Corollary 4

— If $(x_i)_{1 \leq i \leq n}$ and $(y_j)_{1 \leq j \leq m}$ are two finite families of elements of a lattice ordered group G, such that each $x_i$ is coprime to each $y_j$, then $x_1 + \ldots + x_n$ is coprime to $y_1 + \ldots + y_m$. This follows from Cor. 3 by induction on m and n.