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\end{eqnarray}  On the other hand we have for $G^>_{\beta\gamma}(\epsilon)$ (accordingly with J.S note)  \begin{eqnarray}  G^>_{\beta\gamma}(\epsilon) = \sum_{k\alpha\delta} \int \frac{d\epsilon}{2\pi} G^r_{\beta \alpha}(\pesilon) \alpha}(\epsilon)  [V^*_{k\alpha} g^<_{k}(\epsilon)V_{k\delta} + V_{k\alpha} g^{h,<}_{k}(\epsilon) V^*_{k\delta}]G^a_{\delta \gamma}(\epsilon) \end{eqnarray}  We need to compute the following product of Green functions: $G^>_{\beta\gamma}(\epsilon)G_{kq}^{h,<}(\omega+\epsilon)$  \begin{eqnarray} 
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&&+G^r_{\beta \alpha}(\epsilon) [V^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta} + V_{p\alpha} g^{h,<}_{p}(\epsilon) V^*_{p\delta}]G^a_{\delta \gamma}(\epsilon) [g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)]\\ \nonumber  &&+G^r_{\beta \alpha}(\epsilon) [V^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta} + V_{p\alpha} g^{h,<}_{p}(\epsilon) V^*_{p\delta}]G^a_{\delta \gamma}(\epsilon)[g_{k}^{h,<}(\omega+\epsilon) V_{\gamma k} G^{a}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)]\,,  \end{eqnarray}  We now compute separately the different parts of the previous expression for the ac noise