To prove (b), let us consider the moment map

$$\Phi:U\cap U^{\prime}\rightarrow T_{U}\times T_{U^{\prime}},\quad x\mapsto(\varphi_{U}(x),\varphi_{U^{\prime}}(x)).$$

Since $X$ is separated, it is easy to see that $\Phi$ is a closed embedding and hence $U\cap U^{\prime}$ is very affine. We conclude that $U\cap U^{\prime}$ is a very affine open subset of $X$. By definition of a tropical chart, $\Omega:={\rm trop}_{U}(V)$ (resp. $\Omega^{\prime}:={\rm trop}_{U^{\prime}}(V^{\prime})$) is an open subset of ${\rm Trop}(U)$ (resp. ${\rm Trop}(U^{\prime})$). Note that

$$\Omega^{\prime\prime}:=\Phi_{\rm trop}((U\cap U^{\prime})^{\rm an})\cap(\Omega\times\Omega^{\prime})\subset(N_{U})_{\mathbb{R}}\times(N_{U^{\prime}})_{\mathbb{R}}$$

is an open subset of $\Phi_{\rm trop}((U\cap U^{\prime})^{\rm an})$. An easy diagram chase yields $\Phi_{\rm trop}^{-1}(\Omega^{\prime\prime})=V\cap V^{\prime}$. Since $\varphi_{U\cap U^{\prime}}$ refines the moment map $\Phi$, we deduce that $(V\cap V^{\prime},\varphi_{U\cap U^{\prime}})$ is a tropical chart. This proves (b).

Finally, we prove (c). Let $\psi:=\psi_{U,U^{\prime\prime}}:T_{U^{\prime\prime}}\rightarrow T_{U}$ be the canonical affine homomorphism from 4.12. Then we have ${\rm trop}_{U}={\rm Trop}(\psi)\circ{\rm trop}_{U^{\prime\prime}}$ on $(U^{\prime\prime})^{\rm an}$. Since $(V,\varphi_{U})$ is a tropical chart, $\Omega:={\rm trop}_{U}(V)$ is an open subset of ${\rm Trop}(U)$ and $V={\rm trop}_{U}^{-1}(\Omega)$. Using $V\subset(U^{\prime\prime})^{\rm an}$, we get $V={\rm trop}_{U^{\prime\prime}}^{-1}(\Omega^{\prime\prime})$ for the open subset $\Omega^{\prime\prime}:={\rm Trop}(\psi)^{-1}(\Omega)$ of ${\rm Trop}(U^{\prime\prime})$. We conclude that $(V,\varphi_{U^{\prime\prime}})$ is a tropical chart proving (c). $\square$