
theorem
  for T being non empty TopSpace st T is normal for A,B being Subset of
T st A is open & B is closed & B <> {} & B c= A ex C being Subset of T st C is
  open & B c= C & Cl(C) c= A
proof
  let T be non empty TopSpace;
  assume
A1: T is normal;
  let A,B be Subset of T such that
A2: A is open and
A3: B is closed & B <> {} and
A4: B c= A;
  per cases;
  suppose
 A <> [#](T);
    reconsider V = [#](T) \ A as Subset of T;
A5: A = [#](T) \ V by PRE_TOPC:3;
A6: B misses V
    proof
      assume B /\ V <> {};
      then consider z being object such that
A7:   z in B /\ V by XBOOLE_0:def 1;
      z in B & z in V by A7,XBOOLE_0:def 4;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
    V is closed by A2,A5;
    then consider C,Q being Subset of T such that
A8: C is open and
A9: Q is open and
A10: B c= C and
A11: V c= Q and
A12: C misses Q by A1,A3,A6;
    C c= Q` by A12,SUBSET_1:23;
    then Cl(C) c= Q` by A9,TOPS_1:5;
    then Q misses Cl(C) by SUBSET_1:23;
    then
A13: V misses Cl(C) by A11,XBOOLE_1:63;
    take C;
    thus C is open & B c= C by A8,A10;
    A`` = A;
    hence thesis by A13,SUBSET_1:23;
  end;
  suppose
A14: A = [#](T);
    take [#](T);
    thus thesis by A14;
  end;
end;
