= Wolfenstein inequality for normal-valued lattice-ordered groups
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{title2=$|f||g|\leq|g|^2|f|^2$}
A <lattice-ordered group> is normal-valued precisely when the displayed inequality holds for all its elements. The theorem is proved without using it as a definition. In the forward direction a normal value has a linearly ordered quotient of its cover with no proper nontrivial convex subgroup. That quotient is Archimedean, hence Abelian by <Archimedean linearly ordered groups are Abelian>. Apply <squared bound from comparison at values> to $p=b,q=a,x=ab$. For the reverse direction, <power cofinality in the cover of a value> excludes a conjugate $u=a^{-1}va$ outside the value: its right cosets $Vu^n$ are all bounded by $Va$, contradicting cofinality above $Va^2$. Left-coset cofinality treats $ava^{-1}$. Every element is a quotient of nonnegative lattice parts, proving normality in the cover. In a <linearly ordered group>, comparison of $a,b\geq1$ immediately gives $ab\leq\max(a,b)^2\leq b^2a^2$, so every such group is normal-valued.
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