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

theorem
  for A, B being Subset of T st A is condensed holds Cl(Int(A /\ B)) /\
  (A /\ B) = A iff A c= B
proof
  let A, B be Subset of T;
  assume
A1: A is condensed;
  thus Cl(Int(A /\ B)) /\ (A /\ B) = A implies A c= B
  proof
    assume Cl(Int(A /\ B)) /\ (A /\ B) = A;
    then
A2: A c= A /\ B by XBOOLE_1:17;
    A /\ B c= B by XBOOLE_1:17;
    hence thesis by A2;
  end;
  thus A c= B implies Cl(Int(A /\ B)) /\ (A /\ B) = A
  proof
    assume A c= B;
    then
A3: A /\ B = A by XBOOLE_1:28;
    then A c= Cl(Int(A /\ B)) by A1,TOPS_1:def 6;
    hence thesis by A3,XBOOLE_1:28;
  end;
end;
