Riesz decomposition in a lattice-ordered group ($1\le x\le ab⟹x=a^{\prime }{b}^{\prime },1\le a^{\prime }\le a,1\le b^{\prime }\le b$)

For $a,b\ge 1$, take $a^{\prime }=x\wedge a$ and $b^{\prime }=(a^{\prime }{)}^{-1}x$. Then $a^{\prime }\ge 1$ and $b^{\prime }=1\vee a^{-1}x$ is between one and $b$. Thus $x=a^{\prime }{b}^{\prime }$. 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 $x$.