Even Number minus Even Number is Even

Theorem

 * If from an even number an even number be subtracted, the remainder will be even.

Proof
Let $a$ and $b$ be even numbers.

Then by definition of even number:
 * $\exists c \in \Z: a = 2 c$
 * $\exists d \in \Z: b = 2 d$

So:

Hence the result by definition of even number.