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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} \alpha}(\pesilon)  [V^*_{k\alpha} g^<_{k}(\epsilon)V_{k\delta} + V_{k\alpha} g^{h,<}_{k}(\epsilon) V^*_{k\delta}]G^a_{\delta \gamma} \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}  &&G^>_{\beta\gamma}(\epsilon) G_{kq}^{h,<}(\omega+\epsilon) = \sum_{k\alpha\delta} \int \frac{d\omega}{2\pi} G^r_{\beta \alpha} \alpha}(\epsilon)  [V^*_{k\alpha} g^<_{k}(\omega)V_{k\delta} + V_{k\alpha} g^{h,<}_{k}(\omega) V^*_{k\delta}]G^a_{\delta \gamma}] \gamma}(\epsilon)  g_{q}^{h,<}(\omega+\epsilon)\delta_{kq} \\ \nonumber &+& \sum_{p\beta\gamma\alpha\gamma}\int \frac{d\omega}{2\pi} [G^r_{\beta \alpha} \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}] \gamma}(\epsilon)  [g_{k}^{h,r}(\omega+\epsilon) V_{\gamma k} G^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,<}(\omega) \nonumber \\  &&+g_{k}^{h,r}(\omega+\epsilon) &&+[G^r_{\beta \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) [g_{k}^{h,r}(\omega+\epsilon)  V_{\gamma k} G^{<}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,a}(\omega+\epsilon)+g_{k}^{h,<}(\omega+\epsilon) g_{q}^{h,a}(\omega+\epsilon)]\\ \nonumber  &&+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}