Consider the collection $$ \mathcal{T} = \left\{ \begin{matrix} T \subset |X| \text{ nonempty closed such that there exists a coherent sheaf } \\ \mathcal{F} \text{ with } \text{Supp}(\mathcal{F}) = T \text{ for which the lemma is wrong} \end{matrix} \right\} $$ We are trying to show that $\mathcal{T}$ is empty. If not, then because $|X|$ is Noetherian (Properties of Spaces, Lemma \ref{spaces-properties-lemma-Noetherian-topology}) we can choose a minimal element $T \in \mathcal{T}$. This means that there exists a coherent sheaf $\mathcal{F}$ on $X$ whose support is $T$ and for which the lemma does not hold. \medskip\noindent If $T$ is not irreducible, then we can write $T = Z_1 \cup Z_2$ with $Z_1, Z_2$ closed and strictly smaller than $T$. Then we can apply Lemma \ref{lemma-prepare-filter-support} to get a short exact sequence of coherent sheaves $$ 0 \to \mathcal{G}_1 \to \mathcal{F} \to \mathcal{G}_2 \to 0 $$ with $\text{Supp}(\mathcal{G}_i) \subset Z_i$. By minimality of $T$ each of $\mathcal{G}_i$ has $\mathcal{P}$. Hence $\mathcal{F}$ has property $\mathcal{P}$ by (1), a contradiction. \medskip\noindent Suppose $T$ is irreducible. Let $\mathcal{J}$ be the sheaf of ideals defining the reduced induced closed subspace structure on $T$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-reduced-closed-subspace}. By Lemma \ref{lemma-power-ideal-kills-sheaf} we see there exists an $n \geq 0$ such that $\mathcal{J}^n\mathcal{F} = 0$. Hence we obtain a filtration $$ 0 = \mathcal{J}^n\mathcal{F} \subset \mathcal{J}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{J}\mathcal{F} \subset \mathcal{F} $$ each of whose successive subquotients is annihilated by $\mathcal{J}$. Hence if each of these subquotients has a filtration as in the statement of the lemma then also $\mathcal{F}$ does by (1). In other words we may assume that $\mathcal{J}$ does annihilate $\mathcal{F}$. \medskip\noindent Assume $T$ is irreducible and $\mathcal{J}\mathcal{F} = 0$ where $\mathcal{J}$ is as above. Denote $i : Z \to X$ the closed subspace corresponding to $\mathcal{J}$. Then $\mathcal{F} = i_*\mathcal{H}$ for some coherent $\mathcal{O}_Z$-module $\mathcal{H}$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-i-star-equivalence} and Lemma \ref{lemma-coherent-support-closed}. Let $\mathcal{G}$ be the coherent sheaf on $Z$ satisfying (3)(a) and (3)(b). We apply Lemma \ref{lemma-prepare-filter-irreducible} to get injective maps $$ \mathcal{I}_1^{\oplus r_1} \to \mathcal{H} \quad\text{and}\quad \mathcal{I}_2^{\oplus r_2} \to \mathcal{G} $$ where the support of the cokernels are proper closed in $Z$. Hence we find an nonempty open $V \subset Z$ such that $$ \mathcal{H}^{\oplus r_2}_V \cong \mathcal{G}^{\oplus r_1}_V $$ Let $\mathcal{I} \subset \mathcal{O}_Z$ be a quasi-coherent ideal sheaf cutting out $Z \setminus V$ we obtain (Lemma \ref{lemma-homs-over-open}) a map $$ \mathcal{I}^n\mathcal{G}^{\oplus r_1} \longrightarrow \mathcal{H}^{\oplus r_2} $$ which is an isomorphism over $V$. The kernel is supported on $Z \setminus V$ hence annihilated by some power of $\mathcal{I}$, see Lemma \ref{lemma-power-ideal-kills-sheaf}. Thus after increasing $n$ we may assume the displayed map is injective, see Lemma \ref{lemma-Artin-Rees}. Applying (3)(b) we find $\mathcal{G}' \subset \mathcal{I}^n\mathcal{G}$ such that $$ (i_*\mathcal{G}')^{\oplus r_1} \longrightarrow i_*\mathcal{H}^{\oplus r_2} = \mathcal{F}^{\oplus r_2} $$ is injective with cokernel supported in a proper closed subset of $Z$ and such that property $\mathcal{P}$ holds for $i_*\mathcal{G}'$. By (1) property $\mathcal{P}$ holds for $(i_*\mathcal{G}')^{\oplus r_1}$. By (1) and minimality of $T = |Z|$ property $\mathcal{P}$ holds for $\mathcal{F}^{\oplus r_2}$. And finally by (2) property $\mathcal{P}$ holds for $\mathcal{F}$ which is the desired contradiction.