Proposition 18

For a family $(a_t)_{t \in T}$ of a module E to be free, it is necessary and sufficient that every finite subfamily of $(a_t)_{t \in T}$ be free.

This follows immediately from the definition.

Proposition 18 shows that the set of free subsets of E, ordered by inclusion, is inductive (Set Theory, III, § 2, no. 4); as it is non-empty (since $\varnothing$ belongs to it), it has a maximal element $(a_l)_{l \in I}$ by Zorn’s Lemma. It follows (if $A \neq \{0\}$) that for all $x \in E$ there exist an element $\mu \neq 0$ of A and a family $(\xi_l)$ of element A such that $\mu x = \sum_l \xi_l a_l$ (cf. § 7, no. 1).