Lemma 15.7.4. Let $A, A', B, B', C, C', D, D', I, M', M, N, \varphi $ be as in Lemma 15.7.2. If $N$ finite over $D$ and $M'$ finite over $C'$, then $N' = N \times _{\varphi , M} M'$ is finite over $D'$.

**Proof.**
Recall that $D' \to D \times _ C C'$ is surjective by Lemma 15.6.5. Observe that $N' = N \times _{\varphi , M} M'$ is a module over $D \times _ C C'$. We can apply Lemma 15.6.7 to the data $C, C', D, D', IC', M', M, N, \varphi $ to see that $N' = N \times _{\varphi , M} M'$ is finite over $D \times _ C C'$. Thus it is finite over $D'$.
$\square$

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