Example $7 L_{1}$ and $L_{2}$ are two parallel lines in a plane, with $m$ and $n$ points on $L_{1}$ and $L_{2}$ respectively $(m, n \geqslant 2)$. Connect each point on $L_{1}$ with each point on $L_{2}$ with line segments. It is known that any three of the connected line segments have no common points in the plane region $D$ between $L_{1}$ and $L_{2}$. Find the number of different intersection points of these line segments.
Let the set of $m$ points on $L_{1}$ be $Q_{1}$, and the set of $n$ points on $L_{2}$ be $Q_{2}$. The set of intersection points of the line segments within $D$ is $M$. Take any two points $A, B$ from $Q_{1}$ and any two points $C, D$ from $Q_{2}$ (as shown in Figure 6-5), then $A D$ intersects $B C$ at a point $P \in M$. Conversely, for any $P \in M$, since any three of the connected line segments have no common point within $D$, $P$ can only be the intersection point of two line segments $A D$ and $B C$ (where $\left.A, B \in Q_{1}, C, D \in Q_{2}\right)$, let $N=\{(A, B, C, D)$ । $\left.A, B \in Q_{1}, C, D \in Q_{2}\right\}$, and let $(A, B, C, D)$ correspond to the intersection point $P$ of $A D$ and $B C$. Then this correspondence is a one-to-one correspondence from $N$ to $M$, so $|M|=|N|=C_{m}^{2} \cdot C_{n}^{2}$, which means the number of intersection points is $C_{m}^{2} \cdot C_{n}^{2}$.