theorem
  succ X = succ Y implies X = Y
proof
  assume that
A1: succ X = succ Y and
A2: X <> Y;
  Y in X \/ { X } by A1,Th2;
  then
A3: Y in X or Y in { X } by XBOOLE_0:def 3;
  X in Y \/ { Y } by A1,Th2;
  then X in Y or X in { Y } by XBOOLE_0:def 3;
  hence contradiction by A2,A3,TARSKI:def 1;
end;
