Proposition 4

For a non-empty family $(a_i)_{i \in I}$ of points in an affine space E to be affinely related, it is necessary and sufficient that there exist a family $(\lambda_i)_{i \in I}$ of elements not all zero in K, of finite support, such that $\sum_{i \in I} \lambda_i = 0$ and $\sum_{i \in I} \lambda_i a_i = 0$.

Given an index $\kappa \in I$, to say that the family of vectors $(a_i - a_\kappa)$, where $i$ runs through the set of indices $\neq \kappa$, is related in T, means that there exists a family of scalars $(\lambda_i)_{i \neq \kappa}$ not all zero such that $\sum_{i \neq \kappa} \lambda_i (a_i - a_\kappa) = 0$, which may also be written $\sum_{i \in I} \lambda_i a_i = 0$, with $\lambda_\kappa = -\sum_{i \neq \kappa} \lambda_i$, in other words $\sum_{i \in I} \lambda_i = 0$.