intros e n m H' H'0; replace m with (m - n + n).
rewrite pow_add.
pattern (e ^ n)%R at 1 in |- *; replace (e ^ n)%R with (1 * e ^ n)%R;
auto with real.
apply Rminus_lt.
repeat rewrite (fun x : R => Rmult_comm x (e ^ n));
rewrite <- Rmult_minus_distr_l.
replace 0%R with (e ^ n * 0)%R; auto with real.
apply Rmult_lt_compat_l; auto with real.
apply pow_lt; auto with real.
apply Rlt_trans with (r2 := 1%R); auto with real.
apply Rlt_minus; auto with real.
apply Rlt_pow_R1; auto with arith.
apply plus_lt_reg_l with (p := n); auto with arith.
rewrite le_plus_minus_r; auto with arith; rewrite <- plus_n_O; auto.
rewrite plus_comm; auto with arith.
Qed.
Hint Resolve Rlt_pow: real.

Theorem pow_R1 :
forall (r : R) (n : nat), (r ^ n)%R = 1%R -> Rabs r = 1%R \/ n = 0.