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Proof synopsis: After constructing a functorial set up, reduce to the case $X = \mathbb{P}^n$ and $\mathcal{F} = \mathcal{O}(k)$, and use that completions of local rings are faithfully flat, in particular for $\widehat{\mathbb{C}[x_1,...,x_n]} = \mathbb{C}[[x_1,...,x_n]]$.