Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism

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

Let $S'$ be the image $\psi \sqbrk S$ of $S$.

Then $\psi$ is an isomorphism from $S$ onto $S'$.

Proof
From Quotient Mapping is Monomorphism, $\psi: \struct {S, \circ} \to \struct {S', \oplus'}$ is a monomorphism.

Therefore by definition:
 * $\psi$ is a homomorphism
 * $\psi$ is an injection.

Because $S'$ is the image of $\psi \sqbrk S$, by Surjection by Restriction of Codomain $\psi$ is a surjection.

Therefore by definition $\psi: S \to S'$ is a bijection.

A bijective homomorphism is an isomorphism.