Structure Induced by Permutation on Quasigroup is Quasigroup

Theorem
Let $\struct {S, \circ}$ be a quasigroup.

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.

Proof
By definition of quasigroup:


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

Let $a, b \in S$.

As $\sigma$ is a permutation, it is by definition both surjective and injective.

We have that:
 * $\exists ! x: x \circ a = b$

Thus:

Similarly, we have that:
 * $\exists ! x: a \circ x = b$

Thus:

The result follows.