Proof: By Euclid
(related to Proposition: 7.27: Powers of Coprime Numbers are Coprime)
Euclid's' Proof
 For since $A$ and $B$ are prime to one another, and $A$ has made $C$ (by) multiplying itself, $C$ and $B$ are thus prime to one another [Prop. 7.25].
 Therefore, since $C$ and $B$ are prime to one another, and $B$ has made $E$ (by) multiplying itself, $C$ and $E$ are thus prime to one another [Prop. 7.25].
 Again, since $A$ and $B$ are prime to one another, and $B$ has made $E$ (by) multiplying itself, $A$ and $E$ are thus prime to one another [Prop. 7.25].
 Therefore, since the two numbers $A$ and $C$ are both prime to each of the two numbers $B$ and $E$, the (number) created from (multiplying) $A$ and $C$ is thus prime to the (number created) from (multiplying) $B$ and $E$ [Prop. 7.26].
 And $D$ is the (number created) from (multiplying) $A$ and $C$, and $F$ the (number created) from (multiplying) $B$ and $E$.
 Thus, $D$ and $F$ are prime to one another.
 (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"