Associating fraction in subring of finite non-associative ring is at least as much as in whole ring

Statement
Suppose $$R$$ is a finite fact about::non-associative ring (i.e., $$R$$ is a not necessarily associative ring whose underlying set is finite). Suppose $$S$$ is a subring of $$R$$. Then, the associating fraction of $$S$$ is at least as much as that of $$R$$.

In symbols, if $$AT(R) := \{ (x,y,z) \in R^3 \mid (x * y) * z = x * (y * z) \}$$ and $$AT(S) = S^3 \cap CP(R)$$, then:

$$\frac{|AT(S)|}{|S|^3} \ge \frac{|AT(R)|}{|R|^3}$$

In fact, the result also holds if $$S$$ is simply an additive subgroup of $$R$$ and not a subring.

Facts used

 * 1) uses::Associativity is a groupy relation on the additive group of a non-associative ring
 * 2) uses::Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group

Related facts

 * Commuting fraction in subring of finite Lie ring is at least as much as in whole ring
 * Commuting fraction in subring of finite non-associative ring is at least as much as in whole ring
 * Commuting fraction in subgroup is at least as much as in whole group

Proof
The proof follows from facts (1) and (2).