+(* this proof is more logic-oriented than set/lattice oriented *)
+theorem continuous_relation_eqS:
+ ∀o1,o2:basic_topology.∀a,a': continuous_relation_setoid o1 o2.
+ a = a' → ∀X. A ? (extS ?? a X) = A ? (extS ?? a' X).
+ intros;
+ cut (∀a: arrows1 ? o1 ?.∀x. x ∈ extS ?? a X → ∃y:o2.y ∈ X ∧ x ∈ ext ?? a y);
+ [2: intros; cases f; clear f; cases H1; exists [apply w] cases x1; split;
+ try assumption; split; assumption]
+ cut (∀a,a':continuous_relation_setoid o1 o2.eq1 ? a a' → ∀x. x ∈ extS ?? a X → ∃y:o2. y ∈ X ∧ x ∈ A ? (ext ?? a' y));
+ [2: intros; cases (Hcut ?? f); exists; [apply w] cases x1; split; try assumption;
+ apply (. #‡(H1 ?));
+ apply (saturation_expansive ?? (A_is_saturation o1) (ext ?? a1 w) x);
+ assumption;] clear Hcut;
+ split; apply (if ?? (A_is_saturation ???)); intros 2;
+ [lapply (Hcut1 a a' H a1 f) | lapply (Hcut1 a' a (H \sup -1) a1 f)]
+ cases Hletin; clear Hletin; cases x; clear x;
+ cut (∀a: arrows1 ? o1 ?. ext ?? a w ⊆ extS ?? a X);
+ [2,4: intros 3; cases f3; clear f3; simplify in f5; split; try assumption;
+ exists [1,3: apply w] split; assumption;]
+ cut (∀a. A ? (ext o1 o2 a w) ⊆ A ? (extS o1 o2 a X));
+ [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
+ apply Hcut2; assumption.
+qed.
+*)
+
+theorem continuous_relation_eq':
+ ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
+ a = a' → ∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X).
+ intros; split; intro; unfold minus_star_image; simplify; intros;
+ [ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
+ lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
+ cut (A ? (ext ?? a' a1) ⊆ A ? X); [2: apply (. (H ?)‡#); assumption]
+ lapply (fi ?? (A_is_saturation ???) Hcut);
+ apply (Hletin1 x); change with (x ∈ ext ?? a' a1); split; simplify;
+ [ apply I | assumption ]
+ | cut (ext ?? a' a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
+ lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
+ cut (A ? (ext ?? a a1) ⊆ A ? X); [2: apply (. (H ?)\sup -1‡#); assumption]
+ lapply (fi ?? (A_is_saturation ???) Hcut);
+ apply (Hletin1 x); change with (x ∈ ext ?? a a1); split; simplify;
+ [ apply I | assumption ]]
+qed.
+
+theorem extS_singleton:
+ ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
+ intros; unfold extS; unfold ext; unfold singleton; simplify;
+ split; intros 2; simplify; cases f; split; try assumption;
+ [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
+ assumption
+ | exists; try assumption; split; try assumption; change with (x = x); apply refl]
+qed.
+
+theorem continuous_relation_eq_inv':
+ ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
+ (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → a=a'.
+ intros 6;
+ cut (∀a,a': continuous_relation_setoid o1 o2.
+ (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) →
+ ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
+ [2: clear b H a' a; intros;
+ lapply depth=0 (λV.saturation_expansive ??? (extS ?? a V)); [2: apply A_is_saturation;|skip]
+ (* fundamental adjunction here! to be taken out *)
+ cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a (A ? (extS ?? a V)));
+ [2: intro; intros 2; unfold minus_star_image; simplify; intros;
+ apply (Hletin V1 x); whd; split; [ exact I | exists; [apply a1] split; assumption]]
+ clear Hletin;
+ cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a' (A ? (extS ?? a V)));
+ [2: intro; apply (. #‡(H ?)); apply Hcut] clear H Hcut;
+ (* second half of the fundamental adjunction here! to be taken out too *)
+ intro; lapply (Hcut1 (singleton ? V)); clear Hcut1;
+ unfold minus_star_image in Hletin; unfold singleton in Hletin; simplify in Hletin;
+ whd in Hletin; whd in Hletin:(?→?→%); simplify in Hletin;
+ apply (if ?? (A_is_saturation ???));
+ intros 2 (x H); lapply (Hletin V ? x ?);
+ [ apply refl | cases H; assumption; ]
+ change with (x ∈ A ? (ext ?? a V));
+ apply (. #‡(†(extS_singleton ????)));
+ assumption;]
+ split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
+qed.
+
+definition continuous_relation_comp:
+ ∀o1,o2,o3.
+ continuous_relation_setoid o1 o2 →
+ continuous_relation_setoid o2 o3 →
+ continuous_relation_setoid o1 o3.
+ intros (o1 o2 o3 r s); constructor 1;
+ [ apply (s ∘ r)
+ | intros;
+ apply sym1;
+ apply (.= †(image_comp ??????));
+ apply (.= (reduced ?????)\sup -1);
+ [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
+ | apply (.= (image_comp ??????)\sup -1);
+ apply refl1]
+ | intros;
+ apply sym1;
+ apply (.= †(minus_star_image_comp ??????));
+ apply (.= (saturated ?????)\sup -1);
+ [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
+ | apply (.= (minus_star_image_comp ??????)\sup -1);
+ apply refl1]]
+qed.
+