(* KE EP TOGETHER> *)
elim M1_r'a; intro H. tauto. right. unfold q_i. rewrite uhwm_eq. rewrite H. fold M'. rewrite M'_M. trivial. Qed.

(* <KE EP TOGETHER - q_i_eq *)
Lemma q_i_eq :
_Z q_i = x' / 2 ^ p * (2 ^ (p + w) / M mod 2 ^ w) / 2 ^ w.