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
  NTop2Top NT is T_2 non empty strict TopSpace
  proof
    reconsider T = NTop2Top NT as non empty TopSpace;
    now
      let p, q be Point of T;
      assume
A1:   p <> q;
      reconsider p9 = p, q9 = q as Point of NT by FINTOPO7:def 16;
      set Sp = lim_filter U_FMT p9,
          Sq = lim_filter U_FMT q9;
      consider Vx be Element of U_FMT p9, Vy be Element of U_FMT q9 such that
A2:   Vx misses Vy by A1,Th39;
      p9 is_interior_point_of Vx by FINTOPO7:def 5;
      then consider Ox be open Subset of NT such that
A3:   p9 in Ox and
A4:   Ox c= Vx by Lm4;
      q9 is_interior_point_of Vy by FINTOPO7:def 5;
      then consider Oy be open Subset of NT such that
A5:   q9 in Oy and
A6:   Oy c= Vy by Lm4;
      reconsider G1 = Ox, G2 = Oy as open Subset of T by Lm9;
      G1 misses G2 by A4,A6,A2,XBOOLE_1:64;
      hence ex G1,G2 be Subset of T st G1 is open & G2 is open &
        p in G1 & q in G2 & G1 misses G2 by A3,A5;
    end;
    hence thesis by PRE_TOPC:def 10;
  end;
