By Proposition 1.4 $\Pi$ comes as a corner locus of a convex piecewise-linear function $F$ on $\mathbb{R}^{n+1}$. Let $y=a_{j,1}x_{1}+\dots+a_{j,n+1}x_{n+1}$, $j=1,\dots,n+2$ be the equations of the linear functions on the adjacent components of $\mathbb{R}^{n+1}\smallsetminus\Pi$. Then $u_{j}=(a_{j,1},\dots,a_{j,n+1},-1)$ are the vectors in $\mathbb{R}^{n+2}=\mathbb{R}^{n+1}\times\mathbb{R}$ normal to the linear portions of the graph of $F$ adjacent to $B$.

The $\mathbb{R}^{n+2}$-version of the vector product associates a normal vector to $(n+1)$ other vectors in $\mathbb{R}^{n+2}=\mathbb{R}^{n+1}\times\mathbb{R}$. We take all possible such products among $u_{j}$ and project them to $\mathbb{R}^{n+1}$. The result is the vectors which are multiples of $v_{j}$. By linear algebra the sum of these vectors is zero. ∎