  We discuss the mechanisms of photon-photon fusion triggering the dilepton
production in proton-proton collisions with rapidity gap in the main detector
and one forward proton in the forward proton detectors. This correspond to the
LHC measurements made by ATLAS+AFP and CMS+PPS. Transverse momenta of the
intermediate photons and photon fluxes expressed by the form factors and
structure functions are included. Moreover the influence of proton measurement
and cuts on the $\xi_{1/2}$, $M_{ll}$, $Y_{ll}$, $p_{t,ll}$ distributions,
cross section and gap survival factor are considered for both double-elastic
and single-dissociative processes. The analyzes used the SuperChic generator to
calculate the soft gap survival factor and to compare the results obtained with
its use to the results obtained on the owned codes available. It is shown that
the gap survival factor for the single dissociative mechanism is related to the
emission of a (mini)jet into the main detector depending on the type of
contribution. The dependence between this value and the invariant mass as well
as the transverse momentum of the leptons pair and its rapidity was also found.
