Independent Random Variables

Question 1)

http://www.geocities.com/asdfasdf23135/stat2.JPG

Note: iid=idependent and identically dsitributed

http://www.geocities.com/asdfasdf23135/stat3.JPG

I don't get the part underlined in red. Why are they independent? How can we prove that, or at least *intuitively make sense* of it?

Question 2) If X1,X2,...,X6 are independent random varaibles, then my textbook has a theorem saying that g1(X1), g2(X2), ..., g6(X6) are also independent, where the gi's are any function of a single random variable.

But how about f1(X1,X2,...,X5), f2(X6)? If X1,X2,...,X6 are independent, are any function of X1,...,X5 and any function of X6 independent?

Any help is greatly appreciated!:)