Test for Submonoid

Theorem
To show that $\left({T, \circ}\right)$ is a submonoid of a monoid $\left({S, \circ}\right)$, we need to show that:


 * $(1): \quad T \subseteq S$
 * $(2): \quad \left({T, \circ}\right)$ is a magma (i.e. that it is closed)
 * $(3): \quad \left({T, \circ}\right)$ has an identity.

Proof
From Subsemigroup Closure Test, $(1)$ and $(2)$ are sufficient to show that $\left({T, \circ}\right)$ is a subsemigroup of $\left({S, \circ}\right)$.

Demonstrating the presence of an identity is then sufficient to show that it is a monoid.