  The cancellation of infrared (IR) divergences is an old topic in quantum
field theory whose main results are condensed into the celebrated
Kinoshita-Lee-Nauenberg (KLN) theorem. In this paper, we consider
mass-suppressed corrections to the leading (i.e. double-logarithmic) IR
divergences in the context of spontaneously broken gauge theories. We work in a
simplified theoretical set-up based on the spontaneously broken $U'(1)\otimes
U(1)$ gauge group. We analyze, at the one-loop level and including
mass-suppressed terms, the double-logarithmic corrections to the decay channels
of a hypothetical heavy $Z'$ gauge boson coupled to light chiral fermions and
mixed with a light massive $Z$ gauge boson. Limited to this theoretical
framework, only final state IR corrections are relevant. We find that full
exploitation of the KLN theorem requires non-trivial combinations of various
decay channels in order to get rid of the mass-suppressed IR corrections. Based
on this observation we show that, starting from any two-body decay of the heavy
$Z'$ gauge boson, the cancellation of the mass-suppressed double-logarithmic
corrections requires the sum over the full decay width (thus enforcing the
inclusion of final states which are na\"{\i}vely unrelated to the starting
one). En route, we prove a number of technical results that are relevant for
the computation of mass-suppressed double-logarithms of IR origin. Our results
are relevant for models that enlarge the Standard Model by adding a heavy $Z'$.
