map for each state being equal, $\tau$ remains in ${\mathcal{S}}$ during flows. Furthermore, at points $\tau \in {\mathcal{S}}\cap D$, the jump map $G$ is set valued by the definition of $g_i$ in . From these points, $G(\tau) \cap {\mathcal{S}}\neq \emptyset$. In fact, for each $\tau(0,0) \in {\mathcal{S}}$, there exists at least one solution such that $\tau(t,j) \in {\mathcal{S}}$ for all $t + j \geq 0$, with $(t,j) \in {\mathop{\rm dom}\nolimits}\tau$. Consider the case of solutions initialized at $\tau(0,0) \in \startmodif {\mathcal{G}}\stopmodif$ (Note that $\tau(0,0) \in D$). It follows that for some $r \in I$, $\tau_r(0,0) = {\bar{\tau}}$ and $g_r(\tau(0,0)) = 0$. Therefore, after the initial jump, we have that $G(\tau(0,0)) \cap {\mathcal{S}}\neq \emptyset,$ by which using previous arguments implies that $\tau(t,j) \in {\mathcal{S}}$ for all $t + j \geq 1$.

Furthermore, there is a distinct ordering to the jumps. If $\tau$ is such that $\tau_i \neq \tau_r$ for all $i\neq r$ then the ordering of each $\tau_i$ is preserved after $N$ jumps. More specifically, we have the following result.

\[lem:ordering\] For every solution $\tau$ to ${{{\mathcal}H}}_N$ with $\tau(0,0) \notin {\mathcal{X}}$, if at $(t_{j},j) \in {\mathop{\rm dom}\nolimits}\tau$ we have for some sequence of nonrepeated elements $\{i_m\}^N_{m = 1}$ of $I$ (that 