G$ for $j=1$ or $j=2$. By Lemma \[lemm:maximum\_principle\] there are $x_0,x_1 \in G$ such that $u(x_0)\leq s$ and $u(x_1)\geq t$. Moreover, by Proposition \[prop:existence\_paths\] there is a rectifiable curve $\gamma$ joining $x_0$ and $x_1$ in $F$. Since $u||\gamma|$ is continuous, we conclude that $u(x) \in E$ for some $x \in |\gamma|$. This is a contradiction, since $E \cap F = \emptyset$.

Suppose next that the set $G$ in Case \[pring\] does not exist. We then find a subcontinuum $F'$ of $F$ with the following properties: $F' \cap \partial Q$ consists of two distinct points $x_0$ and $x_1$, and $E_1$ and $E_2$ belong to different components, say $\Omega_1$ and $\Omega_2$, of $X \setminus (\partial Q \cup F')$. By Proposition \[prop:existence\_paths\] we may moreover assume that $F'=|\gamma|$, where $\gamma:[0,1]\to Q$ is simple and rectifiable, and $\gamma(0)=x_0$, $\gamma(1)=x_1$.

Suppose that both $x_0$ and $x_1$ belong to $\zeta_j$ for some $j=1,\ldots,4$. Then $\partial \Omega_k \subset |\gamma| \cup \zeta_j$ for $k=1$ or $k=2$. As in Case \[pring\], Lemma \[lemm:maximum\_principle\] and the continuity of $u||\gamma|$ show that there exists $x \in |\gamma|$ such that $u(x) \in [s,t]$. This contradicts the construction of $\gamma$. A similar argument can be applied when $x_0 \in \zeta_i$ and $x_1 \in 