cancelling in a localization

Let \(R\) be commutative ring, \(S \subseteq R\) a multiplicatively closed set and \(S^{-1}R\) be a localization. Then for \(r \in R\) and \(s,s' \in S\) it holds, that

\begin{align*} \frac{r \cdot s'}{s \cdot s'} = \frac{r}{s} \end{align*}

By Definition,

\begin{align*} 1 \cdot \left( \left(r \cdot s' \right) \cdot s - r \cdot \left(s \cdot s' \right) \right) =& 1 \cdot \left(r \cdot s' \cdot s - r \cdot s' \cdot s\right) \\ =& 1 \cdot 0 \\ =& 0 \end{align*}