intros r p q H' H'0 H'1.
cut (powerRZ r (p - q) = 1%R); [ intros Eq0 | idtac ].
case (Rpow_R1 _ _ H' Eq0); auto with zarith.
intros H'2; case H'0; auto.
apply Rmult_eq_reg_l with (r := powerRZ r q); auto with real.
rewrite <- powerRZ_add; auto.
replace (q + (p - q))%Z with p; auto with zarith.
rewrite <- H'1; rewrite Rmult_1_r; auto with arith.
Qed.

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