Proof: By assumption, the support of $\alpha$ is a compact subset of ${X^{\rm an}}$. We conclude that there are finitely many tropical charts $(V_{i},\varphi_{U_{i}})_{i=1,\dots,s}$ covering ${\rm supp}(\alpha)$ such that $\alpha$ is given on $V_{i}$ by the superform $\alpha_{i}\in A^{p,q}({\rm trop}_{U_{i}}(V_{i}))$. Recall that $\Omega_{i}:={\rm trop}_{U_{i}}(V_{i})$ is an open subset of ${\rm Trop}(U_{i})$. By 4.13, $U:=U_{1}\cap\dots\cap U_{s}$ is a non-empty very affine open subset of $X$. We define the open subset $V$ of ${U^{\rm an}}$ by $V:={U^{\rm an}}\cap\bigcup_{i=1}^{s}V_{i}$. Since $\max(p,q)=\dim(X)$, Corollary 5.12 yields ${\rm supp}(\alpha)\subset{U^{\rm an}}$. Using 4.12, we see that ${\rm trop}_{U_{i}}={\rm Trop}(\psi_{i})\circ{\rm trop}_{U}$ for an affine homomorphism $\psi_{i}:T_{U}\rightarrow T_{U_{i}}$ of tori. Then we have

$${\rm trop}_{U}(V_{i}\cap{U^{\rm an}})=({\rm Trop}(\psi_{i}))^{-1}(\Omega_{i})\cap{\rm Trop}(U)$$

and we denote this open subset of ${\rm Trop}(U)$ by $\Omega_{i}^{\prime}$. It follows that the preimage of $\Omega:=\bigcup_{i=1}^{s}\Omega_{i}^{\prime}$ with respect to $(\varphi_{U})_{\rm trop}$ is equal to $V$. We conclude that $(V,\varphi_{U})$ is a tropical chart of ${X^{\rm an}}$. Note that $\alpha$ is given on ${U^{\rm an}}\cap V_{i}$ by $\alpha_{i}^{\prime}:={\rm Trop}(\psi_{i})^{*}(\alpha_{i})\in A^{p,q}(\Omega_{i}^{\prime})$. By Proposition 5.6, $\alpha_{i}^{\prime}$ agrees with $\alpha_{j}^{\prime}$ on $\Omega_{i}^{\prime}\cap\Omega_{j}^{\prime}$ for every $i,j\in\{1,\dots,s\}$ and hence they define a superform $\alpha_{U}\in A^{p,q}(\Omega)$. By construction, $\alpha_{U}$ gives the differential form $\alpha$ on $V$. It follows from Remark 5.7 that $\alpha_{U}$ has compact support in $\Omega$. Since $\alpha$ has compact support in $V$, we conclude that $\alpha_{U}$ is a superform on ${\rm Trop}(U)$ which defines $\alpha$ on ${U^{\rm an}}$. $\square$