and the penultimate map is constructed in the proof.Proof. Let Zp=ker∂:Cp→Cp−1,Bp=im∂:Cp+1→Cp and obtain the short exact sequence0→Zp→Cp→∂Bp−1→0 which may be viewed as a short exact sequence of free chain complexes by giving Z∗ and B∗ the zero dif...