Exchange of Order of Summation

Theorem

 * $\displaystyle \sum_{\Phi_1 \left({i}\right)} \sum_{\Phi_2 \left({j}\right)} a_{i j} = \sum_{\Phi_2 \left({j}\right)} \sum_{\Phi_1 \left({i}\right)} a_{i j}$

Also known as
The word interchange can often be seen for exchange.