Moufang implies diassociative

Statement
Any Moufang loop is a diassociative loop.

Similar facts

 * Artin's theorem on alternative rings

Facts used

 * 1) uses::Moufang implies alternative: A Moufang loop is an alternative loop, i.e., it satisfies the left alternative law $$x * (x * y) = (x * x) * y$$ and the right alternative law $$x * (y * y) = (x * y) * y$$.
 * 2) uses::Moufang's theorem: This states that if $$L$$ is a Moufang loop, and $$x,y,z$$ 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, i.e., it is associative.

Proof
Given: A Moufang loop $$L$$. Elements $$x,y \in L$$ (not necessarily distinct).

To prove: The subloop of $$L$$ generated by $$x,y$$ is associative.

Proof: Setting $$z = y$$, we see that by fact (1), $$x * (y * z) = (x * y) * z$$ (using the right alternative law part). Hence, by fact (2), the subloop generated by $$x,y,z$$ is a group. But since $$z = y$$, this is the same as the subloop generated by $$x$$ and $$y$$, completing the proof.