Displaying the cohomology of toric line bundles

There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $Delta^+-Delta^-$ of polyhedra in the character space $M_{mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $Delta^- setminus Delta^+$. Here, we provide a short and direct proof of this formula.