Exercise 7, § 6

Let $G_0$, $G_1$, $H_0$, $H_1$ be the graphs of four equivalences on a set $E$ such that $G_1 \cap H_0 = G_0 \cap H_1$ and $G_1 \circ H_0 = G_0 \circ H_1$. For each $x \in E$, let $R_0$ (resp. $S_0$) be the relation induced on $G_1(x)$ (resp. $H_1(x)$) by the equivalence relation $(x, y) \in G_0$ (resp. $(x, y) \in H_0$). Show that there exists a bijection of the quotient set $G_1(x)/R_0$ onto the quotient set

(If $A = G_1(x) \cap H_1(x)$, show that both quotient sets are in one-to-one correspondence with the quotient set of $A$ by the equivalence relation induced by $R_0$ on $A$; this relation is equivalent to that induced by $S_0$ on $A$.)