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

theorem Th82:
  Niemytzki-plane is T_1
proof
  set X = Niemytzki-plane;
  let x,y be Point of X;
A1: the carrier of X = y>=0-plane by Def3;
  then
A2: y in y>=0-plane;
  x in y>=0-plane by A1;
  then reconsider a = x, b = y as Point of TOP-REAL 2 by A2;
  assume x <> y;
  then |.a-b.| <> 0 by TOPRNS_1:28;
  then reconsider r = |.a-b.| as positive Real;
  consider q being Point of TOP-REAL 2, U being open Subset of X such that
A3: x in U and
  q in U and
A4: for c being Point of TOP-REAL 2 st c in U holds |.c-q.| < r/2 by Th30;
  consider p being Point of TOP-REAL 2, V being open Subset of X such that
A5: y in V and
  p in V and
A6: for c being Point of TOP-REAL 2 st c in V holds |.c-p.| < r/2 by Th30;
  take U,V;
  thus U is open & V is open & x in U by A3;
  hereby
    assume y in U;
    then |.b-q.| < r/2 by A4;
    then
A7: |.a-q.|+|.b-q.| < (r/2)+r/2 by A3,A4,XREAL_1:8;
    |.a-b.| <= |.a-q.|+|.q-b.| by TOPRNS_1:34;
    hence contradiction by A7,TOPRNS_1:27;
  end;
  thus y in V by A5;
  assume
A8: x in V;
A9: |.a-b.| <= |.a-p.|+|.p-b.| by TOPRNS_1:34;
  |.b-p.| < r/2 by A5,A6;
  then |.a-p.|+|.b-p.| < (r/2)+r/2 by A8,A6,XREAL_1:8;
  hence contradiction by A9,TOPRNS_1:27;
end;
