A worked proof

Theorem 1

Even plus even

If $a$ and $b$ are even integers, then $a + b$ is even.

01Let $a$ and $b$ be even integers.Assumption
02Then $a = 2k$ and $b = 2j$ for some integers $k$ , $j$ .Definition of even
03So $a + b = 2k + 2j = 2(k + j)$ .Factor out $2$
04Since $k + j$ is an integer, $a + b$ is even. $\blacksquare$ Definition of even