- [ apply continuous_relation_comp;
- | intros; simplify; intro x; simplify;
- lapply depth=0 (continuous_relation_eq' ???? H) as H';
- lapply depth=0 (continuous_relation_eq' ???? H1) as H1';
- letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K;
- cut (∀X:Ω \sup o1.
- minus_star_image o2 o3 b (A o2 (minus_star_image o1 o2 a (A o1 X)))
- = minus_star_image o2 o3 b' (A o2 (minus_star_image o1 o2 a' (A o1 X))));
- [2: intro; apply sym1; apply (.= #‡(†((H' ?)\sup -1))); apply sym1; apply (K X);]
- clear K H' H1';
- cut (∀X:Ω \sup o1.
- minus_star_image o1 o3 (b ∘ a) (A o1 X) = minus_star_image o1 o3 (b'∘a') (A o1 X));
- [2: intro;
- apply (.= (minus_star_image_comp ??????));
- apply (.= #‡(saturated ?????));
- [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
- apply sym1;
- apply (.= (minus_star_image_comp ??????));
- apply (.= #‡(saturated ?????));
- [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ]
- apply ((Hcut X) \sup -1)]
- clear Hcut; generalize in match x; clear x;
- apply (continuous_relation_eq_inv');
- apply Hcut1;]
- | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify;
- apply (.= †(ASSOC1‡#));
- apply refl1
- | intros; simplify; intro; unfold continuous_relation_comp; simplify;
- apply (.= †((id_neutral_right1 ????)‡#));
- apply refl1
- | intros; simplify; intro; simplify;
- apply (.= †((id_neutral_left1 ????)‡#));
- apply refl1]
+ [ apply Ocontinuous_relation_comp;
+ | intros; simplify;
+ change with ((b⎻* ∘ a⎻* ) ∘ oA o1 = ((b'⎻* ∘ a'⎻* ) ∘ oA o1));
+ change with (b⎻* ∘ (a⎻* ∘ oA o1) = b'⎻* ∘ (a'⎻* ∘ oA o1));
+ change in e with (a⎻* ∘ oA o1 = a'⎻* ∘ oA o1);
+ change in e1 with (b⎻* ∘ oA o2 = b'⎻* ∘ oA o2);
+ apply (.= e‡#);
+ intro x;
+ change with (eq1 ? (b⎻* (a'⎻* (oA o1 x))) (b'⎻*(a'⎻* (oA o1 x))));
+ apply (.= †(Osaturated o1 o2 a' (oA o1 x) ?)); [
+ apply ((o_saturation_idempotent ?? (oA_is_saturation o1) x)^-1);]
+ apply (.= (e1 (a'⎻* (oA o1 x))));
+ change with (eq1 ? (b'⎻* (oA o2 (a'⎻* (oA o1 x)))) (b'⎻*(a'⎻* (oA o1 x))));
+ apply (.= †(Osaturated o1 o2 a' (oA o1 x):?)^-1); [
+ apply ((o_saturation_idempotent ?? (oA_is_saturation o1) x)^-1);]
+ apply rule #;]
+ | intros; simplify;
+ change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ oA o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ oA o1));
+ apply rule (#‡ASSOC ^ -1);
+ | intros; simplify;
+ change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ oA o1 = a⎻* ∘ oA o1);
+ apply (#‡(id_neutral_right2 : ?));
+ | intros; simplify;
+ change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ oA o1 = a⎻* ∘ oA o1);
+ apply (#‡(id_neutral_left2 : ?));]