
theorem Th2:
  for a being Ordinal holds a +^ a = 2 *^ a
proof
  let a be Ordinal;
  consider fi being Ordinal-Sequence such that
    A1: 2 *^ a = last fi & dom fi = succ 2 & fi.0 = 0 and
    A2: for c being Ordinal st succ c in succ 2
      holds fi.succ c = (fi.c) +^ a and
    for c being Ordinal st c in succ 2 & c <> 0 & c is limit_ordinal
      holds fi.c = union sup(fi|c) by ORDINAL2:def 15;
  succ 0 in succ succ 0 & succ succ 0 in succ 2 by ORDINAL1:6;
  then A3: succ 0 in succ 2 & succ succ 0 in succ 2 by ORDINAL1:10;
  2 *^ a = fi.2 by A1, ORDINAL2:6
    .= (fi.succ 0) +^ a by A2, A3
    .= ((fi.0) +^ a) +^ a by A2, A3
    .= a +^ a by A1, ORDINAL2:30;
  hence thesis;
end;
