reserve n for Nat;

theorem Th5:
  for T being non empty TopSpace holds T is first-countable iff
  the TopStruct of T is first-countable
proof
  let T be non empty TopSpace;
  thus T is first-countable implies the TopStruct of T is first-countable
  proof
    assume
A1: T is first-countable;
    let x be Point of the TopStruct of T;
    reconsider y = x as Point of T;
    consider C being Basis of y such that
A2: C is countable by A1;
    reconsider B = C as Basis of x by Lm1;
    take B;
    thus B is countable by A2;
  end;
  assume
A3: the TopStruct of T is first-countable;
  let x be Point of T;
  reconsider y = x as Point of the TopStruct of T;
  consider C being Basis of y such that
A4: C is countable by A3;
  reconsider B = C as Basis of x by Lm1;
  take B;
  thus B is countable by A4;
end;
