-(* Part of proposition 9.9 *)
-lemma lemmax: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R p ≤ R q.
- intros;
- apply oa_density; intros;
- apply (. (or_prop3 : ?) ^ -1);
- apply
-
-(* Lemma 10.2, to be moved to OA *)
-lemma lemma_10_2_a: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R⎻* (R⎻ p).
- intros;
- apply (. (or_prop2 : ?));
- apply oa_leq_refl.
-qed.
-
-lemma lemma_10_2_b: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* p) ≤ p.
- intros;
- apply (. (or_prop2 : ?) ^ -1);
- apply oa_leq_refl.
+(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
+definition o_basic_topology_of_o_basic_pair: OBP → OBTop.
+ intro t;
+ constructor 1;
+ [ apply (Oform t);
+ | apply (□⎽t ∘ Ext⎽t);
+ | apply (◊⎽t ∘ Rest⎽t);
+ | apply hide; intros 2; split; intro;
+ [ change with ((⊩) \sup ⎻* ((⊩) \sup ⎻ U) ≤ (⊩) \sup ⎻* ((⊩) \sup ⎻ V));
+ apply (. (#‡(lemma_10_4_a ?? (⊩) V)^-1));
+ apply f_minus_star_image_monotone;
+ apply f_minus_image_monotone;
+ assumption
+ | apply oa_leq_trans;
+ [3: apply f;
+ | skip
+ | change with (U ≤ (⊩)⎻* ((⊩)⎻ U));
+ apply (. (or_prop2 : ?) ^ -1);
+ apply oa_leq_refl; ]]
+ | apply hide; intros 2; split; intro;
+ [ change with (◊⎽t ((⊩) \sup * U) ≤ ◊⎽t ((⊩) \sup * V));
+ apply (. ((lemma_10_4_b ?? (⊩) U)^-1)‡#);
+ apply (f_image_monotone ?? (⊩) ? ((⊩)* V));
+ apply f_star_image_monotone;
+ assumption;
+ | apply oa_leq_trans;
+ [2: apply f;
+ | skip
+ | change with ((⊩) ((⊩)* V) ≤ V);
+ apply (. (or_prop1 : ?));
+ apply oa_leq_refl; ]]
+ | apply hide; intros;
+ apply (.= (oa_overlap_sym' : ?));
+ change with ((◊⎽t ((⊩)* V) >< (⊩)⎻* ((⊩)⎻ U)) = (U >< (◊⎽t ((⊩)* V))));
+ apply (.= (or_prop3 ?? (⊩) ((⊩)* V) ?));
+ apply (.= #‡(lemma_10_3_a : ?));
+ apply (.= (or_prop3 : ?)^-1);
+ apply (oa_overlap_sym' ? ((⊩) ((⊩)* V)) U); ]