reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;

theorem
  B = [#] NT \ A implies [#] NT \ Int A = Cl B
  proof
    assume
A1: B = [#] NT \ A;
    reconsider C = [#] NT \ B as Subset of NT by XBOOLE_1:36;
A2: [#] NT \ B = [#] NT /\ A by A1,XBOOLE_1:48
              .= A by XBOOLE_1:28;
    Cl B = [#] NT /\ Cl B by XBOOLE_1:28
        .= [#] NT \ ([#] NT \ Cl B) by XBOOLE_1:48
        .= [#] NT \ Int A by A2,Lm14;
    hence thesis;
  end;
