$\Omega$ is not $x$-simple, we distinguish two cases.

First, we assume that each connected component of $\Omega$ is an $x$-simple set but the entire $\Omega$ is not. This means that there are connected components $U_1$ and $U_2$ of $\Omega$ with overlapping projections ${\mathop{\rm pr}\nolimits}_tU_1$ and ${\mathop{\rm pr}\nolimits}_tU_2$ to the $t$-axis. Suppose that for some $P\in{{\rm DO}}(\Omega)$ of some order $p$ the equation $\mathcal P$: $Pu=0$ possesses a fundamental set of solutions $\varphi^1$, …, $\varphi^p$ on $\Omega$. In view of the previous part of the proof, this equation possesses sets of $p$ solutions with nonzero Wronskians on each connected component of $\Omega$ and hence it does on the entire $\Omega$. Therefore the Wronskian of any fundamental set of solutions of $\mathcal P$ does not vanish on $\Omega$. Using Lemma \[lem:OnOpenX-SimpleRegion\], we fix a function $\theta\in {\rm C}^\infty({\mathop{\rm pr}\nolimits}_tU_1)$ whose graph is contained in $U_1$. There is a solution $\psi$ of $\mathcal P$ such that $\psi=1$ on this graph and $\psi=0$ on $U_2$. By assumption, $\psi=\zeta^s\varphi^s$ for some functions $\zeta^s\in {\rm C}({\mathop{\rm pr}\nolimits}_t\Omega)$. These functions vanish on ${\mathop{\rm pr}\nolimits}_tU_2$ since $\psi\equiv 0$ on $U_2$ and thus the solution $\psi$ vanishes on the intersection of the strip $\{(t,x)\mid t\in{\mathop{\rm pr}\nolimits}_tU_2,\,x\in\mathbb R^2\}$ with $\Omega$. 