Let $\varphi:U\rightarrow{\mathbb{G}}_{m}^{r}$ be a moment map on an open subset $U$ of $X$ and let $U^{\prime}$ be a non-empty open subset of $U$. Then ${\varphi_{\rm trop}}((U^{\prime})^{\rm an})={\varphi_{\rm trop}}({U^{\rm an}})$.