Product of Supremum and Infimum in Lattice-Ordered Group

Theorem
Let $\struct {G, \odot}$ be a group.

Let $\preccurlyeq$ be a lattice ordering on $G$.

Let $x, y \in G$ such that $x$ commutes with $y$.

Then:
 * $\sup \set {x, y} \odot \inf \set {x, y} = x \odot y$

Proof
First we show that $x$ commutes with $\sup \set {x, y}$.

Indeed:

Similarly:
 * $y$ commutes with $\sup \set {x, y}$

Then using Infima in Ordered Group in the same way as above:
 * $x$ commutes with $\inf \set {x, y}$
 * $y$ commutes with $\inf \set {x, y}$

Then we have: