Es sei ${\displaystyle {}R}$ ein kommutativer Halbring und seien ${\displaystyle {}x_{1},\ldots ,x_{k}\in R}$ Elemente und ${\displaystyle {}n\in \mathbb {N} }$. Zeige

${\displaystyle {}{\left(x_{1}+\cdots +x_{k}\right)}^{n}=\sum _{r_{1}+\cdots +r_{k}=n}{\binom {n}{r_{1},\dotsc ,r_{k}}}x_{1}^{r_{1}}\cdots x_{k}^{r_{k}}\,.}$