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 Th88:
  for a, b being Element of Domains_Lattice T for A, B being
  Element of Domains_of T st a = A & b = B holds a [= b iff A c= B
proof
  let a, b be Element of Domains_Lattice T;
  let A, B be Element of Domains_of T;
  assume that
A1: a = A and
A2: b = B;
  B in Domains_of T;
  then B in {C where C is Subset of T : C is condensed} by TDLAT_1:def 1;
  then
A3: ex Q being Subset of T st Q = B & Q is condensed;
  thus a [= b implies A c= B
  proof
    assume a [= b;
    then a "\/" b = b by LATTICES:def 3;
    then Int(Cl(A \/ B)) \/ (A \/ B) = B by A1,A2,Th86;
    hence thesis by A3,Th58;
  end;
  assume A c= B;
  then Int(Cl(A \/ B)) \/ (A \/ B) = B by A3,Th58;
  then a "\/" b = b by A1,A2,Th86;
  hence thesis by LATTICES:def 3;
end;
