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, $\left({T\,', \oplus\,'}\right)$ has an identity, and it is $\left[\!\left[{\left({c, c}\right)}\right]\!\right]_\boxtimes$ for any $c \in C$. Call this identity $e_{T\,'}$.

From Image of Cancellable Elements in Quotient Mapping, $C\,' = \psi \left({C}\right)$.

So:

The inverse of $x'$ is $\left[\!\left[{\left({a, a \circ x}\right)}\right]\!\right]_\boxtimes$, as follows:

... thus showing that the inverse of $\left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes$ is $\left[\!\left[{\left({a, a \circ x}\right)}\right]\!\right]_\boxtimes$.