
theorem Th23:
  for a, b, c being Ordinal st 0 in a & 1 in b & a in exp(b,c)
  holds b -exponent a in c
proof
  let a, b, c be Ordinal;
  assume that
    A1: 0 in a and
    A2: 1 in b and
    A3: a in exp(b,c);
  exp(b,c) = 1*^exp(b,c) & 0 in 1 by CARD_1:49, TARSKI:def 1, ORDINAL2:39;
  then b-exponent(exp(b,c)) = c by A2, ORDINAL5:58;
  then A4: b -exponent a c= c by A3, Th22, ORDINAL1:def 2;
  b -exponent a <> c
  proof
    assume A5: b -exponent a = c;
    exp(b,b -exponent a) c= a by A2, A1, ORDINAL5:def 10;
    hence contradiction by A3, A5, ORDINAL1:5;
  end;
  hence thesis by A4, XBOOLE_0:def 8, ORDINAL1:11;
end;
