Construction of Inverse Completion/Quotient Mapping is Monomorphism

Theorem
Let the mapping $\psi: S \to T'$ be defined as:
 * $\forall x \in S: \psi \left({x}\right) = \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes$

The mapping $\psi: S \to T'$ is a monomorphism.

Proof
We have that this quotient mapping $\psi: S \to T'$ is an injection.

Let $x, y \in S$. Then:

So $\psi \left({x \circ y}\right) = \psi \left({x}\right) \oplus' \psi \left({y}\right)$, and the morphism property is proven.

Thus $\psi$ is an injective homomorphism, and so by definition a monomorphism.