Example 2 Since $(72,157)=1$, by Lemma 1 there must be two integers $x$, $y$ that satisfy
$$72 x+157 y=1$$

established, find $x, y$.
Since
$$\begin{array}{l}
157=2 \times 72+13, \quad 72=5 \times 13+7 \\
13=7+6, \quad 7=6+1
\end{array}$$

we get
$$\begin{aligned}
1 & =7-6=7-(13-7)-2 \times 7-13 \\
& =2 \times(72-5 \times 13)-13=2 \times 72-11 \times 13 \\
& =2 \times 72-11 \times(157-2 \times 72) \\
& =24 \times 72-11 \times 157=72 \times 24-157 \times 11
\end{aligned}$$

Therefore,
$$x=24, \quad y=-11$$