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;

theorem
  A is a_neighborhood of a implies NTop2Top A is a_neighborhood of NTop2Top a
  proof
    reconsider T = NTop2Top NT as non empty TopSpace;
    reconsider TA = NTop2Top A as Subset of T;
    reconsider Tx = NTop2Top a as Point of T;
    assume A is a_neighborhood of a;
    then a is_interior_point_of A;
    then a in Int A;
    then Tx in Int TA by Th34;
    hence thesis by CONNSP_2:def 1;
  end;
