Moufang's theorem

Statement
Suppose $$L$$ is a fact about::Moufang loop. Suppose $$x,y,z$$ are (not necessarily distinct) elements of $$L$$ such that $$\! x * (y * z) = (x * y) * z$$. Then, the subloop of $$L$$ generated by $$x,y,z$$ is a group.

Appplications

 * Moufang implies diassociative

Similar facts

 * Artin's generalized theorem on alternative rings