\[fig:EvPr\]). On the other hand, the short-distance cross sections for ${{\mathbb S}}$ events we started from, eq. (\[Itwodef\]), are associated with two different kinematic configurations, denoted by ${{\bf 1}}^\pm$ in eq. (\[KonefKtwo\]) (i.e. the lower part of fig. \[fig:EvPr\]). Thanks to the procedure described above, we have manipulated eq. (\[Itwodef\]) precisely to be able to associate the two $\pm$ contributions with the same kinematic configuration, denoted by ${{\bf 1}}$ in eq. (\[invmap\]) (i.e. the upper part of fig. \[fig:MCbran\]). Owing to the properties of event projections that we have discussed above, the whole procedure can obviously be carried out for any final-state multiplicity, with ${{\bf 2}}$ and ${{\bf 1}}$ formally replaced by the real-emission and Born-level configurations respectively. Finally, we stress that the manipulations performed here will also be valid in the case of strongly-interacting particles in the final state. The only difference is that, in such a case, eq. (\[Sfunnorm\]) will not hold any longer, and therefore the quantity will appear in eqs. (\[Ithreedef2\]) and (\[Itwodef3\]) as a factor multiplying the real-emission, soft counterterms, soft-virtual, and Born contributions (on the other hand, the purely collinear terms will be unchanged, owing to the fact that eqs. (\[Sfuncolp\]) and (\[Sfuncolm\]) hold 