Proof: By Euclid

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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