Left Distributive and Commutative implies Distributive

Theorem
Let $\left({S, \circ, *}\right)$ be an algebraic structure.

Suppose that the operation $\circ$ is left distributive over the operation $*$.

Suppose also that the operation $\circ$ is commutative.

Then the operation $\circ$ is distributive over the operation $*$.

Proof
Let $a,b,c \in S$.

Then

So $\circ$ is right distributive over $*$.

Since $\circ$ is both left distributive and right distributive over $*$, it is distributive over $*$.