Now that it is even-times odd is manifest; for the half of it, being odd, measures it an even number of times. [[VII. Def. 9](elem.7.def.9)]

I say next that it is also even-times odd only.

For, if A is even-times even also, it will be measured by an even number according to an even number; [[VII. Def. 8](elem.7.def.8)] so that the half of it will also be measured by an even number though it is odd: which is absurd.

Therefore A is even-times odd only. Q. E. D.