**Proof.**
Let $N \to N'$ be an injection of $R$-modules. By the flatness of $S \to S'$ we have

\[ \mathop{\mathrm{Ker}}(N \otimes _ R M \to N' \otimes _ R M) \otimes _ S S' = \mathop{\mathrm{Ker}}(N \otimes _ R M' \to N' \otimes _ R M') \]

If $M$ is flat over $R$, then the left hand side is zero and we find that $M'$ is flat over $R$ by the second characterization of flatness in Lemma 10.39.5. If $M'$ is flat over $R$ then we have the vanishing of the right hand side and if in addition $S \to S'$ is faithfully flat, this implies that $\mathop{\mathrm{Ker}}(N \otimes _ R M \to N' \otimes _ R M)$ is zero which in turn shows that $M$ is flat over $R$.
$\square$

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