+lemma relation_pair_composition_morphism_assoc:
+Πo1:basic_pair
+.Πo2:basic_pair
+ .Πo3:basic_pair
+ .Πo4:basic_pair
+ .Πa12:relation_pair_setoid o1 o2
+ .Πa23:relation_pair_setoid o2 o3
+ .Πa34:relation_pair_setoid o3 o4
+ .relation_pair_composition_morphism o1 o3 o4
+ (relation_pair_composition_morphism o1 o2 o3 a12 a23) a34
+ =relation_pair_composition_morphism o1 o2 o4 a12
+ (relation_pair_composition_morphism o2 o3 o4 a23 a34).
+ intros;
+ change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
+ ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
+ alias symbol "refl" = "refl1".
+ alias symbol "prop2" = "prop21".
+ apply (ASSOC‡#);
+qed.
+
+lemma relation_pair_composition_morphism_respects_id:
+ ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
+ relation_pair_composition_morphism o1 o1 o2 (id_relation_pair o1) a=a.
+ intros;
+ change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
+ apply ((id_neutral_right1 ????)‡#);
+qed.
+
+lemma relation_pair_composition_morphism_respects_id_r:
+ ∀o1,o2:basic_pair.∀a:relation_pair_setoid o1 o2.
+ relation_pair_composition_morphism o1 o2 o2 a (id_relation_pair o2)=a.
+ intros;
+ change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
+ apply ((id_neutral_left1 ????)‡#);
+qed.
+