Besides They can argue  that \(q\left(t,t_{in}\right)\ =\ v_{ex}\left(t=t_{ex},t_{in}\right)\) if we are running  the clock time \(t\) for any of the times larger than \(t_{in}\).  However, it is also,  as we already mentioned \(s\left(t_{in},t_{in}\right)=P\left(t_{in}\right)\) and therefore: