(* KEEP TOGETHER> *)
intros M x. unfold ModRed. rewrite r'''_eq; simpl. rewrite Zdivmod_split. trivial.
exact lt02w. apply Mset_l. apply Mset_r. trivial. elim (Zle_lt_or_eq 1 M). intro H. left.
replace 1 with (0 * M + 1). rewrite Zdiv_mult_plus. ring. omega. omega. omega. ring. intro H. right.
auto. apply Mset_l. rewrite <- modred with (x := 2 ^ (Zlog_sup M + w) / M - 2 ^ w) (m := 2 ^ w).
rewrite Zmod_minus_distr_r. rewrite Z_mod_same. apply f_equal2 with (f := Zmod). ring. trivial.
apply Zlt_gt. exact lt02w. exact lt02w. elim (M'ex M). omega. elim (M'ex M). omega.
apply Zle_lt_trans with ((2 ^ (Zlog_sup M + w) - 1) / 2 ^ w). apply Zdiv_le. exact lt02w.
cut (x < 2 ^ (Zlog_sup M + w)). omega. apply lt_z__Z. rewrite Zpower_exp. rewrite Zdiv_mult_minus.
omega. exact lt02w. omega. cut (0 < 2 ^ w). omega. exact lt02w. apply Zle_ge. apply Zlog_sup_correct1.
apply Zlt_le_trans with 1. omega. apply Mset_l. apply Zle_ge. exact le_0_w. Qed.

(* Specifications: *)
Lemma xheex : forall (M : Mset) (h : Z_ (2 ^ Zlog_sup M)),
in_Z_ (2 ^ w) h.