Construction of Inverse Completion/Quotient Mapping is Injective

Theorem
Let the mapping $\psi: S \to T'$ be defined as:


 * $\forall x \in S: \psi \left({x}\right) = \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes$

Then $\psi: S \to T'$ is an injection, and does not depend on the particular element $a$ chosen.

Proof
The result follows by the definition of injection.