Construction of Inverse Completion/Invertible Elements in Quotient Structure

Theorem
Every cancellable element of $S'$ is invertible in $T'$.

Proof
From Identity of Quotient Structure, $\struct {T', \oplus'}$ has an identity, and it is $\eqclass {\tuple {c, c} } \boxtimes$ for any $c \in C$.

Call this identity $e_{T'}$.

Let the mapping $\psi: S \to T'$ be defined as:
 * $\forall x \in S: \map \psi x = \eqclass {\tuple {x \circ a, a} } \boxtimes$

From Image of Cancellable Elements in Quotient Mapping:
 * $C' = \psi \sqbrk C$

So:

The inverse of $x'$ is $\eqclass {\tuple {a, a \circ x} } \boxtimes$, as follows:

thus showing that the inverse of $\eqclass {\tuple {x \circ a, a} } \boxtimes$ is $\eqclass {\tuple {a, a \circ x} } \boxtimes$.