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:

$$\left[\left[{\left[\left[{x}\right]\right]_{\mathcal{R}_1}}\right]\right]_{\mathcal{R}_2} \mapsto \left[\left[{x}\right]\right]_{\mathcal{R}_3}$$.