Composite of Quotient Mappings

Theorem
Let $\mathcal R_1$ be an equivalence on $S$, and $\mathcal R_2$ be an equivalence on the quotient set $S / \mathcal R_1$.

We can find an equivalence $\mathcal R_3$ on $S$ such that $\left({S / \mathcal R_1}\right) / \mathcal R_2$ is in one-to-one correspondence with $S / \mathcal R_3$ under the mapping:


 * $\left[\!\left[{\left[\!\left[{x}\right]\!\right]_{\mathcal R_1}}\right]\!\right]_{\mathcal R_2} \mapsto \left[\!\left[{x}\right]\!\right]_{\mathcal R_3}$.