Subset Product within Semigroup is Associative/Corollary

Theorem
Let $\left({S, \circ}\right)$ be a magma.

If $\circ$ is associative, then:


 * $x \left({y S}\right) = \left({x y}\right) S$
 * $x \left({S y}\right) = \left({x S}\right) y$
 * $\left({S x}\right) y = S \left({x y}\right)$

Proof
From the definition of Subset Product with Singleton:
 * $x \left({y S}\right) = \left\{{x}\right\} \left({\left\{{y}\right\} S}\right)$
 * $x \left({S y}\right) = \left\{{x}\right\} \left({S \left\{{y}\right\}}\right)$
 * $\left({S x}\right) y = \left({S \left\{{x}\right\}}\right) \left\{{y}\right\}$

The result then follows directly from Subset Product of Associative is Associative

Also see

 * Subset Product of Commutative is Commutative