let T be non empty TopSpace; :: thesis:
let F be Subset-Family of T; :: thesis:
assume A1: F is open-domains-family ; :: thesis:
let X be Subset of (Open_Domains_Lattice T); :: thesis:
assume A2: X = F ; :: thesis:
thus "\/" (X,(Open_Domains_Lattice T)) = Int (Cl (union F)) :: thesis:

proof

set A = Int (Cl (union F));
Int (Cl (union F)) is open_condensed by A1, 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: X is_less_than a

proof

let b be Element of (Open_Domains_Lattice T); :: according to LATTICE3:def 17 :: thesis:
reconsider B = b as Element of Open_Domains_of T by Th102;
assume b in X ; :: thesis:
then B c= Int (Cl (union F)) by A1, A2, Th80, Th83;
hence b [= a by A3, Th105; :: thesis:

end;

A5: 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:
reconsider B = b as Element of Open_Domains_of T by Th102;
assume A6: X is_less_than b ; :: thesis:
A7: for C being Subset of T st C in F holds
C c= B

proof

let C be Subset of T; :: thesis:
reconsider C1 = C as Subset of T ;
assume A8: C in F ; :: thesis:
then C1 is open_condensed by A1;
then C in { P where P is Subset of T : P is open_condensed } ;
then A9: 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 A2, A6, A8;
hence C c= B by A9, Th105; :: thesis:

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 A7, Th83;
hence a [= b by A3, Th105; :: thesis:

end;

Open_Domains_Lattice T is complete by Th107;
hence "\/" (X,(Open_Domains_Lattice T)) = Int (Cl (union F)) by A4, A5, LATTICE3:def 21; :: thesis:

end;