  We study the soft limit of one-photon radiation at next-to-leading power
(NLP) in the framework of heavy-quark effective theory (HQET) to all orders in
perturbation theory. We establish the soft theorem that for unpolarised
scattering the radiative contribution up to NLP is entirely determined by the
non-radiative amplitude. This generalises the Low-Burnett-Kroll (LBK) theorem
for QED to all orders. All hard matching corrections can be calculated by
applying the LBK differential operator to the non-radiative amplitude. The
virtual corrections in the effective theory vanish beyond one loop, resulting
in a one-loop exact soft function. As a first, non-trivial application we
calculate the real-virtual-virtual electron-line corrections to muon-electron
scattering at NLP in the soft limit.
