  We discuss photon-photon fusion mechanisms of dilepton production in
proton-proton collisions with rapidity gap in the main detector and one forward
proton in the forward proton detectors. This is relevant for the LHC
measurements by ATLAS+AFP and CMS+PPS. Transverse momenta of the intermediate
photons are taken into account and photon fluxes are expressed in terms of
proton electromagnetic form factors and structure functions. Differential
distributions in $\xi_{1/2}$, $M_{ll}$, $Y_{ll}$, $p_{t,ll}$, $M_R$ %pamietac
wyjasnic M_R w tekscie are shown and the competition of different mechanisms is
discussed. Both double-elastic and single-dissociative processes are included
in the analysis. Different parametrizations of the structure functions are
used. We discuss also mechanism with one forward $\Delta^+$ isobar, or other
proton resonances in the final state. The role of several cuts is studied. We
also use SuperChic generator and compare corresponding results to the results
of our codes. The soft rapidity gap survival factor is calculated for each
contribution separately. The gap survival factor for the single-dissociative
mechanism due to minijet emission into the main detector is calculated in
addition. It depends on the type of contribution (fully elastic, single
dissociation, double dissociation). The soft rapidity gap survival factor for
the case of single proton measurement is significantly smaller than that for
the inclusive case (no proton measurement). We find only weak dependence on the
invariant mass of the dilepton system as well as the lepton pair transverse
momentum and sizeable dependence on the pair rapidity. The latter effect is
rather difficult to identify experimentally.
