  Let $\alpha$ and $\beta$ be positive real numbers. Let $F(x) \in
K[[x^\Gamma]]$ be a Hahn series. We prove that if $F(x)$ is both
$\alpha$-Mahler and $\beta$-Mahler then it must be a rational function, $F(x)
\in K(x)$, assuming some non-degeneracy conditions on $\alpha$ and $\beta$.
