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_absolu : forall n m:Z, (0 <= m)%Z -> IZR (Zpower_nat n (Z.abs_nat m)) = IZR n ^Z m.