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 Th89:
  for X being Subset of Domains_Lattice T ex a being Element of
Domains_Lattice T st X is_less_than a & for b being Element of Domains_Lattice
  T st X is_less_than b holds a [= b
proof
  let X be Subset of Domains_Lattice T;
  X c= the carrier of Domains_Lattice T;
  then
A1: X c= Domains_of T by Th85;
  then reconsider F = X as Subset-Family of T by TOPS_2:2;
  set A = (union F) \/ (Int Cl(union F));
A2: F is domains-family by A1,Th64;
  then A is condensed by 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: 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
A5: X is_less_than b;
A6: 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
A7:   C in F;
      then C1 is condensed by A2;
      then C in {P where P is Subset of T : P is condensed};
      then
A8:   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 A5,A7;
      hence thesis by A8,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 A6,Th68;
    hence thesis by A3,Th88;
  end;
  take a;
  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 Th68;
    hence thesis by A3,Th88;
  end;
  hence thesis by A4;
end;
