Let us choose an affine open covering $X=\bigcup_{i=1}^{N}U_{i}$ together with a local basis $\omega_{i}$ of $L$ on each $U_{i}$. Let $h_{i}$ be a metric of $L^{\mathrm{an}}$ over $U_{i}^{\mathrm{an}}$ given by $|\omega_{i}|_{h_{i}}(x)=1$ for $x\in U_{i}^{\mathrm{an}}$. As $X^{\mathrm{an}}$ is paracompact (locally compact and $\sigma$-compact), we can find a partition of unity $\{\rho_{i}\}_{i=1,\ldots,N}$ of continuous functions on $X^{\mathrm{an}}$ such that $\mathrm{supp}(\rho_{i})\subseteq U_{i}^{\mathrm{an}}$ for all $i$. If we set $|\raisebox{1.72218pt}{.}|_{h}(x)=\sum_{i=1}^{N}\rho_{i}(x)|\raisebox{1.72218pt}{.}|_{h_{i}}(x)$, then $h=\{|\raisebox{1.72218pt}{.}|_{h}(x)\}_{x\in X^{\mathrm{an}}}$ yields a continuous metric of $L^{\mathrm{an}}$. ∎