If is a subgroup of and is a subgroup of , prove is a subgroup of .
We know , , , and are all groups. Thus, since the Cartesian product of any two groups is a group, and are both groups.
A subgroup is merely a subset of elements from a group which, under the same operation as the original group, form a group. uses the same operation as and uses the same operation as . Therefore, uses the same operation as . All that is left to be shown is that is a subset of .
Pick any element . Well, and so . Thus, every element in is also an element of and our proof is complete.