Proposition 11

If $(x_i), (y_j)$ are two finite families of positive elements of a lattice ordered group, then

Arguing by induction on the number of elements in the families $(x_i)$ and $(y_j)$, it is enough to prove that, if x, y and z are positive elements, then

Indeed, put $t = \inf(x, y + z)$; by VI, p. 12, Cor., we can write $t = t_1 + t_2$ with $0 \leq t_1 \leq y$ and $0 \leq t_2 \leq z$; since $t_1$ and $t_2$ are positive, we also have $t_1 \leq x$ and $t_2 \leq x$, whence $t_1 \leq \inf(x, y)$ and $t_2 \leq \inf(x, z)$.