Entropic Structure with Identity is Commutative Monoid

Theorem
Let $\struct {S, \odot}$ be a magma.

Let $\struct {S, \odot}$ be an entropic structure:
 * $\forall a, b, c, d \in S: \paren {a \odot b} \odot \paren {c \odot d} = \paren {a \odot c} \odot \paren {b \odot d}$

Let $\struct {S, \odot}$ have an identity element $e$.

Then $\struct {S, \odot}$ is a commutative monoid.

Proof
We have that $\struct {S, \odot}$ is a magma.

Thus $\struct {S, \odot}$ is closed, and  is fulfilled.

Then:

Thus $\odot$ is an associative operation and is fulfilled.

We are given that $e$ is an identity element for $\struct {S, \odot}$.

Hence is fulfilled.

Thus we have that $\struct {S, \odot}$ is a monoid.

Then:

Thus $\struct {S, \odot}$ is a commutative monoid.