Equivalence Class is Unique

Theorem
Let $$\mathcal R$$ be an equivalence relation on $$S$$.

For each $$x \in S$$, the one and only one $\mathcal R$-class to which $$x$$ belongs is $$\left[\!\left[{x}\right]\!\right]_{\mathcal R}$$.

Proof
This follows directly from the fact that the set of $$\mathcal R$$-classes forms a partition of $$S$$.