Given a set of points $\pmb{R}$. Let $\pmb{R}(t)$ denote the set of points occupied by $\pmb{R}$ at some time $t \in [a,b]$. Let $\pmb{S}([a,b])$ denote the swept volume generated by the motion of $\pmb{R}$ over the time interval $[a,b]$. $\pmb{S}([a,b])$ can be defined as the infinite union of $\pmb{R}(t)$ in $[a,b]$, such that
$\displaystyle \pmb{S}([a,b]) = \bigcup _{t \in [a,b]} \pmb{R}(t) = \{\vec{p} \mid \vec{p} \in \pmb{R}(t), \exists t \in [a,b] \}.$
Let $I_i$ denote the subinterval of $[a,b]$, such that
$\displaystyle [a,b] = \bigcup^{n}_{i = 1} I_i$
where $I_i$ = $[t_{i-1},t_{i}]$ for $a = t_0 < t_i < \cdots < t_n = b$. Hence, $\pmb{S}([a,b])$ can also be expressed by the union of smaller swept volumes, such that
$\displaystyle \pmb{S}([a,b]) = \bigcup^{n}_{i = 1} \pmb{S} (I_i).$
Given two sets of points $\pmb{R_U}$ and $\pmb{R_S}$. Let $\pmb{R_U}(t)$ and $\pmb{R_S}_(t)$ denote the two sets of points occupied by $\pmb{R_U}$ and $\pmb{R_S}$, respectively, at some time $t \in [a,b]$, and $\pmb{S_U}([a,b])$ and $\pmb{S_S}([a,b])$ denote the swept volume generated by the motion of $\pmb{R_U}$ and $\pmb{R_S}$, respectively, over the time interval $[a,b]$.
It is proven that $\pmb{R_U}$ overlaps $\pmb{R_S}$ in $[a,b]$ if and only if there exists a common point $\vec{p}$ that lies within both $\pmb{R_U}(t)$ and $\pmb{R_S}(t)$ at a certain time $t \in [a,b]$, such that
$\displaystyle \bigcup_{t \in [a,b]} ( \pmb{R_U}(t) \cap \pmb{R_S}(t) ) \neq \emptyset \Leftrightarrow \exists(\vec{p},t) (\vec{p} \in \pmb{R_U}(t) \wedge \vec{p} \in \pmb{R_S}(t), t \in [a,b]).$
Now, prove the following:
$\displaystyle \bigcup_{t \in [a,b]} ( \pmb{R_U}(t) \cap \pmb{R_S}(t) ) \subseteq \bigcup^{n}_{i = 1} ( \pmb{S_U} (I_i) \cap \pmb{S_S} (I_i) )$
$\pmb{R_U}$ overlaps $\pmb{R_S}$ in $[a,b]$ only if $\displaystyle \bigcup^{n}_{i = 1} ( \pmb{S_U} (I_i) \cap \pmb{S_S} (I_i) ) \neq \emptyset$
Many thanks in advance for your help!