Remark 2

Denote by $R_0$ the relation $f \simeq g$ on $\mathscr{H}(\mathfrak{F}, V)$, and by $\mathscr{H}_0(\mathfrak{F}, V)$ the quotient set $\mathscr{H}(\mathfrak{F}, V)/R_0$; note that the relation $R$ implies $R_0$. Passing to the quotient the relation $f \preccurlyeq g$ gives, by prop. 1 of V, p. 213, an order relation on $\mathscr{H}_0(\mathfrak{F}, V)$ (Set Theory, III, p. 134); the preceding example shows that $\mathscr{H}_0(\mathfrak{F}, V)$ is not totally ordered by this relation.