reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;

theorem Th16:
  limit-f = sup dom f
  proof
    per cases;
    suppose
A1:   a nin dom f;
      On dom f c= 0
      proof
        let x be object;
      reconsider xx=x as set by TARSKI:1;
        assume x in On dom f; then
        x in dom f & xx is ordinal by ORDINAL1:def 9;
        hence thesis by A1;
      end; then
      sup dom f c= 0 by ORDINAL2:def 3; then
      sup dom f = 0;
      hence thesis by A1,Def5;
    end;
    suppose
A2:   ex a st a in dom f;
      f is (sup dom f)-limited;
      hence thesis by A2,Def5;
    end;
  end;
