Construction of Inverse Completion/Quotient Mapping is Injective

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


 * $\forall x \in S: \map \psi x = \eqclass {\tuple {x \circ a, a} } \boxtimes$

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.