  We apply a subtracted dispersion relation (DR) formalism with the aim to
improve predictions for the two-photon exchange (TPE) corrections to elastic
electron-proton scattering observables at small momentum transfers. We study
the formalism on the elastic TPE contribution in comparison with existing data
for unpolarized cross sections. We extend the general formalism of TPE to
elastic scattering with massive lepton and perform a numerical estimate of the
muon-proton scattering at low momentum transfer in view of the upcoming
muon-proton scattering experiment (MUSE). We study the influence of the
double-virtual Compton scattering (VVCS) subtraction function on the
unpolarized lepton-proton scattering cross-section. We show that the resulting
TPE correction is negligible in the electron-proton scattering and smaller than
planned uncertainties of the MUSE experiment for the subtraction functions
evaluated in chiral perturbation theory.
