Proof
(related to Lemma: Composition of Relations Preserves Their RightUniqueness Property)
Let \(A,B,C\) be sets and let let \(R_1\subseteq A\times B\) and \(R_2\subseteq B\times C\) be two rightunique relations.
Case \((1)\)
 Suppose, there exist at least one tripple $(a,b,c)$ with $a\in A,$ $b\in B,$ and $c\in C$ such that $aR_1b$ and $bR_2c$.
 From the rightuniqueness of $R_1$ it follows that $b$ is uniquely determined by $a$.
 From the rithtuniqueness of $R_2$ it follows that $c$ is uniquely determined by $b$.
 Alltogether, $c$ is uniquely determined by $a$.
 From the definition of composition of relations, it follows that $(R_2\circ R_1)\subseteq A\times C$ is rightunique.
Case \((2)\):
 Now suppose that such a tripple $(a,b,c)$ does not exist.
 Then the lemma is vacuously true.
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Knauer Ulrich: "Diskrete Strukturen  kurz gefasst", Spektrum Akademischer Verlag, 2001