reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;

theorem
  dom fi = succ A implies last fi is_limes_of fi & lim fi = last fi
proof
  assume
A1: dom fi = succ A;
  then
A2: last fi = fi.A by ORDINAL2:6;
  thus last fi is_limes_of fi
  proof
    per cases;
    case
A3:   last fi = 0;
      take A;
      thus A in dom fi by A1,ORDINAL1:6;
      let B;
      assume that
A4:   A c= B and
A5:   B in dom fi;
      B c= A by A1,A5,ORDINAL1:22;
      hence thesis by A2,A3,A4,XBOOLE_0:def 10;
    end;
    case
      last fi <> 0;
      let B,C such that
A6:   B in last fi and
A7:   last fi in C;
      take A;
      thus A in dom fi by A1,ORDINAL1:6;
      let D;
      assume that
A8:   A c= D and
A9:   D in dom fi;
      D c= A by A1,A9,ORDINAL1:22;
      hence thesis by A2,A6,A7,A8,XBOOLE_0:def 10;
    end;
  end;
  hence thesis by ORDINAL2:def 10;
end;
