  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.
