
theorem
  for a,b be positive Real, m be non positive Real, n be negative Real holds
  a to_power m + b to_power m <= 1 implies
  a to_power (m+n) + b to_power (m+n) < 1
  proof
    let a,b be positive Real, m be non positive Real,n be negative Real;
    reconsider k = a to_power (-1) as positive Real;
    reconsider l = b to_power (-1) as positive Real;
    k to_power -m = a to_power ((-1)*(-m)) &
    k to_power (-m - n) = a to_power ((-1)*(-m -n)) &
    l to_power -m = b to_power ((-1)*(-m)) &
    l to_power (-m -n) = b to_power ((-1)*(-m -n)) by POWER:33;
    hence thesis by Pow;
  end;
