Right Operation is Left Distributive over All Operations

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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.

$\blacksquare$


Sources