For the impatient reader: This proof is a repeat of the proof of Proposition \ref{proposition-generic-flatness} using Algebra, Lemma \ref{algebra-lemma-generic-flatness-reduced} instead of Algebra, Lemma \ref{algebra-lemma-generic-flatness}. \medskip\noindent Since being flat and being of finite presentation is local on the base, see Lemmas \ref{lemma-flat-module-characterize} and \ref{lemma-locally-finite-presentation-characterize}, we may work affine locally on $S$. Thus we may assume that $S = \Spec(A)$, where $A$ is a reduced ring (see Properties, Lemma \ref{properties-lemma-characterize-reduced}). As $f$ is of finite type, it is quasi-compact, so $X$ is quasi-compact. Hence we can find a finite affine open cover $X = \bigcup_{i = 1, \ldots, n} X_i$. Write $X_i = \Spec(B_i)$. Then $B_i$ is a finite type $A$-algebra, see Lemma \ref{lemma-locally-finite-type-characterize}. Moreover there are finite type $B_i$-modules $M_i$ such that $\mathcal{F}|_{X_i}$ is the quasi-coherent sheaf associated to the $B_i$-module $M_i$, see Properties, Lemma \ref{properties-lemma-finite-type-module}. Next, for each pair of indices $i, j$ choose an ideal $I_{ij} \subset B_i$ such that $X_i \setminus X_i \cap X_j = V(I_{ij})$ inside $X_i = \Spec(B_i)$. Set $M_{ij} = B_i/I_{ij}$ and think of it as a $B_i$-module. Then $V(I_{ij}) = \text{Supp}(M_{ij})$ and $M_{ij}$ is a finite $B_i$-module. \medskip\noindent At this point we apply Algebra, Lemma \ref{algebra-lemma-generic-flatness-reduced} the pairs $(A \to B_i, M_{ij})$ and to the pairs $(A \to B_i, M_i)$. Thus we obtain dense opens $U(A \to B_i, M_{ij}) \subset S$ and dense opens $U(A \to B_i, M_i) \subset S$ with notation as in Algebra, Equation (\ref{algebra-equation-good-locus}). Since a finite intersection of dense opens is dense open, we see that $$ U = \bigcap\nolimits_{i, j} U(A \to B_i, M_{ij}) \quad\cap\quad \bigcap\nolimits_i U(A \to B_i, M_i) $$ is open and dense in $S$. We claim that $U$ is the desired open. \medskip\noindent Pick $u \in U$. By definition of the loci $U(A \to B_i, M_{ij})$ and $U(A \to B, M_i)$ there exist $f_{ij}, f_i \in A$ such that (a) $u \in D(f_i)$ and $u \in D(f_{ij})$, (b) $A_{f_{ij}} \to B_{i, f_{ij}}$ is flat and of finite presentation and $M_{ij, f_{ij}}$ is flat over $A_{f_{ij}}$ and of finite presentation over $B_{i, f_{ij}}$, and (c) $B_{i, f_i}$ is flat and of finite presentation over $A_f$ and $M_{i, f_i}$ is flat and of finite presentation over $B_{i, f_i}$. Set $f = (\prod f_i) (\prod f_{ij})$. Now it suffices to prove that $X \to S$ is flat and of finite presentation over $D(f)$ and that $\mathcal{F}$ restricted to $X_{D(f)}$ is flat over $D(f)$ and of finite presentation over the structure sheaf of $X_{D(f)}$. \medskip\noindent Hence we may replace $A$ by $A_f$, i.e., perform the base change by $\Spec(A_f) \to S$. After this base change we see that each of $A \to B_i$ is flat and of finite presentation and that $M_i$, $M_{ij}$ are flat over $A$ and of finite presentation over $B_i$. This already proves that $X \to S$ is quasi-compact, locally of finite presentation, flat, and that $\mathcal{F}$ is flat over $S$ and of finite presentation over $\mathcal{O}_X$, see Lemma \ref{lemma-locally-finite-presentation-characterize} and Properties, Lemma \ref{properties-lemma-finite-presentation-module}. Since $M_{ij}$ is of finite presentation over $B_i$ we see that $X_i \cap X_j = X_i \setminus \text{Supp}(M_{ij})$ is a quasi-compact open of $X_i$, see Algebra, Lemma \ref{algebra-lemma-support-finite-presentation-constructible}. Hence we see that $X \to S$ is quasi-separated by Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated}. This proves the proposition.