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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}(\epsilon) [V^*_{k\alpha} g^<_{k}(\omega)V_{k\delta} + V_{k\alpha} g^{h,<}_{k}(\omega) V^*_{k\delta}]G^a_{\delta \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 G^r_{\beta  \alpha}(\epsilon) [V^*_{k\alpha} g^<_{k}(\epsilon)V_{k\delta} [V^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta}  + V_{k\alpha} g^{h,<}_{k}(\epsilon) V^*_{k\delta}]G^a_{\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^{r}_{\gamma\beta}(\omega+\epsilon)V_{\beta q}^* g_{q}^{h,<}(\omega) \nonumber \\  &&+[G^r_{\beta &&+G^r_{\beta  \alpha}(\epsilon) [V^*_{k\alpha} g^<_{k}(\epsilon)V_{k\delta} [V^*_{p\alpha} g^<_{p}(\epsilon)V_{p\delta}  + V_{k\alpha} g^{h,<}_{k}(\epsilon) V^*_{k\delta}]G^a_{\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_{k}^{h,<}(\omega+\epsilon) &&+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)\,, g_{q}^{h,a}(\omega+\epsilon)]\,,  \end{eqnarray}