reserve T for TopSpace;

theorem
  for A, B being Subset of T st B is closed holds Cl(Int(A /\ B)) = A
  implies A c= B
proof
  let A, B be Subset of T;
  assume
A1: B is closed;
A2: A /\ B c= B by XBOOLE_1:17;
  Int(A /\ B) c= A /\ B by TOPS_1:16;
  then Int(A /\ B) c= B by A2;
  then
A3: Cl Int(A /\ B) c= Cl B by PRE_TOPC:19;
  assume Cl(Int(A /\ B)) = A;
  hence thesis by A1,A3,PRE_TOPC:22;
end;
