
theorem
  for a,b be positive Real st a <> b holds
  ex n,m be Real st a - b = ((a/b) to_power n)*((a/b) to_power m - 1)
  proof
    let a,b be positive Real such that
    A1: a <> b;
    consider x,y be Real such that
    A2: a = (a/b) to_power x & b = (a/b) to_power y by A1,ATB;
    (a/b) to_power x = (a/b) to_power (y + (x-y))
    .= ((a/b) to_power y)*((a/b) to_power (x-y)) by POWER:27; then
    ((a/b) to_power y)*(((a/b) to_power (x-y)) - 1) = a - b by A2;
    hence thesis;
  end;
