Proposition 3

Let $0 \to E' \to E \to E'' \to +0$ be an exact sequence of right $A$-modules. Suppose that $E'$ and $E''$ are flat and that one of them is faithfully flat. Then $E$ is faithfully flat.

We know already that $E$ is flat ($§ 2$, no. 5, Proposition 5). We verify that $E$ has property (b) of Proposition 1. Let $N$ be a left $A$-module. As $E''$ is flat, there is an exact sequence

($§ 2$, no. 5, Proposition 4). If $E \otimes_A N = 0$, it follows that $E' \otimes_A N$ and $E'' \otimes_A N$ are zero; as one of the modules $E'$, $E''$ is faithfully flat, this implies that $N = 0$.