  We relate the forward two-photon exchange (TPE) amplitudes to integrals of
the inclusive lepton-proton scattering cross sections. These relations yield an
alternative way for the evaluation of the TPE correction to hyperfine-splitting
(HFS) in the hydrogen-like atoms with an equivalent to the standard approach
(Iddings, Drell and Sullivan) result implying the Burkhardt-Cottingham sum
rule. For evaluation of the individual effects (e.g., elastic contribution) our
approach yields a distinct result. We compare both methods numerically on
examples of the elastic contribution and the full TPE correction to HFS in
electronic and muonic hydrogen.
