\begin{definition}[Definition:Half-Plane/Open/Lower]
Let $\PP$ be a Cartesian plane.
Let $\LL$ be the $x$-axis.
The '''open lower half-plane''' $\HH_{\text {OL} }$ is the area of $\PP$ below $\LL$.
That is, where $y < 0$:
:$\HH_{\text {OL} } := \set {\tuple {x, y}: y \in \R_{< 0} }$
\end{definition}