Proof: By Euclid
(related to Proposition: 7.37: Integer Divided by Divisor is Integer)
 For as many times as $B$ measures $A$, so many units let there be in $C$.
 Since $B$ measures $A$ according to the units in $C$, and the unit $D$ also measures $C$ according to the units in it, the unit $D$ thus measures the number $C$ as many times as $B$ (measures) $A$.
 Thus, alternately, the unit $D$ measures the number $B$ as many times as $C$ (measures) $A$ [Prop. 7.15].
 Thus, which(ever) part the unit $D$ is of the number $B$, $C$ is also the same part of $A$.
 And the unit $D$ is a part of the number $B$ called the same as it (i.e., a $B$th part).
 Thus, $C$ is also a part of $A$ called the same as $B$ (i.e., $C$ is the $B$th part of $A$).
 Hence, $A$ has a part $C$ which is called the same as $B$ (i.e., $A$ has a $B$th part).
 (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"