Self-Distributive Quasigroup with at least Two Elements is not Associative

Theorem
Let $\struct {S, \odot}$ be a self-distributive quasigroup.

Let $S$ have at least $2$ elements.

Then $\odot$ is not an associative operation.

Proof
$\odot$ is associative operation.

Let $a, b \in S$ such that $a \ne b$.

Then:

which contradicts our supposition that $a \ne b$.