I have a question regarding conditional expectations from this post: https://almostsuremath.com/2010/05/03/girsanov-transformations/
In the final equation in the proof below, I can't see why this follows from the conditional expectation properties. So here, $U$ is a nonnegative martingale, and $X$ is a cadlag adapted process. We are trying to show that $(UX)^\tau$ is a martingale if and only if $UX^\tau$ is a martingale, and so define $M$ to be the difference and show that $M$ is a martingale. Using the tower property and optional sampling for $U$ we are able to obtain $E[M_t|\mathscr{F}_s] = E[(U_{t \wedge (s\vee \tau)} - U_{t\wedge \tau})X_{t \wedge \tau}|\mathscr{F}_s]$. However, from this, how do we get
$$E[(U_{t \wedge (s\vee \tau)} - U_{t\wedge \tau})X_{t \wedge \tau}|\mathscr{F}_s]=(U_s-U_{s\wedge \tau})X_\tau?$$
I cannot think of any properties to derive this. I would greatly appreciate any help.
