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 $$e_S = e_T$$.