let T be non empty TopSpace; :: thesis: for X being Subset of (Open_Domains_Lattice T) ex a being Element of (Open_Domains_Lattice T) st
( X is_less_than a & ( for b being Element of (Open_Domains_Lattice T) st X is_less_than b holds
a [= b ) )
let X be Subset of (Open_Domains_Lattice T); :: thesis: ex a being Element of (Open_Domains_Lattice T) st
( X is_less_than a & ( for b being Element of (Open_Domains_Lattice T) st X is_less_than b holds
a [= b ) )
X c= the carrier of (Open_Domains_Lattice T) ;
then A1: X c= Open_Domains_of T by Th102;
then reconsider F = X as Subset-Family of T by TOPS_2:2;
set A = Int (Cl (union F));
A2: F is open-domains-family by A1, Th78;
then Int (Cl (union F)) is open_condensed by Th82;
then Int (Cl (union F)) in { C where C is Subset of T : C is open_condensed } ;
then A3: Int (Cl (union F)) in Open_Domains_of T by TDLAT_1:def 9;
then reconsider a = Int (Cl (union F)) as Element of (Open_Domains_Lattice T) by Th102;
A4: for b being Element of (Open_Domains_Lattice T) st X is_less_than b holds
a [= b

proof

let b be Element of (Open_Domains_Lattice T); :: thesis: ( X is_less_than b implies a [= b )
reconsider B = b as Element of Open_Domains_of T by Th102;
assume A5: X is_less_than b ; :: thesis: a [= b
A6: for C being Subset of T st C in F holds
C c= B

proof

let C be Subset of T; :: thesis: ( C in F implies C c= B )
reconsider C1 = C as Subset of T ;
assume A7: C in F ; :: thesis: C c= B
then C1 is open_condensed by A2;
then C in { P where P is Subset of T : P is open_condensed } ;
then A8: C in Open_Domains_of T by TDLAT_1:def 9;
then reconsider c = C as Element of (Open_Domains_Lattice T) by Th102;
c [= b by A5, A7;
hence C c= B by A8, Th105; :: thesis: verum

end;

B in Open_Domains_of T ;
then B in { C where C is Subset of T : C is open_condensed } by TDLAT_1:def 9;
then ex C being Subset of T st
( C = B & C is open_condensed ) ;
then Int (Cl (union F)) c= B by A6, Th83;
hence a [= b by A3, Th105; :: thesis: verum

end;

take a ; :: thesis: ( X is_less_than a & ( for b being Element of (Open_Domains_Lattice T) st X is_less_than b holds
a [= b ) )
X is_less_than a

proof

let b be Element of (Open_Domains_Lattice T); :: according to LATTICE3:def 17 :: thesis: ( not b in X or b [= a )
reconsider B = b as Element of Open_Domains_of T by Th102;
assume b in X ; :: thesis: b [= a
then B c= Int (Cl (union F)) by A2, Th80, Th83;
hence b [= a by A3, Th105; :: thesis: verum

end;

hence ( X is_less_than a & ( for b being Element of (Open_Domains_Lattice T) st X is_less_than b holds
a [= b ) ) by A4; :: thesis: verum