definition hos_excess ≝ λO:half_ordered_set.wloss O ? (hos_excess_ O).
+(*
lemma find_leq : half_ordered_set → half_ordered_set.
intro O; constructor 1;
[1: apply (hos_carr O);
rewrite > (H1 ? (hos_excess_ O)); lapply (hos_cotransitive O ?? z H);
[assumption] cases Hletin;[right|left]assumption;]
qed.
+*)
definition dual_hos : half_ordered_set → half_ordered_set.
intro; constructor 1;
constructor 1; [apply hos; | apply (dual_hos hos); | reflexivity]
qed.
-(* coercion half2full. *)
-
definition Type_of_ordered_set : ordered_set → Type.
intro o; apply (hos_carr (os_l o)); qed.