proposition: Lemma MappingCone_Diff3_Diff3 {C1 C2 : Complex A} (f : Morphism C1 C2) (i : hz) : let DS1 := to_BinDirectSums A ((TranslationComplex A C1) i) (C2 i) in let DS2 := to_BinDirectSums A ((TranslationComplex A C1) (i + 1)) (C2 (i + 1)) in let DS3 := to_BinDirectSums A ((TranslationComplex A C1) (i + 1 + 1)) (C2 (i + 1 + 1)) in (MappingConeDiff3 f i) · (MappingConeDiff3 f (i + 1)) = ZeroArrow (Additive.to_Zero A) _ _.
proof: Proof. intros DS1 DS2 DS3. unfold MappingConeDiff3. fold DS1. fold DS2. fold DS3. rewrite assoc. rewrite assoc. rewrite assoc. rewrite <- (assoc _ _ (to_Pr2 DS2)). rewrite (to_IdIn2 DS2). rewrite id_right. rewrite <- (assoc _ _ (Diff C2 (i + 1))). rewrite (DSq A C2 i). rewrite ZeroArrow_comp_right. apply ZeroArrow_comp_left. 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