  We evaluate the two-photon exchange correction to the unpolarized cross
section in the elastic muon-proton scattering within dispersion relations. One
of the six independent invariant amplitudes requires a subtraction. We fix the
subtraction function to the model estimate of the full two-photon exchange at
one of three MUSE beam energies and make a prediction for the two other
energies. Additionally, we present single and double polarization observables
accounting for the lepton mass.
