Notice that in the above we can assume $H^{+}|_{\Gamma}=H^{-}|_{\Gamma}$, therefore we can define $\bar{\Lambda}_{H}=\Lambda_{H^{+}}\mod db_{1}=\Lambda_{H^{-}}\mod db_{1}$. Via the identification in the above Proposition, the space $\bar{Z}^{\#}$ corresponds to $T^{\ast}\Gamma/\penalty\bar{\Lambda}_{H}$ and $\bar{f}^{\#}:\bar{Z}^{\#}\rightarrow\Gamma$ becomes the projection $\bar{\pi}^{\#}$.