
theorem LMN:
  for k be positive Real, n be non positive Real holds
  (k+1) to_power n < (k to_power n) + 1
  proof
    let k be positive Real, n be non positive Real;
    per cases;
    suppose
      n = 0; then
      (k+1) to_power n = 1 & k to_power n = 1 by POWER:24;
      hence thesis;
    end;
    suppose
      n < 0; then
      reconsider n as negative Real;
      (k+1) to_power n is light positive; then
      (k+1) to_power n + 0 < (k+1) to_power n + k to_power n < 1 + k to_power n
        by XREAL_1:6;
      hence thesis by XXREAL_0:2;
    end;
  end;
