single-$\theta$ part of the result, we first group together terms according to the argument of the $\theta(\mu-E_i)$ function they contain, which clearly correspond to all the fermionic propagators in the original graph. Singling out one of them (and the associated $-\frac{1}{2E_i}$ factor), we note that it is multiplied by a sum of terms, each of which contains a product of $N-1$ factors of $1/(2E_j)$ as well as the product of the rest of the propagators with the $E_i$ and $E_j$ lines placed on shell. Recalling the result of the first part of our proof, we recognize this as the result for an $N-1$ -loop $n+2$ -point function that is obtained by removing the line $i$ from the original graph, i.e. $$\begin{aligned}
I&=&\cdots -\frac{\theta(\mu-E_i)}{2E_i} \times \Big(\text{original graph with $Q_i$-propagator removed}\Big)\Big{|}_{q_0^i=iE_i} + \cdots \, , \nonumber \\\end{aligned}$$ so that the sum of all such terms exactly corresponds to the sum of all 1-cut graphs in the cutting rules. In evaluating this expression, the $Q_i$ line clearly must be placed on-shell only after computing the associated $n+2$ -point function, as one of the assumptions of the $\mu=0$ cutting rules above was that all external momenta in the original graph be real-valued (modulo a 