I was reading a research paper titled Purity and Reid's Theorem by A.Blass and J.Irwin and i have the problem with the explanation of the proof of the first theorem, that is theorem 1.1. In the proof of the first theorem, G is a torsion-free abelian group of infinite rank $\kappa$. At the end of the proof of theorem 1.1, the author says that :
Equivalently, $e(\alpha)$ should not be in the affine subspace of $\bar{G}$ spanned by $(f_{\beta}-g_{\alpha}, \beta < \alpha ) \cup (g_{\alpha}) $where $\bar{G}$ is the divisible hull of $G$.
Question:
What does it mean by affine subspace of $\bar{G}$ spanned by $(f_{\beta}-g_{\alpha}, \beta < \alpha ) \cup (g_{\alpha})$?
The research paper can be obtained from the web.