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 Th14:
  for NT be non empty strict NTopSpace
  for A be Subset of NT
  for x be Point of NT st
  A is a_neighborhood of x holds Int A is open a_neighborhood of x
  proof
    let NT be non empty strict NTopSpace;
    let A be Subset of NT;
    let x be Point of NT;
    assume A is a_neighborhood of x;
    then x is_interior_point_of A;
    then x in Int A;
    then x in Int Int A by Th12;
    then ex y be Point of NT st x = y & y is_interior_point_of Int A;
    then reconsider IA = Int A as a_neighborhood of x;
    IA is open a_neighborhood of x;
    hence thesis;
  end;
