To prove equation (4.104) we may assume that $m_{\Lambda}=0$ and $l_{\Lambda}=0$. Let $u\in{\widetilde{N}}(\Lambda)_{\mathbb{R}}$. Then, the function $\operatorname{c}(\psi)|_{{\widetilde{\pi}}^{-1}_{\Lambda}(u)}$ is concave. Let $\Lambda^{\prime}\in\Pi$ such that $\Lambda$ is a face of $\Lambda^{\prime}$ and ${\widetilde{\pi}}^{-1}_{\Lambda}(u)\cap\operatorname{c}(\Lambda^{\prime})\not=\emptyset$. Then, ${\widetilde{\pi}}^{-1}_{\Lambda}(u)\cap\operatorname{c}(\Lambda^{\prime})$ is a polyhedron of maximal dimension of ${\widetilde{\pi}}^{-1}_{\Lambda}(u)$ and the restriction of $\operatorname{c}(\psi)$ to this polyhedron is constant and, by equation (4.90), agrees with $\psi(\Lambda)(u)$. Therefore, by concavity,

agrees with $\psi(\Lambda)(u)$. This proves equation (4.104).

Back in the general case when $m_{\Lambda}$ and $l_{\Lambda}$ may be different from zero, by Proposition 3.78, Proposition 3.40(4) and Lemma 4.102 we have

The remaining statements are clear. ∎