\begin{aligned}
 \sum_{i\in\omega\cdot k+l}(i+\omega)
 = & \sum_{i\in\omega}(i+\omega) + \sum_{i\in\omega}(\omega+i+\omega) + \cdots + \sum_{i\in\omega} \big(\omega\cdot (k-1)+i+\omega\big) \\
 & +(\omega\cdot k+\omega) +(\omega\cdot k + 1 + \omega)+\cdots+\big(\omega\cdot k + (l-1)+\omega\big) \\
 = & \underbrace{\omega^2+\cdots+\omega^2}_k+\omega\cdot(k+1)\cdot l \\
 = & \omega^2\cdot k+\omega\cdot(k+1)\cdot l
\end{aligned}