LC implies left alternative

This article gives the statement and possibly, proof, of an implication relation between two algebra loop properties. That is, it states that every algebra loop satisfying the first algebra loop property (i.e., LC-loop) must also satisfy the second algebra loop property (i.e., left alternative loop)
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Statement

Any LC-loop is a left alternative loop.

Definitions used

LC-loop

Further information: 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

Further information: 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:

$(x*x)*y=x*(x*y)$