intros M h. unfold in_Z_. split. apply le_0__Z. apply Zlt_le_trans with (2 ^ (Zlog_sup M)).
apply lt_z__Z. apply Zle_pow_le. apply Zlog_sup_correct1. apply Zlt_le_trans with 1. omega. apply Mset_l.
apply Zle_trans with (Zlog_sup (2 ^ w)).
apply Zlog_sup_seq. apply Zlt_le_weak. apply Zle_lt_trans with (2 ^ (w - 1)). apply Mset_r.
exact lt_2wm1_2w. rewrite Zlog_sup_pow. omega. exact le_0_w. Qed.

(* Specifications: *)
Lemma xleex : forall (u : Z_ (2 ^ w)),
in_Z_ (2 ^ w) u.