Proof. Let us proceed inductively by codimension of faces. The induction step reduces to the obvious remark that the extension of the standard ${\bf Z}$-affine structure on ${{\bf R}}^{n}\setminus L$ to a neighborhood of point $p\in L$ in ${\bf R}^{n}$ is unique in the case when $L\subset{\bf R}^{n}$ is an affine subspace, $\dim L\leq n-2\,\,$. $\blacksquare$