LC implies middle nuclear square

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

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

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

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

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

Related facts

 * LC implies left alternative
 * LC implies left nuclear square