We prove this by induction on the integer $d = \dim_x(\text{Supp}(\mathcal{F}_s))$. By Lemma \ref{lemma-elementary-devissage-variant} there exists a diagram $$ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (S, s) & (S', s') \ar[l] } $$ of pointed schemes such that the horizontal arrows are elementary \'etale neighbourhoods and such that $g^*\mathcal{F}/X'/S'$ has a one step d\'evissage at $x'$. The local nature of the problem implies that we may replace $(X, x) \to (S, s)$ by $(X', x') \to (S', s')$. Thus after doing so we may assume that there exists a one step d\'evissage $(Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)$ of $\mathcal{F}/X/S$ at $x$. \medskip\noindent We apply Lemma \ref{lemma-existence-alpha} to find a map $$ \alpha_1 : \mathcal{O}_{Y_1}^{\oplus r_1} \longrightarrow \pi_{1, *}\mathcal{G}_1 $$ which induces an isomorphism of vector spaces over $\kappa(\xi_1)$ where $\xi_1 \in Y_1$ is the unique generic point of the fibre of $Y_1$ over $s$. Moreover $\dim_{y_1}(\text{Supp}(\Coker(\alpha_1)_s)) < d$. It may happen that the stalk of $\Coker(\alpha_1)_s$ at $y_1$ is zero. In this case we may shrink $Y_1$ by Lemma \ref{lemma-shrink} (\ref{item-shrink-on-Y}) and assume that $\Coker(\alpha_1) = 0$ so we obtain a complete d\'evissage of length zero. \medskip\noindent Assume now that the stalk of $\Coker(\alpha_1)_s$ at $y_1$ is not zero. In this case, by induction, there exists a commutative diagram \begin{equation} \label{equation-overcome-this} \vcenter{ \xymatrix{ (Y_1, y_1) \ar[d] & (Y'_1, y'_1) \ar[l]^h \ar[d] \\ (S, s) & (S', s') \ar[l] } } \end{equation} of pointed schemes such that the horizontal arrows are elementary \'etale neighbourhoods and such that $h^*\Coker(\alpha_1)/Y'_1/S'$ has a complete d\'evissage $$ (Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 2, \ldots, n} $$ at $y'_1$. (In particular $i_2 : Z_2 \to Y'_1$ is a closed immersion into $Y'_2$.) At this point we apply Lemma \ref{lemma-elementary-etale-neighbourhood} to $S, X, \mathcal{F}, x, s$, the system $(Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)$ and diagram (\ref{equation-overcome-this}). We obtain a diagram $$ \xymatrix{ & & (X'', x'') \ar[lld] \ar[d] & (Z''_1, z''_1) \ar[l] \ar[lld] \ar[d] \\ (X, x) \ar[d] & (Z_1, z_1) \ar[l] \ar[d] & (S'', s'') \ar[lld] & (Y''_1, y''_1) \ar[lld] \ar[l] \\ (S, s) & (Y_1, y_1) \ar[l] } $$ with all the properties as listed in the referenced lemma. In particular $Y''_1 \subset Y'_1 \times_{S'} S''$. Set $X_1 = Y'_1 \times_{S'} S''$ and let $\mathcal{F}_1$ denote the pullback of $\Coker(\alpha_1)$. By Lemma \ref{lemma-base-change-complete-at-x} the system \begin{equation} \label{equation-shrink-this} (Z_k \times_{S'} S'', Y_k \times_{S'} S'', i''_k, \pi''_k, \mathcal{G}''_k, \alpha''_k, z''_k, y''_k)_{k = 2, \ldots, n} \end{equation} is a complete d\'evissage of $\mathcal{F}_1$ to $X_1$. Again, the nature of the problem allows us to replace $(X, x) \to (S, s)$ by $(X'', x'') \to (S'', s'')$. In this we see that we may assume: \begin{enumerate} \item[(a)] There exists a one step d\'evissage $(Z_1, Y_1, i_1, \pi_1, \mathcal{G}_1)$ of $\mathcal{F}/X/S$ at $x$, \item[(b)] there exists an $\alpha_1 : \mathcal{O}_{Y_1}^{\oplus r_1} \to \pi_{1, *}\mathcal{G}_1$ such that $\alpha \otimes \kappa(\xi_1)$ is an isomorphism, \item[(c)] $Y_1 \subset X_1$ is open, $y_1 = x_1$, and $\mathcal{F}_1|_{Y_1} \cong \Coker(\alpha_1)$, and \item[(d)] there exists a complete d\'evissage $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k, z_k, y_k)_{k = 2, \ldots, n}$ of $\mathcal{F}_1/X_1/S$ at $x_1$. \end{enumerate} To finish the proof all we have to do is shrink the one step d\'evissage and the complete d\'evissage such that they fit together to a complete d\'evissage. (We suggest the reader do this on their own using Lemmas \ref{lemma-shrink} and \ref{lemma-shrink-complete} instead of reading the proof that follows.) Since $Y_1 \subset X_1$ is an open neighbourhood of $x_1$ we may apply Lemma \ref{lemma-shrink-complete} (\ref{item-shrink-on-X-complete}) to find a standard shrinking $S', X'_1, Z'_2, Y'_2, \ldots, Y'_n$ of the datum (d) so that $X'_1 \subset Y_1$. Note that $X'_1$ is also a standard open of the affine scheme $Y_1$. Next, we shrink the datum (a) as follows: first we shrink the base $S$ to $S'$, see Lemma \ref{lemma-shrink} (\ref{item-shrink-base}) and then we shrink the result to $S''$, $X''$, $Z''_1$, $Y''_1$ using Lemma \ref{lemma-shrink} (\ref{item-shrink-on-Y}) such that eventually $Y''_1 = X'_1 \times_S S''$ and $S'' \subset S'$. Then we see that $$ Z''_1, Y''_1, Z'_2 \times_{S'} S'', Y'_2 \times_{S'} S'', \ldots, Y'_n \times_{S'} S'' $$ gives the complete d\'evissage we were looking for.