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:

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