Proof: Treating $a$ as a variable and $b, c$ as constants, we denote
$$\begin{aligned} f(a) & =a i x-2-(a+b+c) \\ & =(b c-1) a+2-b-c, \\ & a \in(-1,1) . \end{aligned}$$

Now, we only need to prove $f(a)>0$ based on the properties of a linear function.
$$\begin{array}{l} \because b, c \in(-1,1), b cf(1)=1-b-c+b c \\ =(1-b)(1-c)>0 . \end{array}$$