Let $\mathbb{F}$ be a field. Let $S\subset \mathbb{F}$ be a finite set with a size large enough. Let $f(x,y_1,y_2,\dots,y_n)=f(x,\overline{y})$ be a almost monic and irreducible polynomial in total degree $d$ in $\mathbb{F}[x,\overline{y}]$. Then $$\underset{\overline{a},\overline{b}\in > S^n}{Pr}[f(x,\overline{a}t+\overline{b})\text{ is reducible and > }f(x,\overline{b})\text{ is square-free}]\leq \frac{7d^6}{|S|}$$
Here $S^n$ is the $n-$titmes cartesian product of $S$ and t is a variable. And almost monic means $\deg_x(f)=\deg(f(x,\overline{0})$
Honestly, I have zero idea how to prove this. I found a theorem very near to this: Hilbert Irreducibility Theorem that I can't prove.
The hint I got is that we have to Hensel Lift twice once with respect to modulo the ideal $(t)$ and once replacing the $\overline{a}$ with a formal variable $\overline{z}$ and then lift modulo the ideal $(\overline{z})$
Can you help me.