Complete the proof of Theorem 38.5

Data from Theorem 38.5

If G is a nonzero free abelian group with a basis of r elements, then G is isomorphic to Z x Z x ··· x Z for r factors. 

It is a fact that any two bases of a free abelian group G contain the same number of elements. We shall prove this only if G has a finite basis, although it is also true if every basis of G is infinite. The proof is really lovely; it gives an easy characterization of the number of elements in a basis in terms of the size of a factor group.