
theorem Th20:
  for T being non empty normal TopSpace, A,B being closed Subset
of T st A misses B holds ex F being Function of T,R^1 st F is continuous & 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)
proof
  let T be non empty normal TopSpace;
  let A,B be closed Subset of T;
  assume
A1: A misses B;
  per cases;
  suppose
    A <> {};
    hence thesis by A1,Th19;
  end;
  suppose
A2: A = {};
    set S = the carrier of T, L = the carrier of R^1;
    1 in REAL by XREAL_0:def 1;
    then reconsider r = 1 as Element of L by TOPMETR:17;
    defpred P[set,set] means $2 = r;
A3: for x being Element of S ex y being Element of L st P[x,y];
    ex F being Function of S,L st for x being Element of S holds P[x,F.x]
    from FUNCT_2:sch 3(A3);
    then consider F being Function of S,L such that
A4: for x being Element of S holds F.x = r;
    take F;
A5: dom F = the carrier of T by FUNCT_2:def 1;
    thus F is continuous
    proof
      the carrier of T c= the carrier of T;
      then reconsider O1 = the carrier of T as Subset of T;
      reconsider O2 = {} as Subset of T by XBOOLE_1:2;
      let P be Subset of R^1;
      assume P is closed;
A6:   O2 is closed;
      per cases;
      suppose
        1 in P;
        then for x being object
        holds x in the carrier of T iff x in dom F & F.x
        in P by A4,FUNCT_2:def 1;
        hence thesis by FUNCT_1:def 7;
      end;
      suppose
        not 1 in P;
        then for x being object
        holds x in {} iff x in dom F & F.x in P by A4,A5;
        hence thesis by A6,FUNCT_1:def 7;
      end;
    end;
    let x be Point of T;
    thus thesis by A2,A4;
  end;
end;
