Quotients of equivalence relations

Question

Let $R,S$ and $T$ relations of equivalence in $A$ and suppose that $R\subset S\subset T$ prove that:

If $R \circ T$ is a relation of equivalence in $A$, then $(S/R)\circ (T/R) =(S\circ T)/R$

This is my work that I have done.

$$\begin{array}{crl} ([x]_R,[y]_R)\in (S/R)\circ (T/R) &\iff &\exists_z (x,z)\in(T/R) \wedge(z,y)\in (S/R) \\ &\iff &\ (x,z) \in T \wedge (z,y) \in S \\ &\iff&\ (x,y) \in S \circ T\\ &\iff & ([x]_R,[y]_R) \in (S \circ T)/R \end{array}$$

I appreciate your contributions with the definitions used.

Answer 1

$$ ([x]_R,[y]_R)\in (S/R)\circ (T/R) \\ \Updownarrow 1 \\ \exists [z]_R\in A/R \text{ s.t. } ([x]_R,[z]_R)\in(T/R) \land ([z]_R,[y]_R)\in (S/R) \\ \Updownarrow 2 \\ \exists z\in A \text{ s.t. } (x,z)\in T \land (z,y)\in S \\ \Updownarrow 3 \\ (x,y) \in S\circ T \\ \Updownarrow 4 \\ ([x]_R,[y]_R) \in (S \circ T)/R $$

1. Definition of $\circ$.
2. Definition of quotient relations.
3. Definition of $\circ$.
4. Definition of quotient relations.