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
  (for A being non empty closed Subset of T for f being continuous
Function of T|A, Closed-Interval-TSpace(-1,1) ex g being continuous Function of
  T, Closed-Interval-TSpace(-1,1) st g|A = f) implies T is normal
proof
  assume
A1: for A being non empty closed Subset of T for f being continuous
Function of T|A, Closed-Interval-TSpace(-1,1) ex g being continuous Function of
  T, Closed-Interval-TSpace(-1,1) st g|A = f;
  for C, D being non empty closed Subset of T st C misses D holds ex f
  being continuous Function of T, R^1 st f.:C = {0} & f.:D = {1}
  proof
    set f2 = T --> R^1(1);
    set f1 = T --> R^1(0);
    let C, D being non empty closed Subset of T such that
A2: C misses D;
    set g1 = f1|(T|C), g2 = f2|(T|D);
    set f = g1 union g2;
A3: the carrier of T|D = D by PRE_TOPC:8;
    g2 = f2|the carrier of T|D by TMAP_1:def 4;
    then
A4: rng g2 c= rng f2 by RELAT_1:70;
    g1 = f1|the carrier of T|C by TMAP_1:def 4;
    then rng g1 c= rng f1 by RELAT_1:70;
    then
A5: rng g1 \/ rng g2 c= rng f1 \/ rng f2 by A4,XBOOLE_1:13;
A6: f1 = (the carrier of T) --> 0 by TOPREALB:def 2;
    then
A7: rng f1 = {0} by FUNCOP_1:8;
A8: f2 = (the carrier of T) --> 1 by TOPREALB:def 2;
    then
A9: rng f2 = {1} by FUNCOP_1:8;
A10: the carrier of T|C = C by PRE_TOPC:8;
    then
A11: T|C misses T|D by A2,A3,TSEP_1:def 3;
    then rng f c= rng g1 \/ rng g2 by Th13;
    then
A12: rng f c= rng f1 \/ rng f2 by A5;
A13: rng f c= [. -1,1 .]
    proof
      let x be object;
      assume x in rng f;
      then x in {0} or x in {1} by A12,A7,A9,XBOOLE_0:def 3;
      then x = 0 or x = 1 by TARSKI:def 1;
      hence thesis by XXREAL_1:1;
    end;
    the carrier of T|(C\/D) = C \/ D by PRE_TOPC:8;
    then
A14: T|(C\/D) = (T|C) union (T|D) by A10,A3,TSEP_1:def 2;
A15: f2.:D = {1}
    proof
      thus f2.:D c= {1} by A8,FUNCOP_1:81;
      let y be object;
      consider c being object such that
A16:  c in D by XBOOLE_0:def 1;
      assume y in {1};
      then
A17:  y = 1 by TARSKI:def 1;
      dom f2 = the carrier of T & f2.c = 1 by A8,A16,FUNCOP_1:7,13;
      hence thesis by A17,A16,FUNCT_1:def 6;
    end;
A18: f1.:C = {0}
    proof
      thus f1.:C c= {0} by A6,FUNCOP_1:81;
      let y be object;
      consider c being object such that
A19:  c in C by XBOOLE_0:def 1;
      assume y in {0};
      then
A20:  y = 0 by TARSKI:def 1;
      dom f1 = the carrier of T & f1.c = 0 by A6,A19,FUNCOP_1:7,13;
      hence thesis by A20,A19,FUNCT_1:def 6;
    end;
A21: C \/ D is closed by TOPS_1:9;
    the carrier of Closed-Interval-TSpace(-1,1) = [. -1,1 .] by TOPMETR:18;
    then reconsider
    h = f as Function of T|(C\/D), Closed-Interval-TSpace(-1,1) by A14,A13,
FUNCT_2:6;
    f is continuous Function of (T|C) union (T|D), R^1 by A11,TMAP_1:136;
    then h is continuous by A14,PRE_TOPC:27;
    then consider
    g being continuous Function of T, Closed-Interval-TSpace(-1,1)
    such that
A22: g|(C\/D) = f by A1,A21;
    reconsider F = g as continuous Function of T, R^1 by PRE_TOPC:26,TOPREALA:7
;
    take F;
    thus F.:C = f.:C by A22,FUNCT_2:97,XBOOLE_1:7
      .= g1.:C by A10,A11,Th14
      .= (f1|C).:C by A10,TMAP_1:def 3
      .= {0} by A18,RELAT_1:129;
    thus F.:D = f.:D by A22,FUNCT_2:97,XBOOLE_1:7
      .= g2.:D by A3,A11,Th14
      .= (f2|D).:D by A3,TMAP_1:def 3
      .= {1} by A15,RELAT_1:129;
  end;
  hence thesis by Th18;
end;
