Even Integer Plus 5 is Odd/Indirect Proof

Direct Proof
Let $x$ be an even integer.

Let $y = 2 n + 5$.

Assume $y = x + 5$ is not an odd integer.

Then:
 * $y = x + 5 = 2 n$

where $n \in \Z$.

Then:

Hence $x$ is odd.

That is, it is false that $x$ is even.

It follows by the Rule of Transposition that if $x$ is even, then $y$ is odd.