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 Th41:
  for NT being non empty normal NTopSpace holds NTop2Top NT is normal
  proof
    let NT be non empty normal NTopSpace;
    reconsider T = NTop2Top NT as non empty TopSpace;
    now
      let F1,F2 be Subset of T;
      assume that
A1:   F1 is closed and
A2:   F2 is closed and
A3:   F1 misses F2;
      Top2NTop T = NT by FINTOPO7:25;
      then reconsider F19 = F1,F29 = F2 as closed Subset of NT
        by A1,A2,Lm7;
      consider V be a_neighborhood of F19,
               W be a_neighborhood of F29 such that
A4:   V misses W by A3,Def13;
A5:   NTop2Top F19 c= Int NTop2Top V & NTop2Top F29 c= Int NTop2Top W
        by Th36,CONNSP_2:def 2;
      reconsider G1 = Int NTop2Top V,
                 G2 = Int NTop2Top W as open Subset of T;
      thus ex G1,G2 be Subset of T st G1 is open & G2 is open &
        F1 c= G1 & F2 c= G2 & G1 misses G2 by A4,A5,Th1;
    end;
    hence thesis by PRE_TOPC:def 12;
  end;
