strict inequalities hold for the entire sums.)

\[lem:SimpleConnectnessOfX-SimpleSets\] Any $x$-simple open connected set is simply connected.

Suppose that the set $\Omega\subseteq\mathbb R^2$ is $x$-simple, open and connected. In view of Lemma \[lem:OnOpenX-SimpleRegion\], there exists a smooth function $\theta\in {\rm C}^\infty({\mathop{\rm pr}\nolimits}_t\Omega)$ whose graph is contained in $\Omega$. (In fact, the continuity of $\theta$ is sufficient for the further proof.) Fix a point $t_0\in{\mathop{\rm pr}\nolimits}_t\Omega$ and consider an arbitrary continuous path $\gamma\colon S^1\to\Omega$, $S^1\ni\tau\mapsto(\gamma^1(\tau),\gamma^2(\tau))\in\Omega$, where $S^1$ is the unit circle. The path $\gamma$ can be shrunken to the point $(t_0,\theta(t_0))$ within $\Omega$ using the map from the unit disk to $\Omega$ that is defined by $$(\rho,\tau)\mapsto
\begin{cases}
\Big(\gamma^1(\tau)+(1-2\rho)(t_0-\gamma^1(\tau)),\,\theta\big(\gamma^1(\tau)+(1-2\rho)(t_0-\gamma^1(\tau))\big)\Big),\quad \rho\in\big[0,\tfrac12\big],
\\[1.5ex]
\Big(\gamma^1(\tau),\,\gamma^2(\tau)+2(1-\rho)\big(\theta(\gamma^1(\tau))-\gamma^2(\tau)\big)\Big),\quad \rho\in\big[\tfrac12,1\big],
\end{cases}$$ where $(\rho,\tau)$ are the ‘polar’ coordinates on the disk, $\rho\in[0,1]$ and $\tau\in S^1$. (Roughly speaking, we first shrink the path $\gamma$ along the $x$-direction to the arc $\{(t,\theta(t))\mid t\in{\mathop{\rm pr}\nolimits}_t\gamma(S^1)\}$ of the graph of the function $\theta$ and then shrink this arc along itself to the point $(t_0,\theta(t_0))$.)

Note that an open simply connected set is not in general $x$-simple. An example of such a set is $\mathbb R^2\setminus\big([0,+\infty)\times\{0\}\big)$.

The following lemma introduces an essential technical tool for our further investigation: it identifies, within any non-$x$-simple set, a 