Proof: By Corollary 5.12, ${\rm supp}(\alpha)\subset(U^{\prime})^{\rm an}$ and (a) follows. To prove (b), it is enough to show

$$\int_{{\rm Trop}(U)}\alpha_{U}=\int_{{\rm Trop}(U^{\prime})}\alpha_{U^{\prime}}$$

(3)

for a non-empty very affine open subset $U^{\prime}$ of $U$ by using (a). The differential form $\alpha$ is given on ${U^{\rm an}}$ (resp. $(U^{\prime})^{\rm an}$) by $\alpha_{U}\in A^{n,n}_{c}({\rm Trop}(U))$ (resp. $\alpha_{U^{\prime}}\in A^{n,n}_{c}({\rm Trop}(U^{\prime}))$). By 4.12, there is an affine homomorphism $\psi:T_{U^{\prime}}\rightarrow T_{U}$ of the underlying canonical tori such that $\varphi_{U}={\rm Trop}(\psi)\circ\varphi_{U^{\prime}}$. It follows that $\alpha$ is given on $U^{\prime}$ also by ${\rm Trop}(\psi)^{*}(\alpha_{U})$. By Proposition 5.6, we have $\alpha_{U^{\prime}}={\rm Trop}(\psi)^{*}(\alpha_{U})$. The Sturmfels–Tevelev multiplicity formula shows that ${\rm Trop}(\psi)_{*}({\rm Trop}(U^{\prime}))={\rm Trop}(U)$ (see Proposition 4.11). Then Proposition 3.10 shows that (3) holds. $\square$