  We utilize the DHM integral representation for the BES dressing factor of the
world-sheet S-matrix of the AdS_5xS^5 light-cone string theory, and the
crossing equations to fix the principal branch of the dressing factor on the
rapidity torus. The results obtained are further used, in conjunction with the
fusion procedure, to determine the bound state dressing factor of the mirror
theory. We convincingly demonstrate that the mirror bound state S-matrix found
in this way does not depend on the internal structure of a bound state solution
employed in the fusion procedure. This welcome feature is in perfect parallel
to string theory, where the corresponding bound state S-matrix has no bearing
on bound state constituent particles as well. The mirror bound state S-matrix
we found provides the final missing piece in setting up the TBA equations for
the AdS_5xS^5 mirror theory.
