Structure Induced by Permutation on Commutative Quasigroup is Commutative Quasigroup

Theorem
Let $\struct {S, \circ}$ be a quasigroup such that $\circ$ is a commutative operation.

Let $\sigma: S \to S$ be a permutation on $S$.

Let $\struct {S, \circ_\sigma}$ be the structure induced by $\sigma$ on $\circ$:
 * $\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$

Then $\struct {S, \circ_\sigma}$ is also a quasigroup such that $\circ_\sigma$ is a commutative operation.

Proof
From definition of a quasigroup, we take it for granted that $\struct {S, \circ}$ is closed under $\circ$.

From Structure Induced by Permutation on Quasigroup is Quasigroup we have that $\struct {S, \circ_\sigma}$ is a quasigroup.

Again, from definition of a quasigroup, we take it for granted that $\struct {S, \circ_\sigma}$ is closed under $\circ_\sigma$.

Then we see that: