Identity of Cancellable Monoid is Identity of Submonoid

Theorem
Let $\left({S, \circ}\right)$ be a monoid, all of whose elements are cancellable.

Let $\left({T, \circ}\right)$ be a submonoid of $\left({S, \circ}\right)$.

Then the identity of $T$ is the same element as the identity of $S$.

Proof
Let $e_S$ be the identity of $\left({S, \circ}\right)$, and $e_T$ the identity of $\left({T, \circ}\right)$.

(Note that by Identity is Unique, there is only one identity element of $\left({S, \circ}\right)$.)

From Identity Only Idempotent Cancellable Element, $e_S$ is the only cancellable element of $\left({S, \circ}\right)$ which is idempotent.

As all elements of $S$ are cancellable, $e_S$ is the only idempotent element of $S$.

As all elements of $S$ are cancellable, all elements of $T$ are cancellable, from Cancellable in Subset.

Thus, $e_T$ is the only element of $\left({T, \circ}\right)$ which is idempotent.

Thus, as $e_T \in S$, we have $e_S \circ e_T = e_T = e_T \circ e_T$ and thus $e_S = e_T$.