Corollary 1

If x and y are two coprime elements, and z a positive element, of a lattice ordered group, then $\inf(x, z) = \inf(x, y + z)$.

Indeed $\inf(x, y + z) \leq \inf(x, z)$ by Prop. 11, and $\inf(x, z) \leq \inf(x, y + z)$ since $y \geq 0$, whence the corollary.