Given two sets of points $\pmb{R_U}$, $\pmb{R_S}$ and a time interval $[a,b]$. Let $\pmb{R_U}(t)$, $\pmb{R_S}(t)$ denote the two sets of points occupied by $\pmb{R_U}$ and $\pmb{R_S}$, respectively, at some time $t \in [a,b]$. Then, $\pmb{R_U}$ overlaps $\pmb{R_S}$ in $[a,b]$ if and only if there exists a common point that lies within both $\pmb{R_U}(t)$ and $\pmb{R_S}(t)$ at a certain time $t \in [a,b]$, such that
$\displaystyle \bigcup_{t \in [a,b]} ( \pmb{R_U}(t) \cap \pmb{R_S}(t) ) \neq \emptyset \Leftrightarrow \exists(\vec{p},t) (\vec{p} \in \pmb{R_U}(t) \wedge \vec{p} \in \pmb{R_S}(t), t \in [a,b]).$
How do you prove the above theorem?
Thanks