  The observation of forward proton scattering in association with lepton pairs
($e^+e^-+p$ or $\mu^+\mu^-+p$) produced via photon fusion is presented. The
scattered proton is detected by the ATLAS Forward Proton spectrometer while the
leptons are reconstructed by the central ATLAS detector. Proton-proton
collision data recorded in 2017 at a center-of-mass energy of $\sqrt{s} = 13$
TeV are analyzed, corresponding to an integrated luminosity of 14.6 fb$^{-1}$.
A total of 57 (123) candidates in the $ee+p$ ($\mu\mu+p$) final state are
selected, allowing the background-only hypothesis to be rejected with a
significance exceeding five standard deviations in each channel. Proton-tagging
techniques are introduced for cross-section measurements in the fiducial
detector acceptance, corresponding to $\sigma_{ee+p}$ = 11.0 $\pm$ 2.6 (stat.)
$\pm$ 1.2 (syst.) $\pm$ 0.3 (lumi.) fb and $\sigma_{\mu\mu+p}$ = 7.2 $\pm$ 1.6
(stat.) $\pm$ 0.9 (syst.) $\pm$ 0.2 (lumi.) fb in the dielectron and dimuon
channel, respectively.
