By the definition of the semigroup ${\widetilde{M}}_{\Lambda}$, the condition ${\operatorname{val}}(p)\in\Lambda$ holds if and only in $\langle m,{\operatorname{val}}(p)\rangle+l\geq 0$ for all $(m,l)\in{\widetilde{M}}_{\Lambda}$. This is equivalent to $\log|\chi^{-m}(p)|+\log|\varpi|^{-l}\geq 0$ for all $(m,l)\in{\widetilde{M}}_{\Lambda}$. In turn, this is equivalent to $|\chi^{m}(p)\varpi^{l}|\leq 1$, for all $(m,l)\in{\widetilde{M}}_{\Lambda}$. Hence, ${\operatorname{val}}(p)\in\Lambda$ if and only if $|a(p)|\leq 1$ for all $a\in K^{\circ}[{\mathcal{X}}_{\Lambda}]$, which is exactly the condition ${\operatorname{red}}(p)\in{\mathcal{X}}_{\Lambda}$ (see (2.12)). ∎