Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 27 / 10 / a / Solution

Distributivity makes multiplication order-preserving for the natural order: if $u+v=v$, then $xu+xv=x(u+v)=xv$, so $xu\le xv$. It also expands every generated expression into a finite join of monomials, as used in solution.