Proposition 1

If $a_1, b_1, c_1$ are homologisms, $\psi'(S_1, R_1)$ and $\varphi'(R_1, E_1)$ are the reciprocal bijections of the bijections $\psi(S_1, R_1)$ and $\varphi(R_1, E_1)$ respectively.

In fact, $f = (\beta \otimes \alpha) \circ (\beta_1 \otimes \alpha_1)$ is a morphism of the complex $S_1 \otimes_A R_1$ into itself, and one has $(h_1 \circ \alpha_1) \circ f = f$. By X, p. 49, prop. 3 and 3 bis, $f$ is a homotopism, hence $H(f) = 1$ and

analogously $\psi'(S_1, R_1) \circ \psi(S_1, R_1) = 1$. One reasons in an analogous manner for the mappings $\varphi$ and $\varphi'$.