theorem Th5:
  for P, a holds a in P* iff ex n st a in P^^n
proof
  let P, a;
  set X = the set of all P^^n where n is Nat;
  thus a in P* implies ex n st a in P^^n
  proof
    assume a in P*;
    then consider Y such that A1: a in Y and A2: Y in X by TARSKI:def 4;
    consider n such that A3: Y = P^^n by A2;
    take n;
    thus thesis by A1, A3;
  end;
  given n such that A4: a in P^^n;
  P^^n in X;
  hence thesis by A4, TARSKI:def 4;
end;
