Proposition 5

For a non-empty family $(a_i)_{i \in I}$ of points of an affine space $E$ to be affinely free, it is necessary and sufficient that, for every index $\kappa \in I$, $a_\kappa$ do not belong to the linear variety generated by the $a_i$ of index $\neq \kappa$.

The proposition is obvious if $I$ has only a single element. Otherwise, taking as origin in $E$ one of the $a_i$ of index $\neq \kappa$, the proposition follows from § 7, no. 1, Remark.