described by $\Phi_R(\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})$ must match the $\Phi_B$ kinematics. In the collinear limit, the kinematics of the system obtained by replacing the two collinear partons in $\Phi_R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})$ with a single parton with the appropriate flavour, having momentum equal to the sum of the momenta of the collinear partons, must match the $\Phi_B$ kinematics. One also introduces a simplified approximation to the real cross section, $R_s$, that coincides with $R$ in the soft and collinear singular limits. Eq. (\[eq:observableO\]) is rewritten as $$\begin{aligned}
  \langle \mathcal{O} \rangle & = & \int {\mathrm{d}}\Phi_B \left[ B (\Phi_B) + V
  (\Phi_B) + \int {\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}} R_s (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})
  \right]  \mathcal{O} (\Phi_B) \\
  & + & \int {\mathrm{d}}\Phi_B {\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}} [R (\Phi_R (\Phi_B,
  \Phi_{{\ensuremath{\operatorname{rad}}}})) \mathcal{O} (\Phi_R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})) - R_s
  (\Phi_R (\Phi_B, \Phi_{{\ensuremath{\operatorname{rad}}}})) \mathcal{O} (\Phi_B)] . \nonumber
  \label{eq:subtrmeth} \end{aligned}$$ eq. (\[eq:subtrmeth\]) is clearly identical to eq. (\[eq:observableO\]). It has however the nice property that $1 / \epsilon$ divergences in the square bracket of the first term on the right-hand side of the equation (arising in the virtual term and in the ${\mathrm{d}}\Phi_{{\ensuremath{\operatorname{rad}}}}$ integration of $R_s$) cancel among each other. Furthermore, collinear and soft divergences cancel under the integral sign in the square bracket of the second 