Odd Number minus Even Number is Odd

Theorem

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

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

Then by definition of odd number:
 * $\exists c \in \Z: a = 2 c + 1$

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

So:

Hence the result by definition of odd number.