Choose a toric section $s$ of $L$ whose divisor meets $V(\sigma)$ properly. Let $\Psi$ be the corresponding virtual support function. The condition of proper intersection is equivalent to $\Psi|_{\sigma}=0$. Then $\Psi$ extends to a continuous function ${\overline{\Psi}}$ on $N_{\sigma}$ and the restriction of ${\overline{\Psi}}\to N(\sigma)$ is equal to $\Psi(\sigma)$. Hence the result follows from Proposition 5.22. ∎