Lemma 10.157.5. A regular ring is normal.

**Proof.**
Let $R$ be a regular ring. By Lemma 10.157.4 it suffices to prove that $R$ is $(R_1)$ and $(S_2)$. As a regular local ring is Cohen-Macaulay, see Lemma 10.106.3, it is clear that $R$ is $(S_2)$. Property $(R_1)$ is immediate.
$\square$

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