This proposition also follows from the duality with a unimodular triangulation $\mathcal{D}$ of $\Delta$. Let $U_{j}$ be a primitive piece. It corresponds to a simplex of volume $\frac{1}{(n+1)!}$ in $\mathcal{D}$. There is an element of $SL_{n+1}(\mathbb{Z})$ which takes this simplex to the standard simplex $\Delta_{1}^{n+1}$ (see (1)). Then the image of $U_{j}$ by the adjoint to the inverse of this element is contained in a dual $\Delta_{1}$-complex. Such a complex is the result of a translation of $\Sigma_{n}$ by Proposition 1.8. ∎