Relation Partitions Set iff Equivalence/Proof

Proof
Let $\RR$ be an equivalence relation on $S$.

From the Fundamental Theorem on Equivalence Relations, we have that the equivalence classes of $\RR$ form a partition.

Let $S$ be partitioned into subsets by a relation $\RR$.

From Relation Induced by Partition is Equivalence, $\RR$ is an equivalence relation.