proposition: 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: 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.
imports: UniMath.Foundations.UnivalenceAxiom UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.NaturalNumbers UniMath.MoreFoundations.PartA UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.NumberSystems.Integers UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.TransportMorphisms UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinDirectSums UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.AbelianToAdditive UniMath.CategoryTheory.AdditiveFunctors UniMath.HomologicalAlgebra.Complexes UniMath.HomologicalAlgebra.KA UniMath.HomologicalAlgebra.TranslationFunctors