Cartesian product set difference I know how to handle the 2d case: http://www.proofwiki.org/wiki/Set_Difference_of_Cartesian_Products
But I am having trouble simplifying the following:
Let $X=\prod_{1}^\infty X_i, A_i \subset X_i$
How can I simplify/rewrite $X - (A_1 \times A_2 \times \cdots A_n \times X_{n+1} \times X_{n+2} \cdots)$ with unions/intersections?

Try writing
$$\prod_{k=n+1}^{\infty} X_k = X'$$
then you want the difference of
$$(X_1  \times X_2  \times \cdots  \times X_n  \times X') - (A_1  \times A_2  \times\cdots  \times A_n  \times X')$$
You can use the rule that you linked inductively to this difference. Then note that in some parts of the expression you will get $X' - X' = \emptyset$.
