elementary set theory - cross partition proof

I have been having troubles in how to proof the cross partition. For what I know a partition must hold two properties:

1. The elements of the subsets that form the partition should be
   equal to U, where U is the universe set. b)


2. The subsets that form
   the partition should be disjoint or their intersection should be the
   empty set.

I have seen the following exercise related to cross partition in a book:

I can assume that I have two, or more partitions, of S that holds properties 1 and 2. So if I make a new set of partitions in Ai and Bj then these will be also partitions because I am not adding any new element to any of these subsets. From this point I make the cartesian product between Ai and Bj considering only those subsets that are common, comparing the partitions that I made before of Ai and Bj or that their intersections are not the empty set. These are also partitions because I am not adding any new element that could generate any intersection on the partitions that I have formed.

Is this proof correct?