reserve n, k, r, m, i, j for Nat;

theorem Th4:
  for m being non zero Real, n being Integer holds
  ((-1) * m) to_power n = ((-1) to_power n) * (m to_power n)
proof
  let m be non zero Real, n be Integer;
  per cases;
  suppose
A1: n is odd;
    then (-m) to_power n = -(m to_power n) by POWER:48
      .= (-1) * (m to_power n)
      .= ((-1) to_power n) * (m to_power n) by A1,Th2;
    hence thesis;
  end;
  suppose
A2: n is even;
    then (-m) to_power n = 1 * (m to_power n) by POWER:47
      .= ((-1) to_power n) * (m to_power n) by A2,Th3;
    hence thesis;
  end;
end;
