reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;
reserve NT for T_2 NTopSpace;

theorem Th39:
  for x,y being Point of NT st x <> y ex Vx being Element of U_FMT x,
  Vy being Element of U_FMT y st Vx misses Vy
  proof
    let x,y be Point of NT;
    assume
A1: x <> y;
    now
      assume
A2:   for Vx be Element of U_FMT x, Vy be Element of U_FMT y holds Vx meets Vy;
A3:  now
        let Vx be Element of U_FMT x, Vy be Element of U_FMT y;
        Vx meets Vy by A2;
        hence Vx /\ Vy is non empty;
      end;
      reconsider X = the carrier of NT as non empty set;
      reconsider Ux = U_FMT x as Filter of X;
      reconsider Uy = U_FMT y as Filter of X;
      consider F be Filter of X such that
A4:   F is_filter-finer_than Ux and
A5:   F is_filter-finer_than Uy by A3,CARDFIL2:58;
      reconsider x9 = x, y9 = y as Element of X;
      reconsider F as Filter of the carrier of NT;
A6:   x9 in lim_filter F & y9 in lim_filter F by A4,A5;
      lim_filter F is empty or lim_filter F is trivial by Def12;
      hence contradiction by A1,A6;
    end;
    hence thesis;
  end;
