Quotient Theorem for Surjections

Theorem
For any surjection $$f: S \to T$$, there is one and only one bijection $$r: S / \mathcal{R}_f \to T$$ such that:

$$r \circ q_{\mathcal{R}_f} = f$$

where:
 * $$\mathcal{R}_f$$ is the equivalence induced by $f$;
 * $$r: S / \mathcal{R}_f \to T$$ is the renaming mapping;
 * $$q_{\mathcal{R}_f}: S \to S / \mathcal{R}_f$$ is the quotient mapping induced by $$\mathcal{R}_f$$.

Otherwise known as the factor theorem for surjections.

Proof
From Induced Equivalence, the mapping $$f: S \to T$$ induces an equivalence $$\mathcal{R}_f$$ on its domain.

As $$f: S \to T$$ is a surjection, $$T = \mathrm{Im} \left({f}\right)$$ from Surjection iff Image equals Range.

From Renaming Mapping is a Bijection, the renaming mapping $$r: S / \mathcal{R}_f \to T$$ is a bijection, where $$S / \mathcal{R}_f$$ is the quotient set of $$S$$ by $$\mathcal{R}_f$$.

Clearly:
 * $$r \circ q_{\mathcal{R}_f} = f$$.
 * $$r$$ is the only mapping $$r: S / \mathcal{R}_f \to T$$ that satisfies this equality.