Right Operation is Left Distributive over All Operations

Theorem

Let $\struct {S, \circ, \rightarrow}$ be an algebraic structure where:

$\rightarrow$ is the right operation
$\circ$ is any arbitrary binary operation.

Then $\rightarrow$ is left distributive over $\circ$.

Proof

By definition of the right operation:

 $\ds a \rightarrow \paren {b \circ c}$ $=$ $\ds b \circ c$ $\ds$ $=$ $\ds \paren {a \rightarrow b} \circ \paren {a \rightarrow c}$

The result follows by definition of left distributivity.

$\blacksquare$