cluster with_bottom for Element of K32( the carrier of L);
existence
ex b1 being Subset of L st b1 is with_bottom
proof end;
cluster with_top for Element of K32( the carrier of L);
existence
ex b1 being Subset of L st b1 is with_top
proof end;
end;

registration
let L be continuous sup-Semilattice;
cluster with_bottom for CLbasis of L;
existence
ex b1 being CLbasis of L st b1 is with_bottom
proof end;
cluster with_top for CLbasis of L;
existence
ex b1 being CLbasis of L st b1 is with_top
proof end;
end;

theorem Th42: :: WAYBEL23:42
for L being non empty antisymmetric lower-bounded RelStr
for S being with_bottom Subset of L holds subrelstr S is lower-bounded
proof end;

registration
let L be non empty antisymmetric lower-bounded RelStr ;
let S be with_bottom Subset of L;
coherence by Th42;
end;

registration
let L be continuous sup-Semilattice;
cluster -> join-closed for CLbasis of L;
coherence
for b1 being CLbasis of L holds b1 is join-closed
by Def7;
end;

registration
existence
ex b1 being continuous LATTICE st
( b1 is bounded & not b1 is trivial )
proof end;
end;

registration
let L be non trivial lower-bounded continuous sup-Semilattice;
cluster -> non empty for CLbasis of L;
coherence
for b1 being CLbasis of L holds not b1 is empty
proof end;
end;

theorem Th43: :: WAYBEL23:43
for L being sup-Semilattice holds the carrier of () is join-closed Subset of L
proof end;

theorem Th44: :: WAYBEL23:44
for L being lower-bounded algebraic LATTICE holds the carrier of () is with_bottom CLbasis of L
proof end;

theorem :: WAYBEL23:45
for L being lower-bounded continuous sup-Semilattice st the carrier of () is CLbasis of L holds
L is algebraic
proof end;

theorem Th46: :: WAYBEL23:46
for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L holds
( B is CLbasis of L iff for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b << y ) )
proof end;

theorem Th47: :: WAYBEL23:47
for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L st Bottom L in B holds
( B is CLbasis of L iff for x, y being Element of L st x << y holds
ex b being Element of L st
( b in B & x <= b & b << y ) )
proof end;

Lm2: for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L st Bottom L in B & ( for x, y being Element of L st x << y holds
ex b being Element of L st
( b in B & x <= b & b << y ) ) holds
( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) )

proof end;

Lm3: for L being lower-bounded continuous LATTICE
for B being Subset of L st ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) holds
for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b << y )

proof end;

theorem Th48: :: WAYBEL23:48
for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L st Bottom L in B holds
( B is CLbasis of L iff ( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) ) )
proof end;

theorem :: WAYBEL23:49
for L being lower-bounded continuous LATTICE
for B being join-closed Subset of L st Bottom L in B holds
( B is CLbasis of L iff for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) )
proof end;

theorem Th50: :: WAYBEL23:50
for L being lower-bounded sup-Semilattice
for S being non empty full SubRelStr of L st Bottom L in the carrier of S & the carrier of S is join-closed Subset of L holds
for x being Element of L holds () /\ the carrier of S is Ideal of S
proof end;

definition
let L be non empty reflexive transitive RelStr ;
let S be non empty full SubRelStr of L;
func supMap S -> Function of (InclPoset (Ids S)),L means :Def10: :: WAYBEL23:def 10
for I being Ideal of S holds it . I = "\/" (I,L);
existence
ex b1 being Function of (InclPoset (Ids S)),L st
for I being Ideal of S holds b1 . I = "\/" (I,L)
proof end;
uniqueness
for b1, b2 being Function of (InclPoset (Ids S)),L st ( for I being Ideal of S holds b1 . I = "\/" (I,L) ) & ( for I being Ideal of S holds b2 . I = "\/" (I,L) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines supMap WAYBEL23:def 10 :
for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L
for b3 being Function of (InclPoset (Ids S)),L holds
( b3 = supMap S iff for I being Ideal of S holds b3 . I = "\/" (I,L) );

definition
let L be non empty reflexive transitive RelStr ;
let S be non empty full SubRelStr of L;
func idsMap S -> Function of (InclPoset (Ids S)),(InclPoset (Ids L)) means :Def11: :: WAYBEL23:def 11
for I being Ideal of S ex J being Subset of L st
( I = J & it . I = downarrow J );
existence
ex b1 being Function of (InclPoset (Ids S)),(InclPoset (Ids L)) st
for I being Ideal of S ex J being Subset of L st
( I = J & b1 . I = downarrow J )
proof end;
uniqueness
for b1, b2 being Function of (InclPoset (Ids S)),(InclPoset (Ids L)) st ( for I being Ideal of S ex J being Subset of L st
( I = J & b1 . I = downarrow J ) ) & ( for I being Ideal of S ex J being Subset of L st
( I = J & b2 . I = downarrow J ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines idsMap WAYBEL23:def 11 :
for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L
for b3 being Function of (InclPoset (Ids S)),(InclPoset (Ids L)) holds
( b3 = idsMap S iff for I being Ideal of S ex J being Subset of L st
( I = J & b3 . I = downarrow J ) );

registration
let L be reflexive RelStr ;
let B be Subset of L;
coherence ;
end;

registration
let L be transitive RelStr ;
let B be Subset of L;
coherence ;
end;

registration
let L be antisymmetric RelStr ;
let B be Subset of L;
coherence ;
end;

