Proof: By Euclid
(related to Proposition: 7.17: Multiples of Ratios of Numbers)
 For since $A$ has made $D$ (by) multiplying $B$, $B$ thus measures $D$ according to the units in $A$ [Def. 7.15] .
 And the unit $F$ also measures the number $A$ according to the units in it.
 Thus, the unit $F$ measures the number $A$ as many times as $B$ (measures) $D$.
 Thus, as the unit $F$ is to the number $A$, so $B$ (is) to $D$ [Def. 7.20] .
 And so, for the same (reasons), as the unit $F$ (is) to the number $A$, so $C$ (is) to $E$.
 And thus, as $B$ (is) to $D$, so $C$ (is) to $E$.
 Thus, alternately, as $B$ is to $C$, so $D$ (is) to $E$ [Prop. 7.13].
 (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"