# Subset Product with Identity

## Theorem

Let $\struct {S, \circ}$ be a magma.

Let $\struct {S, \circ}$ have an identity element $e$.

Then $e \circ S = S \circ e = S$, where $\circ$ is understood to be the subset product with singleton.

## Proof

 $\ds e \circ S$ $=$ $\ds \set e \circ S$ Definition of Subset Product with Singleton $\ds$ $=$ $\ds \set {x \circ y: x \in \set e, \, y \in S}$ Definition of Subset Product $\ds$ $=$ $\ds \set {e \circ y: y \in S}$ $\ds$ $=$ $\ds \set {y: y \in S}$ Definition of Identity Element $\ds$ $=$ $\ds S$

Thus:

$e \circ S = S$

A similar argument shows that:

$S \circ e = S$

$\blacksquare$