During this proof we will use without further mention that for a closed immersion $i : Z \to X$ the functor $i_*$ gives an equivalence between the category of coherent modules on $Z$ and coherent modules on $X$ annihilated by the ideal sheaf of $Z$, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-i-star-equivalence}. In particular we think of $$ \textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_e}) \subset \textit{Coh}_{\text{support proper over } A}(\mathcal{O}_X) $$ as the full subcategory of consisting of modules annihilated by $\mathcal{K}^e$ and $$ \textit{Coh}_{\text{support proper over } A}(X_e, \mathcal{I}_e) \subset \textit{Coh}_{\text{support proper over } A}(X, \mathcal{I}) $$ as the full subcategory of objects annihilated by $\mathcal{K}^e$. Moreover (1) tells us these two categories are equivalent under the completion functor (\ref{equation-completion-functor-proper-over-A}). \medskip\noindent Applying this equivalence we get a coherent $\mathcal{O}_X$-module $\mathcal{G}_e$ annihilated by $\mathcal{K}^e$ corresponding to the system $(\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n)$ of $\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})$. The maps $\mathcal{F}_n/\mathcal{K}^{e + 1}\mathcal{F}_n \to \mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n$ correspond to canonical maps $\mathcal{G}_{e + 1} \to \mathcal{G}_e$ which induce isomorphisms $\mathcal{G}_{e + 1}/\mathcal{K}^e\mathcal{G}_{e + 1} \to \mathcal{G}_e$. We obtain an object $(\mathcal{G}_e)$ of the category $\textit{Coh}_{\text{support proper over } A}(X, \mathcal{K})$. The map $\alpha$ induces a system of maps $$ \mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n \longrightarrow \mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H} $$ whence maps $\mathcal{G}_e \to \mathcal{H}/\mathcal{K}^e\mathcal{H}$ (by the equivalence of categories again). Let $t \geq 1$ be an integer, which exists by assumption (2), such that $\mathcal{K}^t$ annihilates the kernel and cokernel of all the maps $\mathcal{F}_n \to \mathcal{H}/\mathcal{I}^n\mathcal{H}$. Then $\mathcal{K}^{2t}$ annihilates the kernel and cokernel of the maps $\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n \to \mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H}$ (details omitted; see Cohomology of Schemes, Remark \ref{coherent-remark-inverse-systems-kernel-cokernel-annihilated-by}). Whereupon we conclude that $\mathcal{K}^{4t}$ annihilates the kernel and the cokernel of the maps $$ \mathcal{G}_e \longrightarrow \mathcal{H}/\mathcal{K}^e\mathcal{H}, $$ (details omitted;  see Cohomology of Schemes, Remark \ref{coherent-remark-inverse-systems-kernel-cokernel-annihilated-by}). We apply Lemma \ref{lemma-existence-easy} to obtain a coherent $\mathcal{O}_X$-module $\mathcal{F}$, a map $a : \mathcal{F} \to \mathcal{H}$ and an isomorphism $\beta : (\mathcal{G}_e) \to (\mathcal{F}/\mathcal{K}^e\mathcal{F})$ in $\textit{Coh}(X, \mathcal{K})$. Working backwards, for a given $n$ the triple $(\mathcal{F}/\mathcal{I}^n\mathcal{F}, a \bmod \mathcal{I}^n, \beta \bmod \mathcal{I}^n)$ is a triple as in the lemma for the morphism $\alpha_n \bmod \mathcal{K}^e : (\mathcal{F}_n/\mathcal{K}^e\mathcal{F}_n) \to (\mathcal{H}/(\mathcal{I}^n + \mathcal{K}^e)\mathcal{H})$ of $\textit{Coh}(X, \mathcal{K})$. Thus the uniqueness in Lemma \ref{lemma-existence-easy} gives a canonical isomorphism $\mathcal{F}/\mathcal{I}^n\mathcal{F} \to \mathcal{F}_n$ compatible with all the morphisms in sight. \medskip\noindent To finish the proof of the lemma we still have to show that the support of $\mathcal{F}$ is proper over $A$. By construction the kernel of $a : \mathcal{F} \to \mathcal{H}$ is annihilated by a power of $\mathcal{K}$. Hence the support of this kernel is contained in the support of $\mathcal{G}_1$. Since $\mathcal{G}_1$ is an object of $\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_1})$ we see this is proper over $A$. Combined with the fact that the support of $\mathcal{H}$ is proper over $A$ we conclude that the support of $\mathcal{F}$ is proper over $A$ by Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-union-closed-proper-over-base}.