Number whose Half is Odd is Even-Times Odd

Theorem
Let $a \in \Z$ be an integer such that $\dfrac a 2$ is an odd integer.

Then $a$ is even-times odd.

Proof
By definition:
 * $a = 2 r$

where $r$ is an odd integer.

Thus:
 * $a$ has an even divisor

and:
 * $a$ has an odd divisor.

Hence the result by definition of even-times odd integer.

As $r$ is an odd integer it follows that $2 \nmid r$.

Thus $a$ is not divisible by $4$.

Hence $a$ is not even-times even.