reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem
  for V being Subset of X holds V is a_neighborhood of {x} iff V is
  a_neighborhood of x
proof
  let V be Subset of X;
  thus V is a_neighborhood of {x} implies V is a_neighborhood of x
  proof
    assume V is a_neighborhood of {x};
    then
A1: {x} c= Int(V) by Def2;
    x in {x} by TARSKI:def 1;
    hence thesis by A1,Def1;
  end;
  assume V is a_neighborhood of x;
  then x in Int (V) by Def1;
  then for p being object holds p in {x} implies p in Int V by TARSKI:def 1;
  then {x} c= Int V;
  hence thesis by Def2;
end;
