  In this paper we give all the details of the calculation that we presented in
our previous paper ArXiv:0908.0387 where the infrared structure of the MHV
gluon amplitudes in the planar limit for ${\cal N}=4$ super Yang-Mills theory
was considered in the next-to-leading order of perturbation theory. Explicit
cancellation of the infrared divergencies in properly defined inclusive
cross-sections is demonstrated first in a toy model example of "conformal QED"
and then in the real ${\cal N}=4$ SYM theory. We give the full-length details
both for the calculation of the real emission and for the diagrams with
splitting in initial and final states. The finite parts for some inclusive
differential cross-sections are presented in an analytical form. In general,
contrary to the virtual corrections, they do not reveal any simple structure.
An example of the finite part containing just the log functions is presented.
The dependence of inclusive cross-section on the external scale related to the
definition of asymptotic states is discussed.
