We assume that $|\raisebox{1.72218pt}{.}|^{\prime}$ is discrete and

$$|a^{\prime}|^{\prime}=\exp(-\alpha\operatorname{ord}_{\mathfrak{o}_{k^{\prime}}}(a^{\prime}))\qquad(a^{\prime}\in k^{\prime})$$

for $\alpha\in\mathbb{R}_{>0}$. If

$$\alpha\not\in\bigcup_{v,v^{\prime}\in V\setminus\{0\}}\mathbb{Q}(\log\|v\|-\log\|v^{\prime}\|),$$

then the assumption (ii) holds. Indeed, we suppose that $|a^{\prime}|^{\prime}=\|v\|/\|v^{\prime}\|$ for some $a^{\prime}\in{k^{\prime}}^{\times}$ and $v,v^{\prime}\in V\setminus\{0\}$. Then

$$-\alpha\operatorname{ord}_{\mathfrak{o}_{k^{\prime}}}(a^{\prime})=\log\|v\|-\log\|v^{\prime}\|,$$

so that $\operatorname{ord}_{\mathfrak{o}_{k^{\prime}}}(a^{\prime})=0$, and hence $\|v\|=\|v^{\prime}\|$, as required.