  We present a formalism which uses fluxes of equivalent photons including
transverse momenta of the intermediate photons. The formalism reminds the
familiar $k_t$-factorization approach used, e.g., to study the two-photon
production of $c\bar{c}$ or $b\bar{b}$ pairs. The results of the new method are
compared with those obtained using the code LPAIR, and a good agreement is
obtained. The inclusion of the photon transverse momenta is necessary in
studies of correlation observables. We present distributions for the dimuon
invariant mass, transverse momentum of the muon pair and relative azimuthal
angle between muons separately for elastic-elastic, elastic-inelastic,
inelastic-elastic and inelastic-inelastic mechanisms. For typical experimental
cuts all mechanisms give similar contributions. The results are shown for
different sets of cuts relevant for the LHC experiments. The cross sections in
different regions of phase space depend on $F_2$ structure function in
different regions of $x$ and $Q^2$. A comment on $F_2$ is made.
