For equation (4.100), we suppose without loss of generality that $m_{\sigma}=0$, and hence $\Psi|_{\sigma}=0$. Let $u\in N(\sigma)_{\mathbb{R}}$. Then, the function $\psi|_{\pi^{-1}_{\sigma}(u)}$ is concave. Let $\Lambda\in\Pi$ such that $\operatorname{rec}(\Lambda)=\sigma$ and $\pi^{-1}_{\sigma}(u)\cap\Lambda\not=\emptyset$. Then, $\pi^{-1}_{\sigma}(u)\cap\Lambda$ is a polyhedron of maximal dimension in $\pi^{-1}_{\sigma}(u)$. The restriction of $\psi$ to this polyhedron is constant and, by (4.88), agrees with $\psi(\sigma)(u)$. Therefore, by concavity,

$$(\pi_{\sigma})_{\ast}\psi(u)=\max_{v\in\pi^{-1}_{\sigma}(u)}\psi(v),$$

agrees with $\psi(\sigma)(u)$. Thus we obtain equation (4.100). Equation (4.101) follows from the previous equation and Proposition 3.78(2). To prove equation (4.101) when $m_{\sigma}\not=0$ we use Proposition 3.40(4). ∎