LC implies left nuclear square

Statement
Any LC-loop is a left nuclear square loop.

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

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

Left nuclear square loop
An algebra loop $$(L,*)$$ is termed a left nuclear square loop if every square element is in the left nucleus. In other words, for all $$x,y,z \in L$$, we have:

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

Related facts

 * LC implies left alternative
 * LC implies middle nuclear square

Facts used

 * 1) uses::LC implies left alternative: This states that any LC-loop is a left alternative loop, i.e., it satisfies the following universal identity:

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

Proof
Given: An algebra loop $$(L,*)$$ that is LC, i.e., for all $$x,y,z \in L$$, we have: $$\! (x * x) * (y * z) = (x * (x * y)) * z$$

To prove: $$L$$ is a left nuclear square loop, i.e., for all $$x,y,z \in L$$, we have: $$\! (x * x) * (y * z) = ((x * x) * y) * z$$

Proof: By the LC condition, we have, for all $$x,y,z \in L$$, that:

$$\! (x * x) * (y * z) = (x * (x * y)) * z \ \qquad (1)$$

Further, by fact (1), $$L$$ is left alternative, so we obtain that:

$$x * (x * y) = (x * x) * y \ \qquad (2)$$

Plugging (2) into the right side of (1) gives:

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

as desired.