theorem
  A is a_neighborhood of x implies Top2NTop A is a_neighborhood of Top2NTop x
  proof
    reconsider NT = Top2NTop T as NTopSpace;
    reconsider NA = Top2NTop A as Subset of NT;
    reconsider Nx = Top2NTop x as Point of NT;
    assume A is a_neighborhood of x;
    then x in Int A by CONNSP_2:def 1;
    then Nx in Int NA by Lm30;
    then ex y be Point of NT st Nx = y & y is_interior_point_of NA;
    hence thesis;
  end;
