Let $s$ be a local section of $L^{{\text{\rm an}}}$ defined on a point $p\in X^{{\text{\rm an}}}$. Let ${\mathcal{U}}\subset{\mathcal{X}}$ be a trivializing open neighbourhood of ${\operatorname{red}}_{{\mathcal{X}}}(p)$, the reduction of $p$ with respect to the model ${\mathcal{X}}$, and $\sigma$ a generator of ${\mathcal{L}}|_{{\mathcal{U}}}$. Let $\lambda$ be an analytic function on $({\mathcal{U}}\cap X)^{{\text{\rm an}}}$ such that $s^{\otimes e}=\lambda\sigma$.

We have that ${\operatorname{red}}_{{\mathcal{X}}^{\prime}}(p)=f^{-1}({\operatorname{red}}(p))$ and ${\mathcal{U}}^{\prime}:=f^{-1}({\mathcal{U}})$ is a trivializing open set of ${\mathcal{L}}^{\prime\otimes e}$ with generator $f^{*}\sigma^{\otimes e^{\prime}}$. Then $s^{\otimes ee^{\prime}}=\lambda^{e^{\prime}}f^{*}\sigma^{\otimes e^{\prime}}$ on $({\mathcal{U}}^{\prime}\cap X)^{{\text{\rm an}}}=({\mathcal{U}}\cap X)^{{\text{\rm an}}}$. Now the proposition follows directly from Definition 2.17. ∎