definition
let L be lower-bounded continuous sup-Semilattice;
let B be with_bottom CLbasis of L;
func baseMap B -> Function of L,(InclPoset (Ids ())) means :Def12: :: WAYBEL23:def 12
for x being Element of L holds it . x = () /\ B;
existence
ex b1 being Function of L,(InclPoset (Ids ())) st
for x being Element of L holds b1 . x = () /\ B
proof end;
uniqueness
for b1, b2 being Function of L,(InclPoset (Ids ())) st ( for x being Element of L holds b1 . x = () /\ B ) & ( for x being Element of L holds b2 . x = () /\ B ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines baseMap WAYBEL23:def 12 :
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L
for b3 being Function of L,(InclPoset (Ids ())) holds
( b3 = baseMap B iff for x being Element of L holds b3 . x = () /\ B );

theorem Th51: :: WAYBEL23:51
for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L holds
( dom () = Ids S & rng () is Subset of L )
proof end;

theorem Th52: :: WAYBEL23:52
for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L
for x being set holds
( x in dom () iff x is Ideal of S )
proof end;

theorem Th53: :: WAYBEL23:53
for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L holds
( dom () = Ids S & rng () is Subset of (Ids L) )
proof end;

theorem Th54: :: WAYBEL23:54
for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L
for x being set holds
( x in dom () iff x is Ideal of S )
proof end;

theorem Th55: :: WAYBEL23:55
for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L
for x being set st x in rng () holds
x is Ideal of L
proof end;

theorem :: WAYBEL23:56
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds
( dom () = the carrier of L & rng () is Subset of (Ids ()) ) by ;

theorem :: WAYBEL23:57
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L
for x being set st x in rng () holds
x is Ideal of ()
proof end;

theorem Th58: :: WAYBEL23:58
for L being non empty up-complete Poset
for S being non empty full SubRelStr of L holds supMap S is monotone
proof end;

theorem Th59: :: WAYBEL23:59
for L being non empty reflexive transitive RelStr
for S being non empty full SubRelStr of L holds idsMap S is monotone
proof end;

theorem Th60: :: WAYBEL23:60
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds baseMap B is monotone
proof end;

registration
let L be non empty up-complete Poset;
let S be non empty full SubRelStr of L;
coherence by Th58;
end;

registration
let L be non empty reflexive transitive RelStr ;
let S be non empty full SubRelStr of L;
coherence by Th59;
end;

registration
let L be lower-bounded continuous sup-Semilattice;
let B be with_bottom CLbasis of L;
coherence by Th60;
end;

theorem Th61: :: WAYBEL23:61
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds idsMap () is sups-preserving
proof end;

theorem Th62: :: WAYBEL23:62
for L being non empty up-complete Poset
for S being non empty full SubRelStr of L holds supMap S = () * ()
proof end;

theorem Th63: :: WAYBEL23:63
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds [(supMap ()),()] is Galois
proof end;

theorem Th64: :: WAYBEL23:64
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds
proof end;

theorem Th65: :: WAYBEL23:65
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds rng (supMap ()) = the carrier of L
proof end;

theorem Th66: :: WAYBEL23:66
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds
( supMap () is infs-preserving & supMap () is sups-preserving )
proof end;

theorem :: WAYBEL23:67
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds baseMap B is sups-preserving by ;

registration
let L be lower-bounded continuous sup-Semilattice;
let B be with_bottom CLbasis of L;
coherence
( supMap () is infs-preserving & supMap () is sups-preserving )
by Th66;
coherence by ;
end;

theorem Th68: :: WAYBEL23:68
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds the carrier of (CompactSublatt (InclPoset (Ids ()))) = { () where b is Element of () : verum }
proof end;

theorem :: WAYBEL23:69
for L being lower-bounded continuous sup-Semilattice
for B being with_bottom CLbasis of L holds CompactSublatt (InclPoset (Ids ())), subrelstr B are_isomorphic
proof end;

Lm4: for L being lower-bounded continuous LATTICE st L is algebraic holds
( the carrier of () is with_bottom CLbasis of L & ( for B being with_bottom CLbasis of L holds the carrier of () c= B ) )

proof end;

theorem Th70: :: WAYBEL23:70
for L being lower-bounded continuous LATTICE
for B being with_bottom CLbasis of L st ( for B1 being with_bottom CLbasis of L holds B c= B1 ) holds
for J being Element of (InclPoset (Ids ())) holds J = (waybelow ("\/" (J,L))) /\ B
proof end;

Lm5: for L being lower-bounded continuous LATTICE st ex B being with_bottom CLbasis of L st
for B1 being with_bottom CLbasis of L holds B c= B1 holds
L is algebraic

proof end;

theorem :: WAYBEL23:71
for L being lower-bounded continuous LATTICE holds
( L is algebraic iff ( the carrier of () is with_bottom CLbasis of L & ( for B being with_bottom CLbasis of L holds the carrier of () c= B ) ) ) by ;

theorem :: WAYBEL23:72
for L being lower-bounded continuous LATTICE holds
( L is algebraic iff ex B being with_bottom CLbasis of L st
for B1 being with_bottom CLbasis of L holds B c= B1 )
proof end;

theorem :: WAYBEL23:73
for T being TopStruct
for b being Basis of T holds weight T c= card b
proof end;

theorem :: WAYBEL23:74
for T being TopStruct ex b being Basis of T st card b = weight T
proof end;