+qed.
+
+lemma o_relation_pair_of_relation_pair_morphism :
+ ∀S,T:category2_of_category1 BP.
+ unary_morphism2 (arrows2 (category2_of_category1 BP) S T)
+ (arrows2 OBP (o_basic_pair_of_basic_pair S) (o_basic_pair_of_basic_pair T)).
+intros (S T);
+ constructor 1;
+ [ apply (o_relation_pair_of_relation_pair S T);
+ | apply (o_relation_pair_of_relation_pair_is_morphism S T)]
+qed.
+
+lemma o_relation_pair_of_relation_pair_morphism_respects_id:
+ ∀o:category2_of_category1 BP.
+ o_relation_pair_of_relation_pair_morphism o o (id2 (category2_of_category1 BP) o)
+ = id2 OBP (o_basic_pair_of_basic_pair o).
+ simplify; intros; whd; split;