Inclusion Mapping is Monomorphism

Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure

Let $\left({T, \circ}\right)$ be an algebraic substructure of $S$.

Let $\iota: T \to S$ be the inclusion mapping from $T$ to $S$.

Then $\iota$ is a monomorphism.

Proof
We have that the inclusion mapping is an injection.

Now let $x, y \in T$:

demonstrating that $\iota$ has the morphism property.

So $\iota$ is a homomorphism which is also an injection.

Thus by definition, $\iota$ is a monomorphism.