Definition of the Deep End: In hyperbolic geometry, for any angle $\angle ABC$, there are points $D$ between the rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ such that none of the straights through $D$ can cross both rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$. These points are told to be in the "Deep End" of the interior angle of $\angle ABC$.
Assume point $D$ is in the Deep End of $\angle ABC$ and $E$ is a point on $\overrightarrow{BD}$ such that $D$ is between $B$ and $E$. Prove $E$ is in the Deep End of $\angle ABC$.
My idea is to suppose that $E$ is not in the deep end. So then the ray containing it will cross both $BA$ and $BC$ which would contradict the assumption that $D$ is in the deep end. I don't feel this is fully justified though.
HINT TO INTUIT THE PROBLEM
The following figure depicts the deep end of the angle $\angle ABC $ in the Beltrami-Klein model.
Here, $p$ is the common parallel to the rays $BC$ and $BA$. The yellow region is the Deep End; $p$ belongs to it. If $D$ is in the Deep End then $E$ is - a fortiory - in it.