Proof: Using Proposition 4.16 and shrinking the open neighbourhood $W$ of $x$, we may assume that $W$ is a tropical chart $(W,\varphi_{U})$ on which $\alpha$ is given by the superform $\alpha_{U}\in A^{p,q}({\rm trop}_{U}(W))$. By Proposition 4.14, there is a very affine open subset $U_{x}$ of $U$ and a compact neighbourhood $V_{x}$ of $x$ in $(U_{x})^{\rm an}$ such that ${\rm trop}_{U_{x}}(V_{x})$ is of dimension $d(x)$. By Proposition 4.16, there is a tropical chart $(V^{\prime},\varphi_{U^{\prime}})$ with $x\in V^{\prime}\subset V_{x}$ and $U^{\prime}\subset U_{x}$. By 4.12, there is an affine homomorphism $\psi:T_{U^{\prime}}\rightarrow T_{U}$ such that $\varphi_{U}=\varphi_{U^{\prime}}\circ\psi$. Using the same factorization for the tropicalizations, we see that the restriction of $\alpha$ to $V^{\prime}$ is given by ${\rm Trop}(\psi)^{*}(\alpha_{U})\in A^{p,q}({\rm trop}_{U^{\prime}}(V^{\prime}))$. The inclusion $U_{x}\subset U$ yields that ${\rm trop}_{U_{x}}$ factorizes through ${\rm trop}_{U}$ (use 4.12). Since $V^{\prime}\subset V_{x}$, we get $\dim({\rm trop}_{U}(V^{\prime}))\leq\dim({\rm trop}_{U_{x}}(V^{\prime}))\leq d(x)<\max(p,q)$. As ${\rm trop}_{U}(V^{\prime})={\rm Trop}(\psi)({\rm trop}_{U^{\prime}}(V^{\prime}))$, we conclude that ${\rm Trop}(\psi)^{*}(\alpha_{U})=0$. This proves $\alpha=0$. $\square$