If $f$ is a map from an Abelian lattice ordered group $G_1$ (endowed with a root function $\gamma_2$ of order two) onto an Archimedean Abelian lattice ordered group $G_2$ with $f(0)=0$ and $|f(x)-f(y)|$ depends functionally on $|x-y|$, then $f$ is additive.