Lemma KAOctaComm3' {x y z : Complex A} (f1 : Morphism x y) (f2 : Morphism y z) : # (ComplexHomotFunctor A) (KAOctaMor3 f1 f2) · # (ComplexHomotFunctor A) (MappingConePr1 A (KAOctaMor1 f1 f2)) = # (ComplexHomotFunctor A) (MappingConePr1 A f2) · # (AddEquiv1 (TranslationHEquiv A)) (# (ComplexHomotFunctor A) (MappingConeIn2 A f1)) · # (AddEquiv1 (TranslationHEquiv A)) (identity ((ComplexHomotFunctor A) (MappingCone A f1))). Proof. rewrite functor_id. rewrite id_right. use (pathscomp0 (! (functor_comp (ComplexHomotFunctor A) _ _))). set (tmp := KAOctaComm3 f1 f2). apply (maponpaths (# (ComplexHomotFunctor A))) in tmp. use (pathscomp0 tmp). clear tmp. use (pathscomp0 ((functor_comp (ComplexHomotFunctor A) _ _))). apply cancel_precomposition. unfold AddEquiv1, TranslationHEquiv. cbn. apply pathsinv0. use TranslationFunctorHImEq. apply idpath. Qed.
Require Import UniMath.Foundations.UnivalenceAxiom. Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.Algebra.Monoids. Require Import UniMath.NumberSystems.Integers. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Kernels. Require Import UniMath.CategoryTheory.Limits.Cokernels. Require Import UniMath.CategoryTheory.Limits.Pushouts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.BinDirectSums. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.PrecategoriesWithAbgrops. Require Import UniMath.CategoryTheory.PreAdditive. Require Import UniMath.CategoryTheory.Additive. Require Import UniMath.CategoryTheory.Abelian. Require Import UniMath.CategoryTheory.AbelianToAdditive. Require Import UniMath.CategoryTheory.AdditiveFunctors. Require Import UniMath.HomologicalAlgebra.Complexes. Require Import UniMath.HomologicalAlgebra.KA. Require Import UniMath.HomologicalAlgebra.TranslationFunctors.