apply Build_preorder.
unfold reflexive.
intros.
unfold Ele.
rewrite Ecmp_refl.
intro.
discriminate.
unfold transitive.
intros.
unfold Ele.
case_eq (Ecmp e x z); do 2 intro; try discriminate.
apply H0.
apply Ecmp_ge_gt_trans with x...
apply Ele_Ege...
Qed.

(* Specifications: *)
Lemma Ege_preorder: preorder _ Ege.
Proof with try assumption.