Lemma 15.127.3. In Situation 15.127.1 let $x_1, \ldots , x_ n \in \Omega $ be pairwise distinct. Let $v_ i \in V(x_ i)$. Then there exists an $s \in M$ such that $s(x_ i)$ maps to $v_ i$ for $i = 1, \ldots , n$.

**Proof.**
Since $x_ i$ is a maximal ideal of $R$ we may use Algebra, Lemma 10.15.4 to see that $M(x_1) \oplus \ldots \oplus M(x_ n)$ is a quotient of $M$.
$\square$

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