
theorem
  for a, b being Ordinal st 1 in a & a in b holds exp(b,a) in b |^|^ a
proof
  let a, b be Ordinal;
  assume A1: 1 in a & a in b;
  then A2: 1 in b by ORDINAL1:10;
  then 0 c< b by XBOOLE_1:2, XBOOLE_0:def 8;
  then 0 in b by ORDINAL1:11;
  then A3: b |^|^ 2 c= b |^|^ a by A1, Lm2, ORDINAL1:21, ORDINAL5:21;
  exp(b,a) in exp(b,b) by A1, A2, ORDINAL4:24;
  then exp(b,a) in b |^|^ 2 by ORDINAL5:18;
  hence thesis by A3;
end;
