Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements

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

Let $S'$ be the image $\psi \left[{S}\right]$ of $S$.

The set $C'$ of cancellable elements of the semigroup $S'$ is $\psi \left[{C}\right]$.

Proof
We have Morphism Property Preserves Cancellability.

Thus:
 * $c \in C \implies \psi \left({c}\right) \in C'$

So by :
 * $\psi \left[{C}\right] \subseteq C'$

From above, $\psi$ is an isomorphism.

Hence, also from Morphism Property Preserves Cancellability:
 * $c' \in C' \implies \psi^{-1} \left({c'}\right) \in C$

So by :
 * $\psi^{-1} \left[{C'}\right] \subseteq C$

Hence by definition of set equality:
 * $\psi \left[{C}\right] = C'$