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 Th96:
  for a, b being Element of Closed_Domains_Lattice T for A, B
  being Element of Closed_Domains_of T st a = A & b = B holds a [= b iff A c= B
proof
  let a, b be Element of Closed_Domains_Lattice T;
  let A, B be Element of Closed_Domains_of T;
  assume that
A1: a = A and
A2: b = B;
  thus a [= b implies A c= B
  proof
    assume a [= b;
    then a "\/" b = b by LATTICES:def 3;
    then A \/ B = B by A1,A2,Th94;
    hence thesis by XBOOLE_1:7;
  end;
  thus A c= B implies a [= b
  proof
    assume A c= B;
    then A \/ B = B by XBOOLE_1:12;
    then a "\/" b = b by A1,A2,Th94;
    hence thesis by LATTICES:def 3;
  end;
end;
