Let $p_{1}, p_{2}, \cdots, p_{m}$ be all the distinct prime divisors of $N$. In Theorem 2, take the sequence $A$ to be $1, 2, \cdots, N, K=p_{1} \cdots p_{m}$. Thus, $\varphi(N)$ is the number of integers in $A$ that are coprime to $K$, i.e., $S(A ; K)$. Note that in this case we have (why)
$$\left[p_{i_{1}}, \cdots, p_{i_{k}}\right]=p_{i_{1}} \cdots p_{i_{k}} \mid N, \quad 1 \leqslant i_{1}<\cdots<i_{k} \leqslant m$$

Therefore, $\left|A_{p_{i_{1}} \cdots p_{i_{k}}}\right|=N /\left(p_{i_{1}} \cdots p_{i_{k}}\right)$, and thus from equation (21) we derive
$$\begin{aligned} \varphi(N)= & N-\sum_{1 \leqslant i_{1} \leqslant m} \frac{N}{p_{i_{1}}}+\sum_{1 \leqslant i_{1}<i_{2} \leqslant m} \frac{N}{p_{i_{1}} p_{i_{2}}}-\cdots \\ & +(-1)^{k} \sum_{1 \leqslant i_{1}<\cdots<i_{k} \leqslant m} \frac{N}{p_{i_{1}} \cdots p_{i_{k}}}+\cdots+(-1)^{m} \frac{N}{p_{1} \cdots p_{m}} \\ = & N\left(1-\frac{1}{p_{1}}\right)\left(1-\frac{1}{p_{2}}\right) \cdots\left(1-\frac{1}{p_{m}}\right) \end{aligned}$$

This proves the desired conclusion. In Chapter 3, §3, $\varphi(N)$ has been discussed in detail, and this conclusion was proven using a different method.