{\widetilde}{C} \cap [L_{u_{{\widetilde}{C}}}(0) \cup (L_{u_{{\widetilde}{D}}}(0) \cap G(L_{u_{{\widetilde}{C}}}(0)))] = L_{V}(r') \cap {\widetilde}{C}$ for $r' \in V({\widetilde}{C})$. Since every maximal solution jumps an infinite number of times, the largest invariant set is given for $r' = 0$ due to the fact that $V(G(\tau)) - V(\tau) = {\varepsilon}V(\tau) < 0$ if $r' > 0$. Then, the largest invariant set is given by $L_{V}(0) \cap {\widetilde}{C} = {\widetilde}{\mathcal{A}}\cap {\widetilde}{C}$ which is identically equal to ${\mathcal{A}}$. Hence, the set ${\mathcal{A}}$ is attractive. Stability is guaranteed from the fact that $V$ is nonincreasing during flows and strictly decreasing during jumps. Then, the set ${\widetilde}{\mathcal{A}}$ is asymptotically stable for the hybrid system ${\widetilde}{{{\mathcal}H}}_{N}$. We have that ${\mathcal{A}}$ is (strongly) forward invariant and from Theorem 3.4 we know that ${\mathcal{A}}$ is uniformly attractive from a neighborhood of itself. Then by Proposition 7.5 in [@teel2012hybrid], it follows that ${\mathcal{A}}$ is asymptotically stable.

Note that the set of solutions to ${\widetilde}{{{\mathcal}H}}_{N}$ coincides with the set of solutions to ${{{\mathcal}H}}_{N}$ from $P_{N} \setminus {\mathcal{X}}_{v}$. Therefore, the set ${\mathcal{A}}$ is asymptotically stable for ${{{\mathcal}H}}_N$ with basin of attraction ${{\mathcal}B}_{{\mathcal{A}}} = P_{N} \setminus {\mathcal{X}}_{v}$. [[ Since $v$ is arbitrary, it follows that the basin of attraction is equal to $P_{N} \stopmodif \setminus {\mathcal{X}}$.]{}]{}

