Symmetric Closure of Relation Compatible with Operation is Compatible

Theorem
Let $(S, \circ)$ be a magma.

Let $\mathcal R$ be a relation compatible with $\circ$.

Let $\mathcal R^\leftrightarrow$ be the symmetric closure of $\mathcal R$.

Then $\mathcal R^\leftrightarrow$ is compatible with $\circ$.

Proof
By the definition of symmetric closure:
 * $\mathcal R^\leftrightarrow = \mathcal R \cup \mathcal R^{-1}$.

Here $\mathcal R^{-1}$ is the inverse of $\mathcal R$.

By Inverse of Relation Compatible with Operation is Compatible, $\mathcal R^{-1}$ is compatible with $\circ$.

Thus by Union of Relations Compatible with Operation is Compatible:
 * $\mathcal R^\leftrightarrow = \mathcal R \cup \mathcal R^{-1}$ is compatible with $\circ$.