  This paper derives a next-to-leading power (NLP) soft theorem for
multi-photon emission to all orders in the electromagnetic coupling constant,
generalising the leading-power theorem of Yennie, Frautschi, and Suura. Working
in the QED version of heavy-quark effective theory, multi-emission amplitudes
are shown to reduce to single- and double-radiation contributions only. Single
soft-photon emission, in turn, is described by the recent all-order extension
of the Low-Burnett-Kroll theorem, where the tree-level formula is supplemented
with a one-loop exact soft function. The same approach is used in this article
to prove that the genuine double-emission contribution is tree-level exact. As
a validation and a first non-trivial application of the multi-photon theorem,
the real-real-virtual electron-line corrections to muon-electron scattering are
calculated at NLP in the soft limit.
