Proof. It suffices to show that the product of two normal rings, say $R$ and $S$, is normal. By Lemma 10.21.3 the prime ideals of $R\times S$ are of the form $\mathfrak {p}\times S$ and $R\times \mathfrak {q}$, where $\mathfrak {p}$ and $\mathfrak {q}$ are primes of $R$ and $S$ respectively. Localization yields $(R\times S)_{\mathfrak {p}\times S}=R_{\mathfrak {p}}$ which is a normal domain by assumption. Similarly for $S$. $\square$

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