Definition powerRZ (x:R) (n:Z) :=
  match n with
    | Z0 => 1
    | Zpos p => x ^ Pos.to_nat p
    | Zneg p => / x ^ Pos.to_nat p
  end.Local Infix "^Z" := powerRZ (at level 30, right associativity) : R_scope.

Lemma Zpower_nat_powerRZ : forall n m:nat, IZR (Zpower_nat (Z.of_nat n) m) = INR n ^Z Z.of_nat m.