# Even Number minus Even Number is Even

## Theorem

In the words of Euclid:

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:

 $\ds a - b$ $=$ $\ds 2 c - 2 d$ $\ds$ $=$ $\ds 2 \left({c - d}\right)$

Hence the result by definition of even number.

$\blacksquare$

## Historical Note

This proof is Proposition $24$ of Book $\text{IX}$ of Euclid's The Elements.