Let $N$ be a positive integer not less than all $a_{j}(1 \leqslant j \leqslant m)$. In Theorem 7, take the sequence $A$ to be $1,2, \cdots, N$, and let property $P_{j}$ be not greater than $a_{j}(1 \leqslant j \leqslant m)$. Thus, we have
$$\begin{array}{c} \left|A\left(i_{1}, \cdots, i_{k}\right)\right|=\min \left(a_{i_{1}}, \cdots, a_{i_{k}}\right) \\ 1 \leqslant i_{1}<\cdots<i_{k} \leqslant m, \quad 1 \leqslant k \leqslant m \end{array}$$

From this and Theorem 7, it follows that the number of elements in $1,2, \cdots, N$ for which no property $P_{j}$ holds, i.e., the number of elements greater than $\max \left(a_{1}, \cdots, a_{m}\right)$, can be expressed as
$$\begin{aligned} N & -\sum_{1 \leqslant i_{1} \leqslant m} a_{i_{1}}+\sum_{1 \leqslant i_{1}<i_{2} \leqslant m} \min \left(a_{i_{1}}, a_{i_{2}}\right)-\cdots \\ & +(-1)^{k} \sum_{1 \leqslant i_{1}<\cdots<i_{k}<m} \min \left(a_{i_{1}}, \cdots, a_{i_{k}}\right)+\cdots \\ & +(-1)^{m} \min \left(a_{1}, \cdots, a_{m}\right) \end{aligned}$$

However, the number of such elements is clearly equal to $N-\max \left(a_{1}, \cdots, a_{m}\right)$. From this and the above expression, the desired conclusion follows.