Proof: By Euclid
(related to Proposition: Prop. 9.25: Even Number minus Odd Number is Odd)
 For let the odd (number) $BC$ have been subtracted from the even number $AB$.
 I say that the remainder $CA$ is odd.
 For let the unit $CD$ have been subtracted from $BC$.
 $DB$ is thus even [Def. 7.7] .
 And $AB$ is also even.
 And thus the remainder $AD$ is even [Prop. 9.24].
 And $CD$ is a unit.
 Thus, $CA$ is odd [Def. 7.7] .
 (Which is) the very thing it was required to show.
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"