Lemma 31.12.7. Let $X$ be an integral locally Noetherian scheme. Let $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}''$ an exact sequence of coherent $\mathcal{O}_ X$-modules. If $\mathcal{F}'$ is reflexive and $\mathcal{F}''$ is torsion free, then $\mathcal{F}$ is reflexive.

**Proof.**
Omitted. See More on Algebra, Lemma 15.23.5.
$\square$

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