Corollary

The relation $f - g \ll f$ is an equivalence relation on $\mathscr{H}(\mathfrak{F}, V)$.

Indeed, if $f - g \ll f$ and $g - h \ll g$ then $f - g \ll g$, whence (V, p. 215, prop. 5) $f - h \ll g$, and consequently, since $g \preceq f$, we have $f - h \ll f$, which shows that this relation is transitive; it is symmetric by prop. 7, and is clearly reflexive, whence the corollary.