+(* the same as Rest for a basic pair *)
+definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+ intros; constructor 1;
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. e ‡#); assumption
+ | apply (. e^ -1‡#); assumption]
+ | intros; split; simplify; intros; [ apply (. #‡e1 ^ -1); | apply (. #‡e1)]
+ apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+qed.
+
+(* the same as Ext for a basic pair *)
+definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+ intros; constructor 1;
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
+ exT ? (λy:V.x ♮r y ∧ y ∈ S) });
+ intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+ [ apply (. (e ^ -1‡#)‡#); assumption
+ | apply (. (e‡#)‡#); assumption]
+ | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+ [ apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
+ apply (if ?? (e ??)); assumption
+ | apply (. #‡(#‡e1)); cases x; split; try assumption;
+ apply (if ?? (e ^ -1 ??)); assumption]]
+qed.
+
+(*