- | 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]
+ | intros; simplify;
+ change with ((b⎻* ∘ a⎻* ) ∘ A o1 = ((b'⎻* ∘ a'⎻* ) ∘ A o1));
+ change with (b⎻* ∘ (a⎻* ∘ A o1) = b'⎻* ∘ (a'⎻* ∘ A o1));
+ change in e with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
+ change in e1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
+ apply (.= e‡#);
+ intro x;
+ change with (b⎻* (a'⎻* (A o1 x)) = b'⎻*(a'⎻* (A o1 x)));
+ alias symbol "trans" = "trans1".
+ alias symbol "prop1" = "prop11".
+ alias symbol "invert" = "setoid1 symmetry".
+ lapply (.= †(saturated o1 o2 a' (A o1 x) : ?));
+ [3: apply (b⎻* ); | 5: apply Hletin; |1,2: skip;
+ |apply ((o_saturation_idempotent ?? (A_is_saturation o1) x)^-1); ]
+ change in e1 with (∀x.b⎻* (A o2 x) = b'⎻* (A o2 x));
+ apply (.= (e1 (a'⎻* (A o1 x))));
+ alias symbol "invert" = "setoid1 symmetry".
+ lapply (†((saturated ?? a' (A o1 x) : ?) ^ -1));
+ [2: apply (b'⎻* ); |4: apply Hletin; | skip;
+ |apply ((o_saturation_idempotent ?? (A_is_saturation o1) x)^-1);]]
+ | intros; simplify;
+ change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
+ apply rule (#‡ASSOC ^ -1);
+ | intros; simplify;
+ change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_right2 : ?));
+ | intros; simplify;
+ change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+ apply (#‡(id_neutral_left2 : ?));]