Given a decreasing function $f(x)$ defined on $[1,3]$, find the solution set of the inequality $f(1-a)-f\left(3-a^{2}\right)$ $>0$.

The function $f(x)$ is decreasing on the interval $[1,3]$. Solve the inequality $f(1-a)-f\left(3-a^{2}\right) > 0$ for $a$.
Solve: From the given, we have $\left\{\begin{array}{l}1 \leqslant 1-a \leqslant 3, \\ 1 \leqslant 3-a^{2} \leqslant 3 \\ 1-a<3-a^{2},\end{array}\right.$ Solving this, we get $-1<a \leqslant 0$, therefore, the solution set of the inequality is $\{a \mid-1<a \leqslant 0\}$.