that the two Born matrix elements appearing implicitly in ${\overline{\Sigma}}_{ab}^{(\pm)}$ are evaluated in two different kinematics configurations, that eventually result after the first branching in the same real-emission kinematics . This situation is precisely the one depicted in fig. \[fig:EvPr\].

A simplification analogous to that of eq. (\[Ithreedef2\]) is also possible in the case of ${{\mathbb S}}$ events, although the argument is slightly more involved. If one fixes (i.e., the real-emission kinematics), the kinematic configurations associated with $I_{{{\mathbb S}}}^{(+)}$ and $I_{{{\mathbb S}}}^{(-)}$ in eq. (\[Itwodef\]) are different, owing to the fact that the two integrals are computed using different event projections. Formally, one can introduce two mappings, ${{\cal P}_{{{\mathbb H}}\to{{\mathbb S}}}^{(+)}}$ and ${{\cal P}_{{{\mathbb H}}\to{{\mathbb S}}}^{(-)}}$, representing these event projections; for a given real-emission kinematic configuration ${{\bf 2}}$, the two Born-like configurations associated with $I_{{{\mathbb S}}}^{(+)}$ and $I_{{{\mathbb S}}}^{(-)}$ can be denoted by [[**1**]{}]{}\^+=[[P]{}\_[[[H]{}]{}]{}\^[(+)]{}]{}[[**2**]{}]{}, [[**1**]{}]{}\^-=[[P]{}\_[[[H]{}]{}]{}\^[(-)]{}]{}[[**2**]{}]{}\[KonefKtwo\] respectively; ${{\bf 1}}^+$ and ${{\bf 1}}^-$ are the hard configurations to be showered by . The ${{\cal P}_{{{\mathbb H}}\to{{\mathbb S}}}^{(\pm)}}$ mappings are pictorially represented in fig. \[fig:EvPr\] by the two thick arrows, with ${{\bf 2}}$, ${{\bf 1}}^+$, and ${{\bf 1}}^-$ the configurations depicted in the upper part, lower left corner, and lower right corner of 