intros e n m; case n; case m; simpl in |- *; auto with real.
intros n1 m1; rewrite Pnat.nat_of_P_plus_morphism; auto with real.
intros n1 m1. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare m1 n1 Datatypes.Eq); simpl in |- *;
auto with real.
intros H' H'0; rewrite BinPos.Pcompare_Eq_eq with (1 := H'); auto with real.
intros H' H'0; rewrite (Pnat.nat_of_P_minus_morphism n1 m1); auto with real.
rewrite
(pow_RN_plus e (BinPos.nat_of_P n1 - BinPos.nat_of_P m1)
(BinPos.nat_of_P m1)); auto with real.
rewrite plus_comm; rewrite le_plus_minus_r; auto with real.
rewrite Rinv_mult_distr; auto with real.
rewrite Rinv_involutive; auto with real.
apply lt_le_weak.
apply Pnat.nat_of_P_lt_Lt_compare_morphism; auto.
apply BinPos.ZC2; auto.
intros H' H'0; rewrite (Pnat.nat_of_P_minus_morphism m1 n1); auto with real.
rewrite
(pow_RN_plus e (BinPos.nat_of_P m1 - BinPos.nat_of_P n1)
(BinPos.nat_of_P n1)); auto with real.
rewrite plus_comm; rewrite le_plus_minus_r; auto with real.
apply lt_le_weak.
change (BinPos.nat_of_P m1 > BinPos.nat_of_P n1) in |- *.
apply Pnat.nat_of_P_gt_Gt_compare_morphism; auto.
intros n1 m1. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare n1 m1 Datatypes.Eq); simpl in |- *;
auto with real.
intros H' H'0; rewrite BinPos.Pcompare_Eq_eq with (1 := H'); auto with real.
intros H' H'0; rewrite (Pnat.nat_of_P_minus_morphism m1 n1); auto with real.
rewrite
(pow_RN_plus e (BinPos.nat_of_P m1 - BinPos.nat_of_P n1)
(BinPos.nat_of_P n1)); auto with real.
rewrite plus_comm; rewrite le_plus_minus_r; auto with real.
rewrite Rinv_mult_distr; auto with real.
apply lt_le_weak.
apply Pnat.nat_of_P_lt_Lt_compare_morphism; auto.
apply BinPos.ZC2; auto.
intros H' H'0; rewrite (Pnat.nat_of_P_minus_morphism n1 m1); auto with real.
rewrite
(pow_RN_plus e (BinPos.nat_of_P n1 - BinPos.nat_of_P m1)
(BinPos.nat_of_P m1)); auto with real.
rewrite plus_comm; rewrite le_plus_minus_r; auto with real.
apply lt_le_weak.
change (BinPos.nat_of_P n1 > BinPos.nat_of_P m1) in |- *.
apply Pnat.nat_of_P_gt_Gt_compare_morphism; auto.
intros n1 m1; rewrite Pnat.nat_of_P_plus_morphism; auto with real.
intros H'; rewrite pow_add; auto with real.
apply Rinv_mult_distr; auto.
apply pow_NR0; auto.
apply pow_NR0; auto.
Qed.
Hint Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add: real.

Theorem powerRZ_Zopp :
forall (e : R) (z : BinInt.Z),
e <> 0%R -> powerRZ e (- z) = (/ powerRZ e z)%R.