reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
reserve T for non empty TopSpace;
reserve T for non empty TopSpace;

theorem Th91:
  for F being Subset-Family of T st F is domains-family for X
being Subset of Domains_Lattice T st X = F holds "\/"(X,Domains_Lattice T) = (
  union F) \/ (Int Cl(union F))
proof
  let F be Subset-Family of T;
  assume
A1: F is domains-family;
  let X be Subset of Domains_Lattice T;
  assume
A2: X = F;
  thus "\/"(X,Domains_Lattice T) = (union F) \/ (Int Cl(union F))
  proof
    set A = (union F) \/ (Int Cl(union F));
    A is condensed by A1,Th67;
    then A in {C where C is Subset of T : C is condensed};
    then
A3: A in Domains_of T by TDLAT_1:def 1;
    then reconsider a = A as Element of Domains_Lattice T by Th85;
A4: X is_less_than a
    proof
      let b be Element of Domains_Lattice T;
      reconsider B = b as Element of Domains_of T by Th85;
      assume b in X;
      then B c= A by A2,Th68;
      hence thesis by A3,Th88;
    end;
A5: for b being Element of Domains_Lattice T st X is_less_than b holds a [= b
    proof
      let b be Element of Domains_Lattice T;
      reconsider B = b as Element of Domains_of T by Th85;
      assume
A6:   X is_less_than b;
A7:   for C being Subset of T st C in F holds C c= B
      proof
        let C be Subset of T;
        reconsider C1 = C as Subset of T;
        assume
A8:     C in F;
        then C1 is condensed by A1;
        then C in {P where P is Subset of T : P is condensed};
        then
A9:     C in Domains_of T by TDLAT_1:def 1;
        then reconsider c = C as Element of Domains_Lattice T by Th85;
        c [= b by A2,A6,A8;
        hence thesis by A9,Th88;
      end;
      B in Domains_of T;
      then B in {C where C is Subset of T : C is condensed } by TDLAT_1:def 1;
      then ex C being Subset of T st C = B & C is condensed;
      then A c= B by A7,Th68;
      hence thesis by A3,Th88;
    end;
    Domains_Lattice T is complete by Th90;
    hence thesis by A4,A5,LATTICE3:def 21;
  end;
end;
