Alternating implies flexible

This article gives the statement and possibly, proof, of an implication relation between two non-associative ring properties. That is, it states that every non-associative ring satisfying the first non-associative ring property (i.e., alternating ring) must also satisfy the second non-associative ring property (i.e., flexible ring)
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Statement

Suppose $R$ is a non-associative ring (i.e., a not necessarily associative ring). Then, if $R$ is an alternative ring (i.e., the square of any element is zero) then $R$ is a flexible ring.

Facts used

1. Alternating implies skew-commutative
2. Skew-commutative implies flexible