Equivalence iff Diagonal and Inverse Composite

Theorem
Let $\RR$ be a relation on $S$.

Then $\RR$ is an equivalence relation on $S$ :
 * $\Delta_S \subseteq \RR$

and:
 * $\RR = \RR \circ \RR^{-1}$

Necessary Condition
Let $\RR$ be an equivalence relation.

Then by definition, it is reflexive, symmetric and transitive.

As $\RR$ is reflexive, we have $\Delta_S \subseteq \RR$ from Relation Contains Diagonal Relation iff Reflexive.

As $\RR$ is transitive:
 * $\RR \circ \RR \subseteq \RR$

from Relation contains Composite with Self iff Transitive.

But as $\RR$ is symmetric:
 * $\RR = \RR^{-1}$

from Relation equals Inverse iff Symmetric.

Thus:
 * $\RR \circ \RR^{-1} \subseteq \RR$

Now we need to show that $\RR \subseteq \RR \circ \RR^{-1}$:

Let $\tuple {x, y} \in \RR$.

Then as $\RR$ is reflexive:
 * $\tuple {x, y} \in \RR \land \tuple {y, y} \in \RR$

and so:
 * $\tuple {x, y} \in \RR \circ \RR$

As $\RR = \RR^{-1}$, it follows that:
 * $\RR \subseteq \RR \circ \RR^{-1}$

So it has been shown:
 * $\Delta_S \subseteq \RR$ and $\RR = \RR \circ \RR^{-1}$

Sufficient Condition
Now, let $\Delta_S \subseteq \RR$ and $\RR = \RR \circ \RR^{-1}$.

From Relation Contains Diagonal Relation iff Reflexive, $\RR$ is reflexive.

Let $\tuple {x, y} \in \RR$.

Then:
 * $\tuple {x, y} \in \RR \circ \RR^{-1}$

So:
 * $\exists z \in S: \tuple {x, z} \in \RR, \tuple {z, y} \in \RR^{-1}$

and so:
 * $\tuple {y, z} \in \RR$

That is, by definition, $\RR$ is transitive.

As $\tuple {x, z} \in \RR$:
 * $\tuple {z, x} \in \RR^{-1}$

and so:
 * $\tuple {y, x} \in \RR$

So it follows that $\RR$ is symmetric.

Thus $\RR$ is reflexive, symmetric and transitive.

Therefore, by definition, $\RR$ is an equivalence relation.