Proof: Let $\sigma^{\prime}$ be an $n$-dimensional polyhedron of ${\mathscr{C}}^{\prime}$. Then $\sigma:=F(\sigma^{\prime})$ is an integral ${\mathbb{R}}$-affine polyhedron in $N_{\mathbb{R}}$. We assume for the moment that $\sigma$ is also $n$-dimensional. As above, we consider the lattice $N_{\sigma}:=N\cap{\mathbb{L}}_{\sigma}$ in $N_{\mathbb{R}}$, where ${\mathbb{L}}_{\sigma}$ is the linear space which is a translate of the affine space generated by $\sigma$. Let $A$ be the matrix of the homomorphism $F:N_{\sigma^{\prime}}\rightarrow N_{\sigma}$ with respect to integral bases. Then we have $|\det(A)|=[N_{\sigma^{\prime}}:N_{\sigma}]$ and hence the transformation formula (1) shows

$$\int_{\sigma^{\prime}}F^{*}\alpha=[N_{\sigma^{\prime}}:N_{\sigma}]\int_{\sigma}\alpha.$$

(2)

If $\dim(\sigma)<n$, then both sides are zero and hence formula (2) is true in any case. Using the weighted sum over all $\sigma^{\prime}$, the claim follows immediately from (2). $\square$