Even Number minus Even Number is Even

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In the words of Euclid:

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

(The Elements: Book $\text{IX}$: Proposition $24$)


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

Then by definition of even number:

$\exists c \in \Z: a = 2 c$
$\exists d \in \Z: b = 2 d$


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

Hence the result by definition of even number.


Historical Note

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