reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem
  Niemytzki-plane is not normal
proof
  reconsider C = (y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> as dense
  Subset of Niemytzki-plane by Th36;
  set T = Niemytzki-plane;
  defpred P[object,object] means
    ex D1 being set, U,V being open Subset of T st D1 = $1 &  $2 = U /\ C & D1
       c= U & y=0-line \ D1 c= V & U misses V;
A1: exp(2, omega) in exp(2, exp(2, omega)) by CARD_5:14;
  card C c= card product <*RAT,RAT*> by CARD_1:11,XBOOLE_1:17;
  then card C c= omega by Th8,CARD_4:6,TOPGEN_3:17;
  then
A2: exp(2, card C) c= exp(2, omega) by CARD_2:93;
  assume
A3: for W, V being Subset of T st W <> {} & V <> {} & W is closed & V is
closed & W misses V ex P, Q being Subset of T st P is open & Q is open & W c= P
  & V c= Q & P misses Q;
A4: for a being object st a in bool y=0-line ex b being object st P[a,b]
  proof
    let a be object;
    assume a in bool y=0-line;
    then reconsider aa = a as Subset of y=0-line;
    reconsider a9 = y=0-line \ aa as Subset of y=0-line by
XBOOLE_1:36;
    reconsider A = aa, B = a9 as closed Subset of T by Th42;
    per cases;
    suppose
A5:   a = {};
      take {};
      take {},{}T,[#]T;
      thus thesis by A5,Def3,Th19;
    end;
    suppose
A6:   a = y=0-line;
      take ([#]T)/\ C;
      take aa,[#]T,{}T;
      thus aa = a;
      thus thesis by A6,Def3,Th19,XBOOLE_1:37;
    end;
    suppose
A7:   a <> {} & a <> y=0-line;
      aa`` = a9`;
      then
A8:   B <> {}y=0-line by A7;
      A misses B by XBOOLE_1:79;
      then consider P, Q being Subset of T such that
A9:   P is open and
A10:  Q is open and
A11:  A c= P and
A12:  B c= Q and
A13:  P misses Q by A8,A3,A7;
      take P /\ C;
      thus thesis by A9,A10,A11,A12,A13;
    end;
  end;
  consider G being Function such that
A14: dom G = bool y=0-line and
A15: for a being object st a in bool y=0-line holds P[a,G.a] from CLASSES1:
  sch 1(A4);
  G is one-to-one
  proof
    let x,y be object;
    assume that
A16: x in dom G and
A17: y in dom G;
    reconsider A = x, B = y as Subset of y=0-line by A16,A17,A14;
    assume that
A18: G.x = G.y and
A19: x <> y;
    consider z being object such that
A20: not (z in A iff z in B) by A19,TARSKI:2;
A21: z in A\B or z in B\A by A20,XBOOLE_0:def 5;
    consider D1 being set,UB,VB being open Subset of T such that
A22:  D1 = B and
A23: G.B = UB /\ C and
A24: D1 c= UB and
A25: y=0-line \ D1 c= VB and
A26: UB misses VB by A15;
    consider D1 being set,UA,VA being open Subset of T such that
A27:  D1 = A and
A28: G.A = UA /\ C and
A29: D1 c= UA and
A30: y=0-line \ D1 c= VA and
A31: UA misses VA by A15;
    B\A = B/\A` by SUBSET_1:13;
    then
A32: B\A c= UB /\ VA by A30,A24,XBOOLE_1:27,A22,A27;
    A\B = A/\B` by SUBSET_1:13;
    then A\B c= UA /\ VB by A29,A25,XBOOLE_1:27,A22,A27;
    then C meets UA /\ VB or C meets UB /\ VA by A32,A21,TOPS_1:45;
    then (ex z being object st z in C & z in UA /\ VB)
or ex z being object st z in
    C & z in UB /\ VA by XBOOLE_0:3;
    then consider z being set such that
A33: z in C and
A34: z in UA /\ VB or z in UB /\ VA;
    z in UA & z in VB or z in UB & z in VA by A34,XBOOLE_0:def 4;
    then z in UA & not z in UB or z in UB & not z in UA by A31,A26,XBOOLE_0:3;
    then z in G.A & not z in G.B or z in G.B & not z in G.A by A28,A23,A33,
XBOOLE_0:def 4;
    hence thesis by A18;
  end;
  then
A35: card dom G c= card rng G by CARD_1:10;
  rng G c= bool C
  proof
    let a be object;
       reconsider aa=a as set by TARSKI:1;
    assume a in rng G;
    then consider b being object such that
A36: b in dom G and
A37: a = G.b by FUNCT_1:def 3;
    P[b,aa] by A14,A15,A36,A37;
    then aa c= C by XBOOLE_1:17;
    hence thesis;
  end;
  then card rng G c= card bool C by CARD_1:11;
  then card bool y=0-line c= card bool C by A35,A14;
  then
A38: exp(2, continuum) c= card bool C by Th16,CARD_2:31;
  card bool C = exp(2, card C) by CARD_2:31;
  then exp(2, continuum) c= exp(2, omega) by A38,A2;
  then exp(2, omega) in exp(2, omega) by A1,TOPGEN_3:29;
  hence contradiction;
end;
