Subset Product within Semigroup is Associative/Corollary

Corollary to Subset Product within Semigroup is Associative
Let $\struct {S, \circ}$ be a semigroup.

Then:


 * $x \paren {y S} = \paren {x y} S$
 * $x \paren {S y} = \paren {x S} y$
 * $\paren {S x} y = S \paren {x y}$

Proof
From the definition of Subset Product with Singleton:
 * $x \paren {y S} = \set x \paren {\set y S}$
 * $x \paren {S y} = \set x \paren {S \set y}$
 * $\paren {S x} y = \paren {S \set x} \set y$

The result then follows directly from Subset Product within Semigroup is Associative.

Also see

 * Subset Product within Commutative Structure is Commutative