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

theorem Th58:
  for A, B being Subset of T st B is condensed holds Int(Cl(A \/ B
  )) \/ (A \/ B) = B iff A c= B
proof
  let A, B be Subset of T;
  assume
A1: B is condensed;
  thus Int(Cl(A \/ B)) \/ (A \/ B) = B implies A c= B
  proof
    assume Int(Cl(A \/ B)) \/ (A \/ B) = B;
    then
A2: A \/ B c= B by XBOOLE_1:7;
    A c= A \/ B by XBOOLE_1:7;
    hence thesis by A2;
  end;
  thus A c= B implies Int(Cl(A \/ B)) \/ (A \/ B) = B
  proof
    assume A c= B;
    then
A3: A \/ B = B by XBOOLE_1:12;
    then Int(Cl(A \/ B)) c= B by A1,TOPS_1:def 6;
    hence thesis by A3,XBOOLE_1:12;
  end;
end;
