(* To define exponential over relative number, we simply do 
   a case analysis on the sign of the number *)
 
Definition powerRZ (e : R) (n : BinInt.Z) :=
  match n with
  | BinInt.Z0 => 1%R
  | BinInt.Zpos p => (e ^ BinPos.nat_of_P p)%R
  | BinInt.Zneg p => (/ e ^ BinPos.nat_of_P p)%R
  end.

Theorem Zpower_nat_powerRZ_absolu : forall n m : BinInt.Z, (0 <= m)%Z -> IZR (Zpower_nat n (Zabs_nat m)) = powerRZ (IZR n) m.