Commutative implies flexible

This article gives the statement and possibly, proof, of an implication relation between two magma properties. That is, it states that every magma satisfying the first magma property (i.e., commutative magma) must also satisfy the second magma property (i.e., flexible magma)
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Statement

Suppose $(S, *)$ is a commutative magma, i.e, we have:

$a * b = b * a \forall a, b \in S$

Then, $(S, *)$ is also a flexible magma, i.e., we have:

$x * (y * x) = (x * y) * x \forall x, y \in S$

This also shows that a commutative non-associative ring must be a Flexible ring (?).

Proof

Given: A magma $(S, *)$ where $a * b = b * a$ for all $a, b \in S$ .

To prove: $x * (y * x) = (x * y) * x$ for all $x, y \in S$ .

Proof:

By commutativity between $x$ and $y$ , we have:

$x * (y * x) = x * (x * y) (†)$

Further, by commutativity of $x$ and $x * y$ , we have:

$x * (x * y) = (X * y) * x († †)$

Combining $(†)$ and $(† †)$ , we obtain what we need to prove.