\begin{definition}[Definition:Horizontal Section of Set]
Let $X$ and $Y$ be sets.
Let $E \subseteq X \times Y$. 
Let $y \in Y$. 
We define the '''$y$-horizontal section''' $E^y$ of $E$ by: 
:$E^y = \set {x \in X : \tuple {x, y} \in E}$
\end{definition}