Proof: By Euclid
(related to Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements)
 For as many as $A$, $B$, $C$, $D$ are in multitude, let so many of the least numbers, $G$, $H$, $K$, $L$, having the same ratio as $A$, $B$, $C$, $D$, have been taken [Prop. 7.33].
 Thus, the outermost of them, $G$ and $L$, are prime to one another [Prop. 8.3].
 And since $A$, $B$, $C$, $D$ are in the same ratio as $G$, $H$, $K$, $L$, and the multitude of $A$, $B$, $C$, $D$ is equal to the multitude of $G$, $H$, $K$, $L$, thus, via equality, as $A$ is to $B$, so $G$ (is) to $L$ [Prop. 7.14].
 And as $A$ (is) to $B$, so $E$ (is) to $F$.
 And thus as $G$ (is) to $L$, so $E$ (is) to $F$.
 And $G$ and $L$ (are) prime (to one another).
 And (numbers) prime (to one another are) also the least (numbers having the same ratio as them) [Prop. 7.21].
 And the least numbers measure those (numbers) having the same ratio (as them) an equal number of times, the greater (measuring) the greater, and the lesser the lesser  that is to say, the leading (measuring) the leading, and the following the following [Prop. 7.20].
 Thus, $G$ measures $E$ the same number of times as $L$ (measures) $F$.
 So as many times as $G$ measures $E$, so many times let $H$, $K$ also measure $M$, $N$, respectively.
 Thus, $G$, $H$, $K$, $L$ measure $E$, $M$, $N$, $F$ (respectively) an equal number of times.
 Thus, $G$, $H$, $K$, $L$ are in the same ratio as $E$, $M$, $N$, $F$ [Def. 7.20] .
 But, $G$, $H$, $K$, $L$ are in the same ratio as $A$, $C$, $D$, $B$.
 Thus, $A$, $C$, $D$, $B$ are also in the same ratio as $E$, $M$, $N$, $F$.
 And $A$, $C$, $D$, $B$ are in continued proportion.
 Thus, $E$, $M$, $N$, $F$ are also in continued proportion.
 Thus, as many numbers as have fallen in between $A$ and $B$ in continued proportion, so many numbers have also fallen in between $E$ and $F$ in continued proportion.
 (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"