Lemma 73.9.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$. Let $P$ be a property of the elements of $\mathcal{B}$. Assume that

every $W \in \mathcal{B}$ is quasi-compact and quasi-separated,

if $W \in \mathcal{B}$ and $U \subset W$ is quasi-compact open, then $U \in \mathcal{B}$,

if $V \in \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$ is affine, then (a) $V \in \mathcal{B}$ and (b) $P$ holds for $V$,

for every elementary distinguished square $(U \subset W, f : V \to W)$ such that

$W \in \mathcal{B}$,

$U$ is quasi-compact,

$V$ is affine, and

$P$ holds for $U$, $V$, and $U \times _ W V$,

then $P$ holds for $W$.

Then $P$ holds for every $W \in \mathcal{B}$.

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