Composition of Symmetric Relation with Itself is Union of Products of Images

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

Then:
 * $\RR \circ \RR = \ds \bigcup_{s \mathop \in S} \map \RR s \times \map \RR s$

where
 * $\RR \circ \RR$ is the composition of $\RR$ with itself
 * $\map \RR s$ is the image of $s$ under $\RR$
 * $\map \RR s \times \map \RR s$ is the Cartesian product of $\map \RR s$ with itself

Proof
We have: