Lemma 66.18.4. In diagram (66.18.0.1) the set (66.18.0.2) is finite if $f$ is of finite type and $f$ is quasi-finite at $x$.

**Proof.**
The morphism $q$ is quasi-finite at every $w \in F_{x, z}$, see Morphisms of Spaces, Lemma 65.27.2. Hence the lemma follows from Morphisms of Spaces, Lemma 65.27.9.
$\square$

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