Equivalence of definitions of alternative ring

Statement
The following (all definitions of alternative ring) are equivalent for a non-associative ring $$R$$ with multiplication denoted by $$*$$:


 * 1) $$(R,*)$$ is an fact about::alternative magma, i.e., it satisfies the identities $$x * (x * y) = (x * x) * y$$ and $$x * (y * y) = (x * y) * y$$ for all $$x,y \in R$$.
 * 2) $$(R,*)$$ is both a fact about::left-alternative magma and a fact about::flexible magma, i.e., it satisfies the identities $$x * (x * y) = (x * x) * y$$ and $$x * (y * x) = (x * y) * x$$ for all $$x,y \in R$$.
 * 3) $$(R,*)$$ is both a fact about::right-alternative magma and a fact about::flexible magma, i.e., it satisfies the identities $$x * (y * y) = (x * y) * y$$ and $$x * (y * x) = (x * y) * x$$ for all $$x,y \in R$$.
 * 4) The associator function on $$R$$ is an alternating function on any two of its variables.

Facts used

 * 1) uses::Alternating function condition is transitive

Proof
The associator function:

$$a(x,y,z) = ((x * y) * z) - (x * (y * z))$$

is additive separately in each variable, i.e., it is multi-additive. Here is what each condition says:

Fact (1) now tells us that any two of these conditions must imply the third, completing the proof.