Proof: By Euclid
(related to Proposition: 7.36: Least Common Multiple of Three Numbers)
 For let the least (number), $D$, measured by the two (numbers) $A$ and $B$ have been taken [Prop. 7.34].
 So $C$ either measures, or does not measure, $D$.
 Let it, first of all, measure ($D$).
 And $A$ and $B$ also measure $D$.
 Thus, $A$, $B$, and $C$ (all) measure $D$.
 So I say that ($D$ is) also the least (number measured by $A$, $B$, and $C$).
 For if not, $A$, $B$, and $C$ will (all) measure [some] number which is less than $D$.
 Let them measure $E$ (which is less than $D$).
 Since $A$, $B$, and $C$ (all) measure $E$ then $A$ and $B$ thus also measure $E$.
 Thus, the least (number) measured by $A$ and $B$ will also measure [$E$] [[Prop. 7.35]]bookofproofs$2365.
 And $D$ is the least (number) measured by $A$ and $B$.
 Thus, $D$ will measure $E$, the greater (measuring) the lesser.
 The very thing is impossible.
 Thus, $A$, $B$, and $C$ cannot (all) measure some number which is less than $D$.
 Thus, $A$, $B$, and $C$ (all) measure the least (number) $D$.
 So, again, let $C$ not measure $D$.
 And let the least number, $E$, measured by $C$ and $D$ have been taken [Prop. 7.34].
 Since $A$ and $B$ measure $D$, and $D$ measures $E$, $A$ and $B$ thus also measure $E$.
 And $C$ also measures [$E$].
 Thus, $A$, $B$, and $C$ [also] measure $E$.
 So I say that ($E$ is) also the least (number measured by $A$, $B$, and $C$).
 For if not, $A$, $B$, and $C$ will (all) measure some (number) which is less than $E$.
 Let them measure $F$ (which is less than $E$).
 Since $A$, $B$, and $C$ (all) measure $F$, $A$ and $B$ thus also measure $F$.
 Thus, the least (number) measured by $A$ and $B$ will also measure $F$ [Prop. 7.35].
 And $D$ is the least (number) measured by $A$ and $B$.
 Thus, $D$ measures $F$.
 And $C$ also measures $F$.
 Thus, $D$ and $C$ (both) measure $F$.
 Hence, the least (number) measured by $D$ and $C$ will also measure $F$ [Prop. 7.35].
 And $E$ is the least (number) measured by $C$ and $D$.
 Thus, $E$ measures $F$, the greater (measuring) the lesser.
 The very thing is impossible.
 Thus, $A$, $B$, and $C$ cannot measure some number which is less than $E$.
 Thus, $E$ (is) the least (number) which is measured by $A$, $B$, and $C$.
 (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"