Corollary

— Let $y, x_1, x_2, \ldots, x_n$ be $n + 1$ positive elements of $G$ such that $y \leq \sum_{i=1}^n x_i$; then there exist $n$ positive elements $y_i$ ($1 \leq i \leq n$) such that $y_i \leq x_i$ and $y = \sum_{i=1}^n y_i$.

It is sufficient to apply theorem 1 to the sequence $(x_i)$ and the sequence consisting of the two elements $y$ and $z = \left( \sum_{i=1}^n x_i \right) - y$.