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

theorem Th11:
  (-1) to_power (-n) = (-1) to_power n
proof
  per cases;
  suppose
    n is odd;
    then reconsider n as odd Integer;
    (-1) to_power (-n) = -1 by Th2
      .= (-1) to_power n by Th2;
    hence thesis;
  end;
  suppose
    n is even;
    then reconsider n as even Integer;
    (-1) to_power (-n) = 1 by Th3
      .= (-1) to_power n by Th3;
    hence thesis;
  end;
end;
