Condition for Existence of Epimorphism from Quotient Structure to Epimorphic Image

Theorem
Let $\struct {A, \odot}$ and $\struct {B, \otimes}$ be algebraic structures.

Let $\RR$ be a congruence relation on $\struct {A, \odot}$.

Let $f: \struct {A, \odot} \to \struct {B, \otimes}$ be an epimorphism.

Let $\struct {A / \RR, \odot_\RR}$ denote the quotient structure defined by $\RR$.

Let $q_\RR: A \to A / \RR$ denote the quotient mapping induced by $\RR$:
 * $\forall x \in A: \map {q_\RR} x = \eqclass x \RR$

where $\eqclass x \RR$ denotes the equivalence class of $x$ under $\RR$.

Then:
 * there exists an epimorphism $g$ from $\struct {A / \RR, \odot_\RR}$ to $\struct {B, \otimes}$ which satisfies $g \circ q_\RR = f$


 * $\RR \subseteq \RR_f$
 * $\RR \subseteq \RR_f$