Localization of Ring Exists/Lemma 2

Lemma
The operations $+$ and $\cdot$ are well defined on $A_S$.

Proof
Let $a/s = c/u$, $b/t = d/v$ be two sets of representatives for two distinct equivalence classes in $A_S$.

We have $w, z \in S$ such that $(au - cs)w = 0$ and $(bv - dt)z = 0$. Therefore


 * $zw\left[(au - cs)w - (bv - dt)z\right] = 0$

and


 * $zw\left[(at + bs)uv - (cv + du)st\right]$. So


 * $(at + bs,st) \sim (cv + du,uv)$. That is,


 * $\displaystyle \frac{at + bs}{st} = \frac{cv + du}{uv}$

For multiplication, with $z,w$ as above we have $(abuv - cdst)zw = 0$.

So $(ab,st) \sim (dc,uv)$ and $\displaystyle \frac{ab}{st} = \frac{dc}{uv}$.