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;
reserve NT for T_2 NTopSpace;
reserve T   for non empty TopSpace,
        A,B for Subset of T,
        F   for closed Subset of T,
        O   for open Subset of T;
reserve T   for non empty strict TopSpace,
        A,B for Subset of T,
        x   for Point of T;
reserve A for Subset of FMT_R^1,
        x for Point of FMT_R^1,
        y for Point of RealSpace,
        z for Point of TopSpaceMetr RealSpace,
        r for Real;

theorem
  for NT being non empty normal NTopSpace, A,B being closed Subset of NT st
  A misses B holds ex F being Function of NT,FMT_R^1 st
  F is continuous & for
  x being Point of NT holds 0 <= F.x & F.x <= 1 & (x in A implies F.x = 0) &
  (x in B implies F.x = 1)
  proof
    let NT be non empty normal NTopSpace;
    let NA,NB be closed Subset of NT;
    assume
A1: NA misses NB;
    reconsider T = NTop2Top NT as non empty TopSpace;
    reconsider T as non empty normal TopSpace;
    reconsider A = NA, B = NB as closed Subset of T by Lm29;
    consider F being Function of T,R^1 such that
A2: F is continuous and
A3: for x being Point of T holds 0 <= F.x & F.x <= 1 &
    (x in A implies F.x = 0) & (x in B implies F.x = 1) by A1,URYSOHN3:20;
    reconsider TTX = T as non empty TopSpace;
    reconsider TTY = R^1 as non empty strict TopSpace;
    reconsider G = F as continuous Function of TTX,TTY by A2;
A4: NT = Top2NTop T & FMT_R^1 = Top2NTop R^1 by FINTOPO7:25;
    then reconsider F9 = Top2NTop G as Function of NT,FMT_R^1;
    now
      thus F9 is continuous by A4;
      now
        let x be Point of NT;
        reconsider x9 = x as Point of T by FINTOPO7:def 16;
        F9.x = F.x & 0 <= F.x9 by A3;
        hence 0 <= F9.x;
        F9.x = F.x & F.x9 <= 1 by A3;
        hence F9.x <= 1;
      end;
      hence for x being Point of NT holds 0 <= F9.x & F9.x <= 1 &
        (x in NA implies F9.x = 0) & (x in NB implies F9.x = 1) by A3;
    end;
    hence thesis;
  end;
