Proof: By Euclid

For since $A$ and $B$ are each prime to $C$, the (number) created from (multiplying) $A$ and $B$ will thus also be prime to $C$ [Prop. 7.24].
And $E$ is the (number) created from (multiplying) $A$ and $B$.
Thus, $E$ and $C$ are prime to one another.
So, for the same (reasons), $E$ and $D$ are also prime to one another.
Thus, $C$ and $D$ are each prime to $E$.
Thus, the (number) created from (multiplying) $C$ and $D$ will also be prime to $E$ [Prop. 7.24].
And $F$ is the (number) created from (multiplying) $C$ and $D$.
Thus, $E$ and $F$ are prime to one another.
(Which is) the very thing it was required to show.
∎

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