Lemma 34.10.5. Let $T$ be an affine scheme. Let $\{ T_ j \to T\} _{j = 1, \ldots , m}$ be a standard V covering. Let $\{ T_{ji} \to T_ j\} _{i = 1, \ldots n_ j}$ be a standard V covering. Then $\{ T_{ji} \to T\} _{i, j}$ is a standard V covering.

**Proof.**
This follows formally from the observation that if $V \subset W$ and $W \subset \Omega $ are extensions of valuation rings, then $V \subset \Omega $ is an extension of valuation rings.
$\square$

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