Proof: By Euclid
(related to Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd)
 For let any multitude whatsoever of odd numbers, $AB$, $BC$, $CD$, lie together, and let the multitude of them be odd.
 I say that the whole, $AD$, is also odd.
 For let the unit $DE$ have been subtracted from $CD$.
 The remainder $CE$ is thus even [Def. 7.7] .
 And $CA$ is also even [Prop. 9.22].
 Thus, the whole $AE$ is also even [Prop. 9.21].
 And $DE$ is a unit.
 Thus, $AD$ 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"