Lemma 29.16.5. Let $f : T \to S$ be a morphism of schemes. If $f$ is locally of finite type, then $f(T_{\text{ft-pts}}) \subset S_{\text{ft-pts}}$.

**Proof.**
If $T$ is the spectrum of a field this is Lemma 29.16.1. In general it follows since the composition of morphisms locally of finite type is locally of finite type (Lemma 29.15.3).
$\square$

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