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;
