\begin{aligned}
\sum_{i\in\omega\cdot 2} (i+\omega\cdot 2)
& = \sum_{i\in\omega} (i+\omega\cdot 2) + \sum_{i\in\omega} (\omega+i+\omega\cdot 2) \\
& = \sup_{n\in\omega}\sum_{i\in n} (i+\omega\cdot 2) + \sup_{n\in\omega}\sum_{i\in n} (\omega+i+\omega\cdot 2) \\
& = \sup_{n\in\omega}(\omega\cdot 2\cdot n)+\sup_{n\in\omega}(\omega\cdot 3\cdot n) \\
& = \omega^2+\omega^2 \\
& = \omega^2\cdot 2
\end{aligned}
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