Left Operation is Right Distributive over All Operations

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Theorem

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

$\leftarrow$ is the left operation
$\circ$ is any arbitrary binary operation.

Then $\leftarrow$ is right distributive over $\circ$.


Proof

\(\ds \forall a, b, c \in S: \paren {a \circ b} \leftarrow c\) \(=\) \(\ds a \circ b\) Definition of Left Operation
\(\ds \) \(=\) \(\ds \paren {a \leftarrow c} \circ \paren {b \leftarrow c}\) Definition of Left Operation

The result follows by definition of right distributivity.

$\blacksquare$


Sources