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

theorem
  Niemytzki-plane is T_1
proof
  set T = Niemytzki-plane;
  let p,q be Point of T such that
A1: p <> q;
A2: q in the carrier of T;
A3: the carrier of T = y>=0-plane by Def3;
  p in the carrier of T;
  then reconsider p9 = p, q9 = q as Point of TOP-REAL 2 by A2,A3;
  p9-q9 <> 0.TOP-REAL 2 by A1,RLVECT_1:21;
  then |.p9-q9.| <> 0 by EUCLID_2:42;
  then reconsider r = |.p9-q9.| as positive Real;
  consider ap being Point of TOP-REAL 2, Up being open Subset of T such that
A4: p in Up and
  ap in Up and
A5: for b being Point of TOP-REAL 2 st b in Up holds |.b-ap.| < r/2 by Th30;
  consider aq being Point of TOP-REAL 2, Uq being open Subset of T such that
A6: q in Uq and
  aq in Uq and
A7: for b being Point of TOP-REAL 2 st b in Uq holds |.b-aq.| < r/2 by Th30;
  take Up,Uq;
  thus Up is open & Uq is open & p in Up by A4;
  thus not q in Up
  proof
    assume q in Up;
    then
A8: |.q9-ap.| < r/2 by A5;
    |.q9-ap.| = |.ap-q9.| by TOPRNS_1:27;
    then |.p9-ap.|+|.ap-q9.| < r/2+r/2 by A8,A4,A5,XREAL_1:8;
    hence contradiction by TOPRNS_1:34;
  end;
  thus q in Uq by A6;
  assume
A9: p in Uq;
A10: |.q9-aq.| = |.aq-q9.| by TOPRNS_1:27;
  |.q9-aq.| < r/2 by A6,A7;
  then |.p9-aq.|+|.aq-q9.| < r/2+r/2 by A10,A9,A7,XREAL_1:8;
  hence contradiction by TOPRNS_1:34;
end;
