  Let $\Omega$ and $\Omega'$ be open subsets of a flat $(2,3,5)$-distribution.
We show that a $C^1$-smooth contact mapping $f : \Omega \to \Omega'$ is a
$C^\infty$-smooth contact mapping. Ultimately, this is a consequence of the
rigidity of the associated stratified Lie group (the Tanaka prolongation of the
Lie algebra is of finite-type). The conclusion is reached through a careful
study of some differential identities satisfied by components of the
Pansu-derivative of a $C^1$-smooth contact mapping.
