Left Bol implies flexible

Verbal statement
Any left Bol loop is a flexible loop.

Slightly more general version for magmas
Any fact about::left Bol magma with neutral element (i.e., a fact about::left Bol magma with a fact about::neutral element) is a fact about::flexible magma.

Related facts

 * Right Bol implies flexible
 * Left Bol implies left alternative
 * Right Bol implies right alternative

Proof
We prove the statement for magmas; the same proof works for loops, since the other conditions needed to make a loop are undisturbed.

Given: A magma $$(L,*)$$ with neutral element $$e$$ satisfying the following identity for all $$x,y,z \in L$$:

$$\! x * (y * (x * z)) = (x * (y * x)) * z$$.

To prove: For all $$x,y \in L$$, we have $$(x * y) * x = x * (y * x)$$.

Proof: Set $$z = e$$ in the identity, and we obtain:

$$\! x * (y * (x * e)) = (x * (y * x)) * e$$

Using that $$e$$ is a neutral element, we obtain:

$$\! x * (y * x) = (x * y) * x$$

as desired.