  Hard spin-independent three-loop radiative contributions to energy levels in
positronium generated by the two-photon-exchange diagrams with one-loop
radiative insertions in the fermion lines and exchanged photons are calculated.
This is a next step in calculations of all corrections of order $m\alpha^7$
inspired by the new generation of precise $1S-2S$ and $2S-2P$ measurements in
positronium.
