When I wrote that, I litteraly meant that

holds.

The term

should be obvious: it is a function of the variable we are conditioning by, therefore the conditional expectation acts trivially. (Conditioning by

is like conditioning by the

-valued random variable

, or by the sigma-algebra of events depending on N(1) and N(2))

The second term is because

is independent of

(it is even independent of

). There is no mutual independence between

: it would mean that

are independent...

As you see, these are the same arguments as for 1 random variable.