I am studying "Noncommutative localization in Noncommutative Geometry" by Zoran Skoda, and, given a monoid $R$ with a set of left denominators $S$, he is constructing a monoid of left fractions.
Given a monoid $R$, a set of left denominators is a set $S\subseteq R$ such that:
$\bullet$ $1\in S$.
$\bullet$ $x,y\in S$ imply $xy\in S$.
$\bullet$ $\forall s \in S:\forall r \in R:\exists s' \in S: \exists r' \in R:(r's = s'r)$ (left Ore condition).
$\bullet$ $\forall n_1, n_2 \in R: \forall s \in S:(n_1s = n_2s \Rightarrow \exists s' \in S, s'n_1 = s'n_2)$ (left reversibility).
Let us define the relation $\sim$ on $S \times R$ by
$(s, r) \sim (s', r') \Leftrightarrow \exists\tilde{s} \in S: \exists\tilde{r} \in R:(\tilde{s}s' = \tilde{r}s$ and $\tilde{s}r' = \tilde{r}r)$.
Skoda claimed and proved that $\sim$ is an equivalence relation. I understood the proof of the reflexivity and the transitivity, but I really do not understand the proof of the symmetry, and I do not know how to correct it. It was written like this:
"If $(s,r)\sim(s',r')$, then there are $\tilde{s} \in S,\tilde{r} \in R$ such that $\tilde{s}s' = \tilde{r}s$ and $\tilde{s}r' = \tilde{r}r$, thus, by Ore, $\exists r_1 \in R, s_1 \in S$ with $r_1s = s_1s'$. Also $\exists r_2 \in R, s_2 \in S$ with $r_1s = s_1s'$. Thus
$r_2\tilde{r}s' = r_2\tilde{s}s = s_2r_1s = s_2s_1s'$.
Thus by the left reversibility, $\exists t \in S$ with $tr_2\tilde{r} = ts_2s_1$. Therefore $tr_2\tilde{r}r' = ts_2s_1r'$, hence $ts_2s_1r' = tr_2\tilde{s}r$. Compare with $ts_2s_1s' = tr_2\tilde{s}s$ to see that $(s', r') \sim (s, r)$."
I do not know how to correct the passage "$\exists r_2 \in R, s_2 \in S$ with $r_1s = s_1s'$". Moreover, in the end, the equalities $ts_2s_1r' = tr_2\tilde{s}r$ and $ts_2s_1s' = tr_2\tilde{s}s$ imply $(s,r)\sim(s',r')$ again, but I cannot see why $tr_2\tilde{s}$ would belong to $S$, so I do not know how to conclude that $(s',r')\sim(s,r)$.