\lambda_k \mu_h}{m} \right) \left \| x_{k,i} - x_{\lambda_k}^* \right \|^2 - 2\gamma_k \left( f_{i+1}(x_{k,i}) + \frac{\lambda_k}{m} h(x_{k,i}) \right) \\& + 2\gamma_k \left(f_{i+1}(x_{\lambda_k}^*)+ \frac{\lambda_k}{m} h(x_{\lambda_k}^*) \right) +2 \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2).
    \end{aligned}$$ Taking summation from both sides over $i$, using $x_{k,0}=x_k,  x_{k,m}=x_{k+1}$, and that $\gamma_k \lambda_k \mu_h >0$, we obtain [[$$\begin{aligned}
    &\sum_{i=0}^{m-1} \left \|x_{k,i+1} - x_{\lambda_k}^* \right \|^2 \nonumber \\
                &\leq \left( 1- \frac{\gamma_k \lambda_k \mu_h}{m} \right) \left \|x_k - x_{\lambda_k}^* \right \|^2 + \sum_{i=1}^{m-1} \left \| x_{k,i} - x_{\lambda_k}^* \right \|^2 \nonumber \\& + 2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2) 
    - 2\gamma_k \sum_{i=0}^{m-1} \left( f_{i+1}(x_{k,i}) + \frac{\lambda_k}{m} h(x_{k,i}) \right) \nonumber\\&+ 2\gamma_k \left(f(x_{\lambda_k}^*)+ \lambda_k h(x_{\lambda_k}^*) \right), \label{ineq1 INC}
    \end{aligned}$$]{}]{} where we used the definition of function $f$ in the second inequality. Now by rearranging the terms and adding and subtracting $f(x_k)+\lambda_k h(x_k)$ we obtain [[$$\begin{aligned}
    &\left \|x_{k+1} - x_{\lambda_k}^* \right \|^2 \\ &\leq \left( 1- \frac{\gamma_k \lambda_k \mu_h}{m} \right) \left \|x_k - x_{\lambda_k}^* \right \|^2 + 2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2)\\
    & - 2\gamma_k \sum_{i=0}^{m-1} \left((f_{i+1}(x_{k,i}) - f_{i+1}(x_{k}) ) + \frac{\lambda_k}{m} (h(x_{k,i})  -h(x_{k}))  \right) \\& + 2\gamma_k \underbrace{\left(f(x_{\lambda_k}^*)+ \lambda_k h(x_{\lambda_k}^*) -f(x_k)-\lambda_k h(x_k) \right)}_{Term1}\\
    &\leq \left( 1- \frac{\gamma_k \lambda_k \mu_h}{m} \right) \left \|x_k - x_{\lambda_k}^* \right \|^2 + 