theorem
  sup { A } = succ A
proof
A1: On { A } c= succ A
  proof
    let x be object;
    assume x in On { A };
    then x in { A } by ORDINAL1:def 9;
    then x = A by TARSKI:def 1;
    hence thesis by ORDINAL1:6;
  end;
  now
    A in { A } by TARSKI:def 1;
    then
A2: A in On { A } by ORDINAL1:def 9;
    let B;
    assume On { A } c= B;
    hence succ A c= B by A2,ORDINAL1:21;
  end;
  hence thesis by A1,Def3;
end;
