For $x\in\operatorname{ri}(\Delta)$ and $F$ a facet of $\Gamma$, we deduce from equation (7.32) that $P(\beta_{x}=F)=\ell_{F}(x)/(n!\operatorname{vol}_{n}(\Gamma))$. Hence,

	$\displaystyle{\mathcal{E}}(x)$	$\displaystyle=-\sum_{F}\frac{\ell_{F}(x)}{n!{\operatorname{vol}_{n}}(\Gamma)}\log\Big(\frac{\ell_{F}(x)}{n!{\operatorname{vol}_{n}}(\Gamma)}\Big)$
		$\displaystyle=\frac{1}{n!{\operatorname{vol}_{n}}(\Gamma)}\bigg(-\sum_{F}{\ell_{F}(x)}\log({\ell_{F}(x)})-\log({n!{\operatorname{vol}_{n}}(\Gamma)})\Big(\sum_{F}\lambda_{F}\Big)\bigg)$
		$\displaystyle=\frac{1}{n!{\operatorname{vol}_{n}}(\Gamma)}\bigg(\frac{\vartheta(x)}{c}-\log({n!{\operatorname{vol}_{n}}(\Gamma)})\Big(\sum_{F}\lambda_{F}\Big)\bigg).$

The result then follows from Theorem 6.37. ∎