Factoring Mapping into Quotient and Injection

Theorem
Any mapping $$f: S \to T$$ can be uniquely factored into a quotient mapping, followed by an injection.

Thus:
 * $$f = h \circ q_{\mathcal R_f}$$

where:


 * $$q_{\mathcal R_f}: S \to S / \mathcal R_f : q_{\mathcal R_f} \left({s}\right) = \left[\!\left[{s}\right]\!\right]_{\mathcal R_f}$$
 * $$h: S / \mathcal R_f \to T : h \left({\left[\!\left[{s}\right]\!\right]_{\mathcal R_f}}\right) = f \left({s}\right)$$

This can be illustrated using a commutative diagram as follows:


 * QuotientInjection.png

Proof
The mapping $$q_{\mathcal R_f}: S \to S / \mathcal R_f$$ follows from the definition of quotient mapping.

The mapping $$h$$ is justified by Existence of Renaming Mapping.