Let $f:X^{\prime}\to X$ be the projection. For $x^{\prime}\in{X^{\prime}}^{\mathrm{an}}$, we set $x=f^{\mathrm{an}}(x^{\prime})$. Then $\hat{\kappa}(x)\subseteq\hat{\kappa}(x^{\prime})$ and $(L\otimes_{k}\hat{\kappa}(x))\otimes_{\hat{\kappa}(x)}\hat{\kappa}(x^{\prime})=L^{\prime}\otimes_{k^{\prime}}\hat{\kappa}(x^{\prime})$, that is, $L(x)\otimes_{\hat{\kappa}(x)}\hat{\kappa}(x^{\prime})=L^{\prime}(x^{\prime})$. Moreover, $V^{\prime}\otimes_{k^{\prime}}\hat{\kappa}(x^{\prime})=(V\otimes_{k}\hat{\kappa}(x))\otimes_{\hat{\kappa}(x)}\hat{\kappa}(x^{\prime})$, and by Lemma 1.10, $\|\raisebox{1.72218pt}{.}\|^{\prime}_{\hat{\kappa}(x^{\prime})}=\|\raisebox{1.72218pt}{.}\|_{\hat{\kappa}(x^{\prime})}=\|\raisebox{1.72218pt}{.}\|_{\hat{\kappa}(x),\hat{\kappa}(x^{\prime})}$. Thus the assertion follows from Lemma 1.11. ∎