Left Bol implies left-inverse property

Statement
Any left Bol loop is an inverse property loop.

Left Bol loop
An algebra loop $$(L,*)$$ is termed a left Bol loop if it satisfies the following for all $$x,y,z$$:

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

Left-inverse property loop
An algebra loop $$(L,*)$$ is termed a left-inverse property loop if there exists a bijection $$\lambda:L \to L$$ such that:

$$\lambda(x) * (x * y) = y \ \forall \ x,y \in L$$

Proof
Given: A left Bol loop $$(L,*)$$

To prove: We can define an inverse map on $$L$$ satisfying the conditions.

Proof: For any $$x$$, define $$\lambda(x)$$ as the unique element $$y$$ such that $$y * x = e$$. Now, plug in $$y = \lambda(x)$$ but $$z$$ arbitrary in the identityfor the left Bol loop, we obtain:

$$\! x * (\lambda(x) * (x * z)) = (x * (x^{-1} * x)) * z$$

This simplifies to:

$$\! x * (\lambda(x) * (x * z)) = x * z$$

Canceling $$x$$ from both sides, we obtain that:

$$\! \lambda(x) * (x * z) = z$$

Replacing the arbitrary letter $$z$$ by the arbitrary letter $$y$$, we obtain:

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

This completes the proof.