
theorem Th22:
  for a, b, c being Ordinal st a c= c holds b -exponent a c= b -exponent c
proof
  let a, b, c be Ordinal;
  assume A1: a c= c;
  per cases;
  suppose A2: 1 in b & 0 in a & 0 in c;
    then exp(b,b -exponent a) c= a by ORDINAL5:def 10;
    then exp(b,b -exponent a) c= c by A1, XBOOLE_1:1;
    hence thesis by A2, ORDINAL5:def 10;
  end;
  suppose not 1 in b;
    then b-exponent a = 0 & b-exponent c = 0 by ORDINAL5:def 10;
    hence thesis;
  end;
  suppose not 0 in a or not 0 in c;
    then not 0 in a by A1;
    then b-exponent a = {} by ORDINAL5:def 10;
    hence thesis;
  end;
end;
