Proof: By Euclid
(related to Proposition: 7.11: Proportional Numbers have Proportional Differences)
- (For) since as $AB$ is to $CD$, so $AE$ (is) to $CF$, thus which(ever) part, or parts, $AB$ is of $CD$, $AE$ is also the same part, or the same parts, of $CF$ [Def. 7.20] .
- Thus, the remainder $EB$ is also the same part, or parts, of the remainder $FD$ that $AB$ (is) of $CD$ [Prop. 7.7], [Prop. 7.8].
- Thus, as $EB$ is to $FD$, so $AB$ (is) to $CD$ [Def. 7.20] .
- (Which is) the very thing it was required to show.
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Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"