# Identity of Submagma containing Identity of Magma is Same Identity

## Theorem

Let $\struct {S, \circ}$ be a magma which has an identity $e$.

Let $\struct {T, \circ}$ be a submagma of $\struct {S, \circ}$ such that $e \in T$.

Then $e$ is the identity of $T$.

## Proof

From Identity is Unique, there can be only one identity $e$ of $\struct {S, \circ}$.

We have that:

 $\ds x$ $\in$ $\ds T$ $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds S$ $\ds \leadsto \ \$ $\ds x \circ e$ $=$ $\ds x = e \circ x$ $\ds \leadsto \ \$ $\ds \forall x \in T: \,$ $\ds x \circ e$ $=$ $\ds x = e \circ x$

That is, $e$ is the (unique} identity of $T$.

$\blacksquare$