Definition pow_N (x:R) (p:N) :=
  match p with
  | N0 => rI
  | Npos p => pow_pos x p
  end.Definition id_phi_N (x:N) : N := x.

(* Specifications: *)
Lemma pow_N_pow_N x n : pow_N x (id_phi_N n) == pow_N x n.