theorem
  P is open iff for x holds x in P iff ex Q st Q is open & Q c= P & x in Q
proof
  thus P is open implies for x holds x in P iff ex Q st Q is open & Q c= P & x
  in Q;
  assume
A1: for x holds x in P iff ex Q st Q is open & Q c= P & x in Q;
  now
    let x be object;
    x in P iff ex Q st Q is open & Q c= P & x in Q by A1;
    hence x in P iff x in Int P by Th22;
  end;
  hence thesis by TARSKI:2;
end;
