${\overline{\Sigma}}_{ab}^{(\alpha)}$ terms, see eq. (4.18) of [@Frixione:2002ik]. In general, this is not true for contributions with a purely collinear structure (such as the collinear counterterms to the real-emission matrix elements). Equations (\[vchange\]), (\[bStohS\]), and (\[bStohSsoft\]) allow us to rewrite I\_[[[S]{}]{}]{}\^[()]{}&=&\_[ab]{}d[x\_[1]{}\^[(s)]{}]{}d[x\_[2]{}\^[(s)]{}]{}d\_2\^. \[Itwodef2\] At this point, the two integrals in eq. (\[Itwodef2\]) correspond by construction to the same Born-like kinematic configuration (to be showered by ), and can therefore be integrated together. It is thus convenient to [*rename*]{} the integration variables ([x\_[1]{}\^[(s)]{}]{},[x\_[2]{}\^[(s)]{}]{},\_2\^) ([\_1]{},[\_2]{},\_2), and use I\_[[[S]{}]{}]{}I\_[[[S]{}]{}]{}\^[(+)]{}+I\_[[[S]{}]{}]{}\^[(-)]{} \[ISsum\] for the generation of ${{\mathbb S}}$ events. One observes that in the integrand of eq. (\[ISsum\]) there will be a term \_[L=]{}( +)= ( +), where we have made use of eq. (\[Sfunnorm\]). The same kind of simplification will occur in the sum of the two soft counterterms. As far as the collinear counterterms are concerned, we stress that the ${{{\cal S}}_{\pm}}$ terms that multiply them are equal to one (by construction of the ${{\cal S}}$ functions – see eqs. (\[Sfuncolp\]) and (\[Sfuncolm\])). We arrive therefore at the following form: I\_[[[S]{}]{}]{}&=&\_[ab]{}d[\_1]{}d[\_2]{}d\_2 . \[Itwodef3\] Equations (\[Ithreedef2\]) and (\[Itwodef3\]) are our final expressions, used for the generation of ${{\mathbb H}}$ and ${{\mathbb S}}$ events respectively. The net 