Let (A, ∂) and (A', ∂') be chain complexes, and let f and g be collections of homomorphisms f_k : A_k → A_k' and g_k : A_k → A_k' such that both f and g commute with ∂. An algebraic homotopy between f and g is a collection D of homomorphisms D_k : A_k →A~+i such that for all c ∈ A_k, we have f_k(c) - g_k(c) = ∂'_k+1(D_k(c)) + D_k-1(∂_k(c)). 

(One abbreviates this condition by f - g =∂D + D∂) Show that if there exists an algebraic homotopy between f and g, that is, if f and g are homotopic, then f_*k and g_*k are the same homomorphism of H_k(A) into H_k(A').