intros n m; elim m; simpl in |- *; auto with real.
intros m1 H'; rewrite nat_of_P_o_P_of_succ_nat_eq_succ; simpl in |- *.
replace (Zpower_nat n (S m1)) with (n * Zpower_nat n m1)%Z.
rewrite Rmult_IZR; auto with real.
rewrite H'; simpl in |- *.
case m1; simpl in |- *; auto with real.
intros m2; rewrite nat_of_P_o_P_of_succ_nat_eq_succ; auto.
unfold Zpower_nat in |- *; auto.
Qed.

Theorem powerRZ_lt : forall (e : R) (z : Z), (0 < e)%R -> (0 < powerRZ e z)%R.
intros e z; case z; simpl in |- *; auto with real.