Renaming Mapping is Well-Defined/Proof 2

Proof
From Condition for Mapping from Quotient Set to be Well-Defined:
 * there exists a mapping $\phi: S / \RR \to T$ such that $\phi \circ q_\RR = f$


 * $\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$
 * $\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$

But by definition of the equivalence induced by the mapping $f$:


 * $\forall x, y \in S: \tuple {x, y} \in \RR_f \implies \map f x = \map f y$

The result follows directly.