Lemma 1

Let (a_i)_{i \in I} be a free family of elements of E; if b \in E does not belong to the subspace F generated by (a_i), the subset of E consisting of the a_i and b is free.

Suppose that there were a relation $\mu b + \sum_i \lambda_i a_i = 0$ with $\mu \in K$ and $\lambda_i \in K$ for all $i \in I$, the family $(\lambda_i)$ having finite support; if $\mu \neq 0$, it would follow that $b = -\sum_i (\mu^{-1} \lambda_i) a_i$ and hence $b \in F$ contrary to the hypothesis; hence $\mu = 0$ and the relation becomes $\sum_i \lambda_i a_i = 0$, which implies $\lambda_i = 0$ for all $i \in I$ by hypothesis; whence the lemma.