By Cohomology of Schemes, Lemma \ref{coherent-lemma-existence-easy} to prove the lemma, we may replace $(\mathcal{F}_n)$ by an object differing from it by $\mathcal{I}$-torsion (see below for more precision). Let $T' = \{q \in Q \mid \dim(\overline{\{q\}}) = 0\}$ and $T = \{q \in Q \mid \dim(\overline{\{q\}}) \leq 1\}$. The assumption in the proposition is exactly that $Q \subset P$, $(\mathcal{F}_n)$, and $T' \subset T \subset Q$ satisfy the conditions of Lemma \ref{lemma-improvement-formal-coherent-module-better} with $d = 1$; besides trivial manipulations of inequalities, use that $V(\mathfrak p) \cap V(\mathcal{I}^\wedge_y) = \{\mathfrak m^\wedge_y\} \Leftrightarrow \dim(\mathcal{O}_{P, q}^\wedge/\mathfrak p) = 1$ as $\mathcal{I}_y^\wedge$ is generated by $1$ element. Combining these two remarks, we may replace $(\mathcal{F}_n)$ by the object $(\mathcal{H}_n)$ of $\textit{Coh}(P, \mathcal{I})$ found in Lemma \ref{lemma-improvement-formal-coherent-module-better}. Thus we may and do assume $(\mathcal{F}_n)$ is pro-isomorphic to an inverse system $(\mathcal{F}_n'')$ of coherent $\mathcal{O}_P$-modules such that $\text{depth}(\mathcal{F}''_{n, q}) + \dim(\overline{\{q\}}) \geq 2$ for all $q \in Q$. \medskip\noindent We will use More on Morphisms, Lemma \ref{more-morphisms-lemma-apply-proj-spec} and we will use the notation used and results found More on Morphisms, Section \ref{more-morphisms-section-proj-spec} without further mention; this proof will not make sense without at least understanding the statement of the lemma. Observe that in our case $A = \bigoplus_{m \geq 0} \Gamma(P, \mathcal{L}^{\otimes m})$ is a finite type $k$-algebra all of whose graded parts are finite dimensional $k$-vector spaces, see Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-proper-ample}. \medskip\noindent By Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-pullback} the pull back by $\pi|_U : U \to P$ is an object $(\pi|_U^*\mathcal{F}_n)$ of $\textit{Coh}(U, f\mathcal{O}_U)$ which is pro-isomorphic to the inverse system $(\pi|_U^*\mathcal{F}_n'')$ of coherent $\mathcal{O}_U$-modules. We claim $$ \text{depth}(\pi|_U^*\mathcal{F}''_{n, y}) + \delta_Z^Y(y) \geq 3 $$ for all $y \in U \cap Y$. Since all the points of $Z$ are closed, we see that $\delta_Z^Y(y) \geq \dim(\overline{\{y\}})$ for all $y \in U \cap Y$, see Lemma \ref{lemma-discussion}. Let $q \in Q$ be the image of $y$. Since the morphism $\pi : U \to P$ is smooth of relative dimension $1$ we see that either $y$ is a closed point of a fibre of $\pi$ or a generic point. Thus we see that $$ \text{depth}(\pi^*\mathcal{F}''_{n, y}) + \delta_Z^Y(y) \geq \text{depth}(\pi^*\mathcal{F}''_{n, y}) + \dim(\overline{\{y\}}) = \text{depth}(\mathcal{F}''_{n, q}) + \dim(\overline{\{q\}}) + 1 $$ because either the depth goes up by $1$ or the dimension. This proves the claim. \medskip\noindent By Lemma \ref{lemma-cd-1-canonical} we conclude that $(\pi|_U^*\mathcal{F}_n)$ canonically extends to $X$. Observe that $$ M_n = \Gamma(U, \pi|_U^*\mathcal{F}_n) = \bigoplus\nolimits_{m \in \mathbf{Z}} \Gamma(P, \mathcal{F}_n \otimes_{\mathcal{O}_P} \mathcal{L}^{\otimes m}) $$ is canonically a graded $A$-module, see More on Morphisms, Equation (\ref{more-morphisms-equation-cohomology-torsor}). By Properties, Lemma \ref{properties-lemma-quasi-coherent-quasi-affine} we have $\pi|_U^*\mathcal{F}_n = \widetilde{M_n}|_U$. Thus we may apply Lemma \ref{lemma-Gm-equivariant-extend-canonically} to find a finite graded $A$-module $N$ such that $(M_n)$ and $(N/I^nN)$ are pro-isomorphic in the category of graded $A$-modules modulo $A_+$-torsion modules. Let $\mathcal{F}$ be the coherent $\mathcal{O}_P$-module associated to $N$, see Cohomology of Schemes, Proposition \ref{coherent-proposition-coherent-modules-on-proj-general}. The same proposition tells us that $(\mathcal{F}/\mathcal{I}^n\mathcal{F})$ is pro-isomorphic to $(\mathcal{F}_n)$. Since both are objects of $\textit{Coh}(P, \mathcal{I})$ we win by Lemma \ref{lemma-recognize-formal-coherent-modules}.