Proposition 13

Let G be a group with operators and $(H_i)_{i \in I}$ a family of stable subgroups of G such that every element of $H_i$ is permutable with every element of $H_j$ for $j \neq i$. For G to be the restricted sum of the family of subgroups $(H_i)_{i \in I}$, it is necessary and sufficient that every element x of G be expressible uniquely in the form $\prod_{i \in I} y_i$, where $(y_i)_{i \in I}$ is a family with finite support of elements of G with $y_i \in H_i$ for all i.

Obvious.