result of the various manipulations carried out in this section is that it is still possible to treat simultaneously the two initial-state collinear singularities, as was the case when matching with . We point out that the fact that eqs. (\[Ithreedef2\]) and (\[Itwodef3\]) do not depend upon the ${{\cal S}}$ functions is not a property of $V$-boson production, but applies as well to all processes whose Born-level final-state particles are all colour singlets[^2]. In fact, for this independence of ${{\cal S}}$ functions to occur, we only need eq. (\[Sfunnorm\]) to hold, which is true in the cases just mentioned (see ref. [@Frederix:2009yq] for further details on the construction of ${{\cal S}}$ functions). Clearly, in order to make use of eq. (\[Sfunnorm\]), ${{{\cal S}}_{+}}$ and ${{{\cal S}}_{-}}$ need be computed for the same kinematic configuration. This is in fact what happens, in spite of the fact that in the intermediate steps of the event-projection procedure we had to treat separately the $+$ and $-$ contributions. For ${{\mathbb H}}$ events this is trivially true by construction – both $I_{{{\mathbb H}}}^{(\pm)}$ in eq. (\[Ithreedef\]) are associated with the kinematic configuration denoted by ${{\bf 2}}$ in eq. (\[KonefKtwo\]) (i.e. the upper part of fig. 