+
+lemma ap_rewl: ∀A:apartness.∀x,z,y:A. x ≈ y → y # z → x # z.
+intros (A x z y Exy Ayz); cases (ap_cotransitive ???x Ayz); [2:assumption]
+cases (Exy (ap_symmetric ??? a));
+qed.
+
+lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x.
+intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy);
+apply ap_symmetric; assumption;
+qed.