I would like to solve Problem 1.7.6 from Dugundji's Topology:
Let $S$, $R$ be two equivalence relations in $A$, with $S \subset R$. Let $1^* : A/S \to A/R$ be the map induced by relation preserving map $1_A$. Define $(S a) R/S (Sb)$ if $1^*(S a)=1^*(Sb)$.
Show that $R/S$ is an equivalence relation, and there is a bijection of $(A/S)/(R/S)$ onto $A/R$
If $f:X\rightarrow Y$ is a function then the relation $\sim$ defined by $x\sim x'\iff f\left(x\right)=f\left(x'\right)$ is always an equivalence relation.
Verification:
reflexive: $f\left(x\right)=f\left(x\right)$,
symmetric: $f\left(x\right)=f\left(x'\right)\Rightarrow f\left(x'\right)=f\left(x\right)$
transitive: $f\left(x\right)=f\left(x'\right)\wedge f\left(x'\right)=f\left(x''\right)\Rightarrow f\left(x\right)=f\left(x''\right)$.
The map $\phi:\left(A/S\right)/\left(R/S\right)$ defined by $\left[\left[a\right]_{S}\right]_{R/S}\mapsto\left[a\right]_{R}$ is welldefined and is a bijection.