The intersection multiplicity and the degree in the above Proposition only depend on the isomorphism class of the line bundles ${\mathcal{O}}(D_{\Psi_{i}})$ and not on the $\mathbb{T}$-Cartier divisors themselves. It is easy to check directly that the right-hand sides of (4.38) and (4.39) only depends on the isomorphism class of the line bundles. In fact, let $L$ be a toric line bundle generated by global sections and $s_{1}$, $s_{2}$ two toric sections. For $i=1,2$, set $D_{i}=\operatorname{div}(s_{i})$ and let $\Psi_{i}$ be the corresponding support function and $\Delta_{i}$ the associated polytope. Then $s_{2}=\chi^{m}s_{1}$ for some $m\in M$. Thus $\Psi_{2}=\Psi_{1}-m$ and $\Delta_{2}=\Delta_{1}-m$. Since the volume and the mixed volume are invariant under translation, we see that these formulae do not depend on the choice of sections.