reserve r for Real,
  X for set,
  f, g, h for real-valued Function;
reserve T for non empty TopSpace,
  A for closed Subset of T;

theorem Th18:
  (for A, B being non empty closed Subset of T st A misses B ex f
  being continuous Function of T, R^1 st f.:A = {0} & f.:B = {1}) implies T is
  normal
proof
  assume
A1: for A, B being non empty closed Subset of T st A misses B ex f being
  continuous Function of T, R^1 st f.:A = {0} & f.:B = {1};
  let W, V be Subset of T;
  assume W <> {} & V <> {} & W is closed & V is closed & W misses V;
  then consider f being continuous Function of T, R^1 such that
A2: f.:W = {0} and
A3: f.:V = {1} by A1;
  set Q = f"R^1(right_open_halfline(1/2));
  set P = f"R^1(left_open_halfline(1/2));
  take P, Q;
  [#]R^1 <> {};
  hence P is open & Q is open by TOPS_2:43;
A4: R^1(left_open_halfline(1/2)) = left_open_halfline(1/2) by TOPREALB:def 3;
A5: dom f = the carrier of T by FUNCT_2:def 1;
  thus W c= P
  proof
    let a be object;
A6: 0 in left_open_halfline(1/2) by XXREAL_1:233;
    assume
A7: a in W;
    then f.a in f.:W by FUNCT_2:35;
    then f.a = 0 by A2,TARSKI:def 1;
    hence thesis by A4,A5,A7,A6,FUNCT_1:def 7;
  end;
A8: R^1(right_open_halfline(1/2)) = right_open_halfline(1/2) by TOPREALB:def 3;
  thus V c= Q
  proof
    let a be object;
A9: 1 in right_open_halfline(1/2) by XXREAL_1:235;
    assume
A10: a in V;
    then f.a in f.:V by FUNCT_2:35;
    then f.a = 1 by A3,TARSKI:def 1;
    hence thesis by A8,A5,A10,A9,FUNCT_1:def 7;
  end;
  thus thesis by A4,A8,FUNCT_1:71,XXREAL_1:275;
end;
