Associative Operation/Examples/Associative/x circ a circ y

Example of Associative Operation
Let $\struct {S, \circ}$ be an algebraic structure where $\circ$ is an associative operation.

Let $a \in S$ be an arbitrary element of $S$.

Let $*$ be the operation defined on $S$ by:


 * $\forall x, y \in S: x * y := x \circ a \circ y$

Then $*$ is associative on $S$.

Proof
Let $x, y, z \in S$ be arbitrary.