Lemma 26.11.6. Let $X$ be a scheme. Let $X = \bigcup _ i U_ i$ be an affine open covering. Let $V \subset X$ be an affine open. There exists a standard open covering $V = \bigcup _{j = 1, \ldots , m} V_ j$ (see Definition 26.5.2) such that each $V_ j$ is a standard open in one of the $U_ i$.
Proof. Pick $v \in V$. Then $v \in U_ i$ for some $i$. By Lemma 26.11.5 above there exists an open $v \in W_ v \subset V \cap U_ i$ such that $W_ v$ is a standard open in both $V$ and $U_ i$. Since $V$ is quasi-compact the lemma follows. $\square$
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