Let $S' \subset S$ be the integral closure of $R$ in $S$. Let $\mathfrak q' = S' \cap \mathfrak q$. By Zariski's Main Theorem \ref{theorem-main-theorem} there exists a $g \in S'$, $g \not \in \mathfrak q'$ such that $S'_g \cong S_g$. Consider the fibre rings $F = S \otimes_R \kappa(\mathfrak p)$ and $F' = S' \otimes_R \kappa(\mathfrak p)$. Denote $\overline{\mathfrak q}'$ the prime of $F'$ corresponding to $\mathfrak q'$. Since $F'$ is integral over $\kappa(\mathfrak p)$ we see that $\overline{\mathfrak q}'$ is a closed point of $\Spec(F')$, see Lemma \ref{lemma-integral-over-field}. Note that $\mathfrak q$ defines an isolated closed point $\overline{\mathfrak q}$ of $\Spec(F)$ (see Definition \ref{definition-quasi-finite}). Since $S'_g \cong S_g$ we have $F'_g \cong F_g$, so $\overline{\mathfrak q}$ and $\overline{\mathfrak q}'$ have isomorphic open neighbourhoods in $\Spec(F)$ and $\Spec(F')$. We conclude the set $\{\overline{\mathfrak q}'\} \subset \Spec(F')$ is open. Combined with $\mathfrak q'$ being closed (shown above) we conclude that $\overline{\mathfrak q}'$ defines an isolated closed point of $\Spec(F')$ as well. \medskip\noindent An additional small remark is that under the map $\Spec(F) \to \Spec(F')$ the point $\overline{\mathfrak q}$ is the only point mapping to $\overline{\mathfrak q}'$. This follows from the discussion above. \medskip\noindent By Lemma \ref{lemma-disjoint-implies-product} we may write $F' = F'_1 \times F'_2$ with $\Spec(F'_1) = \{\overline{\mathfrak q}'\}$. Since $F' = S' \otimes_R \kappa(\mathfrak p)$, there exists an $s' \in S'$ which maps to the element $(r, 0) \in F'_1 \times F'_2 = F'$ for some $r \in R$, $r \not \in \mathfrak p$. In fact, what we will use about $s'$ is that it is an element of $S'$, not contained in $\mathfrak q'$, and contained in any other prime lying over $\mathfrak p$. \medskip\noindent Let $f(x) \in R[x]$ be a monic polynomial such that $f(s') = 0$. Denote $\overline{f} \in \kappa(\mathfrak p)[x]$ the image. We can factor it as $\overline{f} = x^e \overline{h}$ where $\overline{h}(0) \not = 0$. By Lemma \ref{lemma-factor-mod-lift-etale} we can find an \'etale ring extension $R \to R'$, a prime $\mathfrak p'$ lying over $\mathfrak p$, and a factorization $f = h i$ in $R'[x]$ such that $\kappa(\mathfrak p) = \kappa(\mathfrak p')$, $\overline{h} = h \bmod \mathfrak p'$, $x^e = i \bmod \mathfrak p'$, and we can write $a h + b i = 1$ in $R'[x]$ (for suitable $a, b$). \medskip\noindent Consider the elements $h(s'), i(s') \in R' \otimes_R S'$. By construction we have $h(s')i(s') = f(s') = 0$. On the other hand they generate the unit ideal since $a(s')h(s') + b(s')i(s') = 1$. Thus we see that $R' \otimes_R S'$ is the product of the localizations at these elements: $$ R' \otimes_R S' = (R' \otimes_R S')_{i(s')} \times (R' \otimes_R S')_{h(s')} = S'_1 \times S'_2 $$ Moreover this product decomposition is compatible with the product decomposition we found for the fibre ring $F'$; this comes from our choices of $s', i, h$ which guarantee that $\overline{\mathfrak q}'$ is the only prime of $F'$ which does not contain the image of $i(s')$ in $F'$. Here we use that the fibre ring of $R'\otimes_R S'$ over $R'$ at $\mathfrak p'$ is the same as $F'$ due to the fact that $\kappa(\mathfrak p) = \kappa(\mathfrak p')$. It follows that $S'_1$  has exactly one prime, say $\mathfrak r'$, lying over $\mathfrak p'$ and that this prime lies over $\mathfrak q$. Hence the element $g \in S'$ maps to an element of $S'_1$ not contained in $\mathfrak r'$. \medskip\noindent The base change $R'\otimes_R S$ inherits a similar product decomposition $$ R' \otimes_R S = (R' \otimes_R S)_{i(s')} \times (R' \otimes_R S)_{h(s')} = S_1 \times S_2 $$ It follows from the above that $S_1$ has exactly one prime, say $\mathfrak r$, lying over $\mathfrak p'$ (consider the fibre ring as above), and that this prime lies over $\mathfrak q$. \medskip\noindent Now we may apply Lemma \ref{lemma-produce-finite} to the ring maps $R' \to S'_1 \to S_1$, the prime $\mathfrak p'$ and the element $g$ to see that after replacing $R'$ by a principal localization we can assume that $S_1$ is finite over $R'$ as desired.