to re-write the soft and collinear counterterms in a pure-NLO computation, so as all counterterms associated with a given ${{\cal S}}$ contribution at the real-emission level (in the present case, ${{{\cal S}}_{+}}d\Sigma^{(f)}$ or ${{{\cal S}}_{-}}d\Sigma^{(f)}$) have the same kinematics. In order to achieve this, it turns out to be convenient to define the counterterm kinematics starting from a fixed real-emission kinematics. This situation is depicted in fig. \[fig:EvPr\] for soft counterterms (the case of collinear counterterms is essentially identical, the only differences being in the assignments of the momentum fractions of the incoming partons, which we shall specify in what follows). Since the event projections we define here are ultimately motivated by the branching strategy of , the procedure of fig. \[fig:EvPr\] is by construction the inverse of that depicted in fig. \[fig:MCbran\]. We shall return to this point in sect. \[sec:xsec\].

In order to actually obtain event projections, we use the master equations (A.36) and (A.37) of ref. [@Frixione:2002ik]. For an emission from leg 1, these equations read in our case O\_1([[**2**]{}]{}([z\_1]{},[z\_2]{},\_2))&=& O\_1([[**1**]{}]{}\_s([x\_[1+]{}\^[(s)]{}]{},[x\_[2+]{}\^[(s)]{}]{},\_2\^[(s)]{})), \[Oonecond\]\
O\_2([[**2**]{}]{}([z\_1]{},[z\_2]{},\_2))&=& O\_2([[**1**]{}]{}\_s([x\_[1+]{}\^[(s)]{}]{},[x\_[2+]{}\^[(s)]{}]{},\_2\^[(s)]{})), \[Otwocond\] and their analogues for the collinear case (where one formally replaces $s$ with $c+$ in the r.h.s. of eqs. (\[Oonecond\]) and (\[Otwocond\])). 