Left Distributive and Commutative implies Distributive

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Theorem

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

Let the operation $\circ$ be left distributive over the operation $*$.

Let $\circ$ be commutative.


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


Proof

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

Then

\(\displaystyle \left({a * b}\right) \circ c\) \(=\) \(\displaystyle c \circ \left({a * b}\right)\) $\circ$ is commutative
\(\displaystyle \) \(=\) \(\displaystyle \left({c \circ a}\right) * \left({c \circ b}\right)\) $\circ$ is left distributive over $*$
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ c}\right) * \left({b \circ c}\right)\) $\circ$ is commutative

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

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

$\blacksquare$