Let $X$ be a set of points.
Def 1: A subgraph of a real function on $X$ is the set
$$G_f:= \{ (t,x) : t\leq f(x) \quad t \in \mathbb{R} ,\quad x \in X\}$$
Def 2 A class of functions $\mathcal{F}$ is a VC subgraph of index $v$ if the class of sets $C:= \{ G_f : f \in \mathcal{F}\}$ is a VC ( Vapnik-Chervonenkis) of dimension $v$.
My question:
If $\mathcal{F}$ and $\mathcal{G}$ are VC subgraph on $X$, is $\mathcal{F} \mathcal{G}:= \{ fg : f\in \mathcal{F} , g \in \mathcal{G} \}$ is also a VC subgraph?