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;
reserve T   for non empty TopSpace,
        A,B for Subset of T,
        F   for closed Subset of T,
        O   for open Subset of T;
reserve T   for non empty strict TopSpace,
        A,B for Subset of T,
        x   for Point of T;

theorem TH:
  for T being non empty normal strict TopSpace holds Top2NTop T is normal
  proof
    let T be non empty normal strict TopSpace;
    reconsider NT = Top2NTop T as NTopSpace;
    now
      let NA,NB be closed Subset of NT;
      assume
A1:   NA misses NB;
      NTop2Top NT = T by FINTOPO7:24;
      then reconsider A = NTop2Top NA,
      B = NTop2Top NB as closed Subset of T by Lm29;
      consider G1,G2 be Subset of T such that
A2:   G1 is open and
A3:   G2 is open and
A4:   A c= G1 and
A5:   B c= G2 and
A6:   G1 misses G2 by A1,PRE_TOPC:def 12;
      reconsider V = Top2NTop G1, W = Top2NTop G2 as open Subset of NT
        by Lm1,A2,A3;
A7:   A c= Int G1 & B c= Int G2 by A2,A3,A4,A5,TOPS_1:23;
      reconsider V = G1,
                 W = G2 as open Subset of NT by A2,A3,Lm1;
      Top2NTop G1 is a_neighborhood of Top2NTop A &
      Top2NTop G2 is a_neighborhood of Top2NTop B
        by A7,CONNSP_2:def 2,Lm31;
      hence ex V be a_neighborhood of NA, W be a_neighborhood of NB st
        V misses W by A6;
    end;
    hence thesis;
  end;
