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;
reserve T for non empty TopSpace;

theorem
  for F being Subset-Family of T st F is closed-domains-family for X
  being Subset of Closed_Domains_Lattice T st X = F holds "\/"(X,
  Closed_Domains_Lattice T) = Cl(union F)
proof
  let F be Subset-Family of T;
  assume
A1: F is closed-domains-family;
  let X be Subset of Closed_Domains_Lattice T;
  assume
A2: X = F;
  thus "\/"(X,Closed_Domains_Lattice T) = Cl(union F)
  proof
    set A = Cl(union F);
    A is closed_condensed by A1,Th75;
    then A in {C where C is Subset of T : C is closed_condensed};
    then
A3: A in Closed_Domains_of T by TDLAT_1:def 5;
    then reconsider a = A as Element of Closed_Domains_Lattice T by Th93;
A4: X is_less_than a
    proof
      let b be Element of Closed_Domains_Lattice T;
      reconsider B = b as Element of Closed_Domains_of T by Th93;
      assume b in X;
      then B c= A by A2,Th76;
      hence thesis by A3,Th96;
    end;
A5: for b being Element of Closed_Domains_Lattice T st X is_less_than b
    holds a [= b
    proof
      let b be Element of Closed_Domains_Lattice T;
      reconsider B = b as Element of Closed_Domains_of T by Th93;
      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 closed_condensed by A1;
        then C in {P where P is Subset of T : P is closed_condensed};
        then
A9:     C in Closed_Domains_of T by TDLAT_1:def 5;
        then reconsider c = C as Element of Closed_Domains_Lattice T by Th93;
        c [= b by A2,A6,A8;
        hence thesis by A9,Th96;
      end;
      B in Closed_Domains_of T;
      then B in {C where C is Subset of T : C is closed_condensed} by
TDLAT_1:def 5;
      then ex C being Subset of T st C = B & C is closed_condensed;
      then A c= B by A7,Th76;
      hence thesis by A3,Th96;
    end;
    Closed_Domains_Lattice T is complete by Th98;
    hence thesis by A4,A5,LATTICE3:def 21;
  end;
end;
