We study the inclusive production of isolated prompt photons within the framework of the quasi-multi-Regge-kinematic approach, applying the quark Reggeization hypothesis. We describe accurately and without free parameters the transverse momentum and pseudorapidity spectra of prompt photons in the inclusive photoproduction at the HERA Collider. It is shown that the main mechanism of the inclusive prompt photon production in the gamma p collisions is the fusion of a Reggeized quark (antiquark) from the proton and a collinear antiquark (quark) from the photon into a photon, via the effective Reggeon-quark-gamma vertex. The fragmentation of the quark, which is produced via the gamma-Reggeon-quark and quark-Reggeon-quark vertices, into a photon is strongly suppressed by the isolation cone condition and it gives a significant contribution in the region of a large negative pseudorapidity only. At the stage of numerical calculations we use the Kimber-Martin-Ryskin prescription for unintegrated quark and gluon distribution functions, with the following collinear parton densities as input: MRST for a proton and GRV for a photon.
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