Strengthen a lemma

Thanks to Manuel Hoff
https://stacks.math.columbia.edu/tag/0C6H#comment-3952

algebra.tex

@@ -29266,23 +29266,27 @@ \section{Quasi-finite maps}
 
 \begin{lemma}
 \label{lemma-quasi-finite-permanence}
-Let $A \to B$ and $B \to C$ be finite type ring homomorphisms.
-Let $\mathfrak r$ be a prime of $C$ lying over
+Let $A \to B$ and $B \to C$ be ring homomorphisms such that $A \to C$
+is of finite type. Let $\mathfrak r$ be a prime of $C$ lying over
 $\mathfrak q \subset B$ and $\mathfrak p \subset A$.
 If $A \to C$ is quasi-finite at $\mathfrak r$, then
 $B \to C$ is quasi-finite at $\mathfrak r$.
 \end{lemma}
 
 \begin{proof}
-Using property (3) of Lemma \ref{lemma-isolated-point-fibre}:
-By assumption there exists some $c \in C$ such that
+Observe that $B \to C$ is of finite type
+(Lemma \ref{lemma-compose-finite-type})
+so that the statement makes sense.
+Let us use characterization (3) of Lemma \ref{lemma-isolated-point-fibre}.
+If $A \to C$ is quasi-finite at $\mathfrak r$, then
+there exists some $c \in C$ such that
 $$
 \{\mathfrak r' \subset C \text{ lying over }\mathfrak p\} \cap D(c) =
 \{\mathfrak{r}\}.
 $$
 Since the primes $\mathfrak r' \subset C$ lying over $\mathfrak q$
 form a subset of the primes $\mathfrak r' \subset C$ lying over
-$\mathfrak p$ we conclude.
+$\mathfrak p$ we conclude $B \to C$ is quasi-finite at $\mathfrak r$.
 \end{proof}
 
 \noindent