Corollary

If $V$ is a normed algebra, if $f_1, f_2$ are two functions in $\mathscr{H}(\mathfrak{F}, V)$, and $g_1, g_2$ are two functions in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$, then the relations $f_1 \preccurlyeq g_1, f_2 \preccurlyeq g_2$ imply $f_1 f_2 \preccurlyeq g_1 g_2$.

The relation $f \preccurlyeq g$ between functions in $\mathscr{H}(\mathfrak{F}, V)$ is transitive by prop. 1; since it is reflexive the relation "*f \preccurlyeq g and g \preccurlyeq f*" is an equivalence relation on $\mathscr{H}(\mathfrak{F}, V)$ (Set Theory, II, p. 113).