Identity of Submagma containing Identity of Magma is Same Identity

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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 $T$.


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 $T$.