
theorem
  for T being non empty TopSpace, p being Point of T, P being Basis of p
  holds P is basis of p
proof
  let T be non empty TopSpace, p be Point of T, P be Basis of p;
  now
    let A be Subset of T;
    assume p in Int A;
    then consider V being Subset of T such that
A1: V in P and
A2: V c= Int A by YELLOW_8:def 1;
    take V;
    V is open by A1,YELLOW_8:12;
    then Int A c= A & Int V = V by TOPS_1:16,23;
    hence V in P & p in Int V & V c= A by A1,A2,YELLOW_8:12;
  end;
  hence thesis by Def1;
end;
