
theorem ATB:
  for a,b be positive Real st a <> b holds
  ex n, m be Real st a = (a/b) to_power n & b = (a/b) to_power m
  proof
    let a,b be positive Real such that
    A1: a <> b;
    reconsider x = a/b as positive Real;
    x <> 1 by A1,XCMPLX_1:58; then
    x to_power (log (x,a)) = a & x to_power (log (x,b)) = b by POWER:def 3;
    hence thesis;
  end;
