We have already shown that a $\mathbb{T}$-Cartier divisor produces a toric line bundle with a toric section. Let now $((L,z),s)$ be a toric line bundle equipped with a toric section and $\Sigma$ the fan that defines $X$. Since every line bundle on an affine toric variety is trivial, for each $\sigma\in\Sigma$ we can find a section $s_{\sigma}$ that generates $L$ on $X_{\sigma}$ and such that $s_{\sigma}(x_{0})=z$. Since $s$ is regular and nowhere vanishing on $X_{0}$ and $s(x_{0})=z$, we can find elements $m_{\sigma}\in M$ such that $s=\chi^{-m_{\sigma}}s_{\sigma}$, because any regular nowhere vanishing function on a torus is a constant times a monomial. The elements $m_{\sigma}$ glue together to define a virtual support function $\Psi$ on $\Sigma$ that does not depend on the chosen trivialization. It is easy to see that the correspondence $(L,s)\mapsto D_{\Psi}$ is the inverse of the previous one, which proves the theorem. ∎