Even Number minus Odd Number is Odd

Theorem

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

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

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

and by definition of odd number:
 * $\exists d \in \Z: b = 2 d + 1$

So:

Hence the result by definition of odd number.