The law of total probability with extra conditioning

[red65]

TL;DR Summary

the proof of a theorem

Hello, I am studying probability and came across this theorem, it's the law of total probability with extra conditioning, I tried to work out a proof but couldn't ,does anyone know the proof for this :

thanks!

 

[FactChecker]

1) If you are looking for a proof, you should be very careful about the exact statement. There is more to that statement, right? Don't the $A_i$ need to be a partitioning?
2) If all probabilities are conditional on $E$, isn't that just another probability where $E$ is the universe and does not need to be included in the notation?

 

[PeroK]

TL;DR Summary: the proof of a theorem

Hello, I am studying probability and came across this theorem, it's the law of total probability with extra conditioning, I tried to work out a proof but couldn't ,does anyone know the proof for this :
View attachment 319864
thanks!

That's just the usual equation with a restriction to $E$ as the universal set or sample space. Given the proviso, as above, that the $A_i$ (when restricted to $E$) partition $E$.