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Proof: By Euclid

(related to Proposition: Prop. 9.27: Odd Number minus Even Number is Odd)

• For let the even (number) $BC$ have been subtracted from the odd (number) $AB$.
• I say that the remainder $CA$ is odd.

• For let the unit $AD$ have been subtracted (from $AB$).
• $DB$ is thus even [Def. 7.7] .
• And $BC$ is also even.
• Thus, the remainder $CD$ is also even [Prop. 9.24].
• $CA$ (is) thus 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)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"