Lang's Theorem 7.3 wording, what does he mean?

Question

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Above is the first part of proof Theorem 7.3, Lang's Algebra, p41.

Why could we apply Lemma 7.2? Isn't this only a consequence of this line? (Which was stated after application?)

Since $f(B)$ is either $0$ or infinite cyclic...$B$ is free.

EDIT: I clarified question a bit.

Answer 1

The free group on the empty set is $0$.
The free group on $\{g\}$ is the infinite cyclic group generated by $g$.

So in both cases considered $f(B)$ is free.

So $B_1$ is isomorphic to a restricted direct sum of copies of the group of integers because it is free by induction and abelian. Furthermore $\mathbb{Z}_1$ is not induced in this decomposition.

On the other hand $f(B)=C_1$ is isomorphic to $\mathbb{Z}_1$ or $\{0\}$, so $B$ is isomorphic to a direct sum of copies of the group of integers, so $B$ is free.