. Now, since $\lim_{\varepsilon \rightarrow 0^+} \Delta \tilde{E}^{H^\varepsilon}_\eta (\gamma_x) = \Delta \tilde{E}^{H^0}_\eta (\gamma_x)$ for every $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$, it follows that there exists (random) $\varepsilon_0 > 0$ such that if $0 \leq \varepsilon < \varepsilon_0$ then $$\label{ancesinc