Lemma 13.9.10. Let $\mathcal{A}$ be an additive category. Let $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ be termwise split exact sequences as in Definition 13.9.9. Let $(\pi ')^ n$, $(s')^ n$ be a second collection of splittings. Denote $\delta ' : C^\bullet \longrightarrow A^\bullet [1]$ the morphism associated to this second set of splittings. Then

is an isomorphism of triangles in $K(\mathcal{A})$.

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