The assumptions yield two charts for the $\mathbb{Z}$-affine structure on $B$: the $\mathbb{Z}$-affine subsets $B^{\prime}$ and $\tau_{1}$. These two charts are glued along $B^{\prime}\cap\tau_{1}$ which is a simplex and thus has no non-trivial $\mathbb{Z}$-automorphisms preserving the vertices, hence the affine structure on $B$ is uniquely determined. The set $B^{\prime}\subset\mathbb{R}^{n}$ can be obtained as the result of the same star subdivisions of a subset $\tilde{B}\subset\mathbb{R}^{n}$, and uniqueness of the affine structure ensures a $\mathbb{Z}$-affine isomorphism $B\simeq\tilde{B}$. ∎