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...
\end{eqnarray}
We need to compute the following product of Green functions: $G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$
\begin{eqnarray}
&&G^>_{\beta\gamma}(\epsilon) G_{kq}^{h,<}(\omega+\epsilon) = \sum_{k\alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{k\alpha}
g^<_{k}(\omega)V_{\delta g^>_{k}(\omega)V_{\delta k} + V_{\alpha k}
g^{h,<}_{k}(\omega) g^{h,>}_{k}(\omega) V^*_{\delta k}]G^a_{\delta \gamma}(\epsilon) g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} \\ \nonumber
&+& \sum_{p\beta\gamma\alpha\gamma} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p}
g^<_{p}(\epsilon)V_{ g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p}
g^{h,<}_{p}(\epsilon) g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [g_{q}^{h,r}(\omega+\epsilon) V_{\gamma q} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta k}^* g_{k}^{h,<}(\omega) \nonumber
\\
&&+G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p}
g^<_{p}(\epsilon)V_{ g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p}
g^{h,<}_{p}(\epsilon) g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [g_{q}^{h,r}(\omega+\epsilon) V_{\gamma q} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta k}^* g_{k}^{h,a}(\omega+\epsilon)]\\ \nonumber
&&+G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p}
g^<_{p}(\epsilon)V_{ g^>_{p}(\epsilon)V_{ \delta}p + V_{\alpha p}
g^{h,<}_{p}(\epsilon) g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon)[g_{q}^{h,<}(\omega+\epsilon) V_{\gamma q} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta k}^* g_{k}^{h,a}(\omega+\epsilon)]\,,
\end{eqnarray}
We now compute separately the different parts of the previous expression for the ac noise
\begin{eqnarray}
S^{>,1}(\omega)= \frac{e^2}{\hbar^2}\int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p}
g^<_{p}(\epsilon)V_{ g^>_{p}(\epsilon)V_{ \delta p} + V_{\alpha p}
g^{h,<}_{p}(\epsilon) g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon)[V_{\beta k} g_{q}^{h,<}(\omega+\epsilon)V^*_{\gamma q}\delta_{kq}
\end{eqnarray}
\begin{eqnarray}
S^{>,2}(\omega) = \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) [V_{\alpha p}^*
g^<_{p}(\epsilon)V_{\delta g^>_{p}(\epsilon)V_{\delta p} + V_{\alpha p}
g^{h,<}_{p}(\epsilon) g^{h,>}_{p}(\epsilon) V_{\delta p}^*]G^a_{\delta \gamma}(\epsilon) [V_{\gamma q}^* g_{q}^{h,r}(\omega+\epsilon) V_{\gamma q} G^{<}_{\gamma\beta}(\omega+\epsilon) V_{\beta k} g_{k}^{h,a}(\omega+\epsilon) V_{\beta k}^*]
\end{eqnarray}
\begin{eqnarray}
S^{>,3}(\omega)= \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \sum_{k,q,p\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon) [V^*_{\alpha p}
g^<_{p}(\epsilon)V_{ g^>_{p}(\epsilon)V_{ \delta p} + V_{\alpha p}
g^{h,<}_{p}(\epsilon) g^{h,>}_{p}(\epsilon) V^*_{\delta p}]G^a_{\delta \gamma}(\epsilon) [V_{\gamma q}^* g_{k}^{h,r}(\omega+\epsilon) V^*_{\gamma q} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\gamma q} g_{q}^{h,a}(\omega+\epsilon) V^*_{\gamma q}]
\end{eqnarray}
\begin{eqnarray}
S^{>,4}(\omega)= \frac{e^2}{\hbar^2} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \sum_{k,q,p\beta\gamma\alpha\delta} G^r_{\beta \alpha}(\epsilon)[V^*_{\alpha p}
g^<_{p}(\epsilon)V_{ g^>_{p}(\epsilon)V_{ \delta p } + V_{\alpha p}
g^{h,<}_{p}(\epsilon) g^{h,>}_{p}(\epsilon) V^*_{\delta p}] G^a_{\delta \gamma}(\epsilon)[V_{\beta k} g_{k}^{h,<}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)V^*_{\gamma q}]
\end{eqnarray}
