Let $R$ be a reflexive and transitive relation on a set $A$. Define a new relation $E$ on $A$ as
$E=\{(a, b) \mid (a, b) \in R \text{ and } (b, a) \in R \}$
Prove that $E$ is an equivalence relation on $A$.
Define a relation $\leq$ on the equivalence classes of $E$ as $E_1 \leq E_2$ if $\exists a, b$ such that $a \in E_1, b \in E_2 \text{ and } (a, b) \in R$. Prove that $\leq$ is a partial order.