Riesz decomposition in a lattice-ordered group
= Riesz decomposition in a lattice-ordered group
{c}
{title2=$1\le x\le ab\implies x=a'b',\quad1\le a'\le a,\quad1\le b'\le b$}
For $a,b\ge1$, take $a'=x\wedge a$ and $b'=(a')^{-1}x$. Then $a'\ge1$ and $b'=1\vee a^{-1}x$ is between one and $b$. Thus $x=a'b'$. Repetition splits a positive element bounded by a finite product into factors bounded by the corresponding factors. Every resulting positive factor is also bounded by $x$.