proposition: Lemma X_aux_type_center_of_contr_proof (b : B) (anot : A) (hnot : z_iso (H anot) b) : ∏ (t t' : total2 (λ a : A, z_iso (H a) b)) (f : pr1 t --> pr1 t'), #H f· pr2 t' = pr2 t -> #F f· #F (fH^-1 (pr2 t'· inv_from_z_iso hnot)) = #F (fH^-1 (pr2 t· inv_from_z_iso hnot)).
proof: Proof. intros t t' f. destruct t as [a h]. destruct t' as [a' h']. simpl in *. intro star. rewrite <- functor_comp. apply maponpaths. apply (invmaponpathsweq (weq_from_fully_faithful fH a anot)). simpl. rewrite functor_comp. inv_functor fH a' anot. rewrite assoc. inv_functor fH a anot. rewrite <- star. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Univalence UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering