Jordan implies power-associative

Statement
Any Jordan ring is a power-associative ring. In other words, if $$R$$ is a commutative possibly non-associative ring satisfying the following

$$x * (y * (x * x)) = (x * y) * (x * x) \ \forall \ x,y \in R$$

Related facts
The analogous statement is not necessarily true for Jordan magmas, i.e., a Jordan magma need not be a power-associative magma. However, it is true that Jordan implies powers up to the fifth are well-defined even in the magma context.

Facts used

 * 1) uses::Jordan implies powers up to the fifth are well-defined: Actually, we only need the fourth power to be well-defined.
 * 2) uses::Equivalence of definitions of power-associative ring: This in particular implies that a non-associative ring is power-associative iff fourth powers are well-defined in the ring.

Proof
The proof combines facts (1) and (2).