Right Quasigroup if (1-3) Parastrophe of Magma is Magma

Theorem

Let $\left({S, \circ}\right)$ be a magma.

If the $(1-3)$parastrophe of $\left({S, \circ}\right)$ is a magma then $\left({S, \circ}\right)$ is a right quasigroup.

Proof

By the definition of a right quasigroup it must be shown that:

$\forall a, b \in S: \exists ! x \in S: x \circ a = b$

Suppose there exists $a, b \in S$ such that $x \circ a = b$ does not have a unique solution for $x$.

Then in the $(1-3)$parastrophe of $(S, \circ )$ we see that $\circ$ as a mapping either fails to be left-total or many-to-one for $b \circ a$.

So $(S, \circ)$ is not a magma which contradicts our assumption.

$\blacksquare$