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

Theorem

Let $\struct {S, \circ}$ be a magma.

Let the $(1-3)$ parastrophe of $\struct {S, \circ}$ be a magma.

Then $\struct {S, \circ}$ is a right quasigroup.

Proof

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

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

Aiming for a contradiction, 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 $\struct {S, \circ}$ we see that $\circ$ as a mapping either fails to be left-total or many-to-one for $b \circ a$.

So $\struct {S, \circ}$ is not a magma which contradicts our assumption.

$\blacksquare$