A special case of Tonelli's theorem is in the interchange of the summations, as in formula_9, where formula_10 are non-negative for all "x" and "y". The crux of the theorem is that the interchange of order of summation holds even if the series diverges. In effect, the only way a change in order of summation can change the sum is when there exist some subsequences that diverge to formula_11 and others diverging to formula_12. With all elements non-negative, this does not happen in the stated example.