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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\"ıvely 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 $Z'$.