IX.33

If a number have its half odd, it is even-times odd only.

Proof

For let the number A have its half odd; I say that A is even-times odd only.

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

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 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.