Left Quasigroup if (2-3) Parastrophe of Magma is Magma

Theorem

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

Let the $\paren {2 - 3}$ parastrophe of $\struct {S, \circ}$ be a magma.

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

Proof

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

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

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

Then in the $\paren {2 - 3}$ parastrophe of $\struct {S, \circ}$ we see that $\circ$ as a mapping either fails to be left-total or many-to-one for $a \circ b$.

So $\struct {S, \circ}$ is not a magma.

This contradicts our assumption.

$\blacksquare$