Lemma 86.24.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. If $Y$ is locally Noetherian and $f$ locally of finite type, then $X$ is locally Noetherian.

**Proof.**
Pick $\{ Y_ j \to Y\} $ and $\{ X_{ij} \to Y_ j \times _ Y X\} $ as in Lemma 86.24.6. Then it follows from Lemma 86.19.9 that each $X_{ij}$ is Noetherian. This proves the lemma.
$\square$

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