Lemma 15.32.5. Let $A$ be a ring. Let $I \subset J \subset A$ be ideals. Assume that $J/I \subset A/I$ is a $H_1$-regular ideal. Then $I \cap J^2 = IJ$.

**Proof.**
Follows immediately from Lemma 15.30.9 by localizing.
$\square$

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