Suppose that $\phi$ is the pointwise limit on $X^{\mathrm{an}}$ of $\{\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})\}$, where $\{\lVert\mathord{\cdot}\rVert_{n}\}$ are norms on $V_{n}(L)$. Then $L|_{Y}$ is the pointwise limit of $\{\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n,X|Y})\}$ on $Y^{\mathrm{an}}$. ∎