(\Lambda_f \times G)$ is contained in $\Lambda' \times G$. Furthermore, since $\eta$ is also of finite local interaction range, by taking any $\Lambda \in \mathcal{B}^0_{S}$ sufficiently large so that $\Lambda' \subseteq \Lambda$ and $\eta( I(\Lambda' \times G) \cap (\Lambda^c \times G) ) = 0$ we 