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G^{<,h}_{\gamma k}(\omega+\epsilon) = \sum_\beta [G_{\gamma\beta}^r(\omega+\epsilon) V^*_{\beta k} g^{<,h}_{k}(\omega+\epsilon)+ G_{\gamma\beta}^<(\omega+\epsilon) V^*_{\beta k} g^{a,h}_{k}(\omega+\epsilon)]
\end{eqnarray}
Then, we have
\begin{eqnarray}
&&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} V_{\beta k} V^*_{\gamma q}
[G_{\beta\nu}^r(\omega+\epsilon) V_{\nu q} g^{>}_{q}(\omega+\epsilon)+ G_{\beta\nu}^>(\omega+\epsilon) V_{\nu q} g^{a}_{q}(\omega+\epsilon)]
\\ \nonumber
&&[G_{\gamma\mu}^r(\epsilon) &[G_{\gamma\mu}^r(\epsilon) V^*_{\mu k} g^{<,h}_{k}(\epsilon)+ G_{\gamma\mu}^<(\epsilon) V^*_{\mu k} g^{a,h}_{k}(\epsilon)]
\end{eqnarray} \end{align*}
\begin{eqnarray}
N^>(\omega)&=&(e^2/h)\sum_{k\beta,q\gamma, \begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\epsilon) V_{\nu q} g^{>}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k} g^{<,h}_{k}(\omega+\epsilon) V_{\beta k}]
\\
\nonumber
&&[G_{\beta\nu}^r(\epsilon) &[G_{\beta\nu}^r(\epsilon) V_{\nu q} g^{>}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\omega+\epsilon) V_{\beta k}]
\\
\nonumber
&&[G_{\beta\nu}^>(\epsilon) V_{\nu q} g^{a}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^r(\omega+\epsilon) V^*_{\mu k} g^{<,h}_{k}(\omega+\epsilon) V_{\beta k}]
\\
\nonumber
&&[G_{\beta\nu}^>(\epsilon) V_{\nu q} g^{a}_{q}(\epsilon) V^*_{\gamma q} G_{\gamma\mu}^<(\omega+\epsilon) V^*_{\mu k} g^{a,h}_{k}(\omega+\epsilon) V_{\beta k}]
\end{eqnarray} \end{align*}
Inserting the expressions for the self-energies we get
\begin{eqnarray}
N^>(\omega)&=&(e^2/h)\sum_{k\beta,q\gamma, \begin{align*}
&N^>(\omega)=(e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi}
[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^r(\epsilon+\omega) \Sigma_{0,\mu\beta}^{<,h}(\epsilon+\omega)]
\\
\nonumber
&&[G_{\beta\nu}^r(\epsilon) &[G_{\beta\nu}^r(\epsilon) \Sigma_{0,\nu\gamma}^>(\epsilon) G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)]
\\
\nonumber
&&[G_{\beta\nu}^>(\epsilon) &[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^r(\omega+\epsilon)\Sigma_{0,\mu\beta}^{<,h}(\omega+\epsilon)]
\\
\nonumber
&&[G_{\beta\nu}^>(\epsilon) &[G_{\beta\nu}^>(\epsilon) \Sigma_{0,\nu\gamma}^a(\epsilon) G_{\gamma\mu}^<(\epsilon+\omega) \Sigma_{0,\mu\beta}^{a,h}(\epsilon+\omega)]
\end{eqnarray} \end{align*}
\begin{align*}
&N^>(\omega)=(4e^2/h)\sum_{k\beta,q\gamma, \nu\mu} \int \frac{d\epsilon}{2\pi} \times \Biggr\{
