# Left Distributive and Commutative implies Distributive

## Theorem

Let $\struct {S, \circ, *}$ 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

 $\ds \paren {a * b} \circ c$ $=$ $\ds c \circ \paren {a * b}$ $\circ$ is commutative $\ds$ $=$ $\ds \paren {c \circ a} * \paren {c \circ b}$ $\circ$ is left distributive over $*$ $\ds$ $=$ $\ds \paren {a \circ c} * \paren {b \circ c}$ $\circ$ is commutative

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

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

$\blacksquare$