LC implies left alternative

Statement
Any LC-loop is a left alternative loop.

LC-loop
An algebra loop $$(L,*)$$ is termed a LC-loop if it satisfies the following identity for all $$x,y,z \in L$$:

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

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

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

Proof
Given: An algebra loop $$(L,*)$$ such that $$\! (x * x) * (y * z) = (x * (x * y)) * z$$ for all $$x,y,z \in L$$.

To prove: $$\! (x * x) * y = x * (x * y)$$ for all $$x,y \in L$$

Proof: We set $$z$$ to be the identity element $$e$$ for $$L$$, and obtain that, for all $$x,y \in L$$, we have:

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

Simplifying this gives:

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