I am studying Positive Model Theory and from this context, I bumped into this question. It should make sense, considering the Chang-Los-Suszko theorem. However I fail to rewrite it in that way.
So, a h-inductive sentence is a finite conjunction of sentences of the form $$\forall\bar x(\exists\bar y\phi(\bar x,\bar y)\rightarrow\exists\bar z\psi(\bar x,\bar z)),$$ where $\phi$ and $\psi$ are quantifier free.
I suppose that if I can show this form arises in a conjunction of two such sentences, I am done. So, consider $$(\forall\bar x_1(\exists\bar y_1\phi(\bar x_1,\bar y_1)\rightarrow\exists\bar z_1\psi(\bar x_1,\bar z_1)))\wedge(\forall\bar x_2(\exists\bar y_2\phi(\bar x_2,\bar y_2)\rightarrow\exists\bar z_2\psi(\bar x_2,\bar z_2))).$$
Is it possible to rewrite this as $\forall\bar p\exists\bar qf(\bar p,\bar q)$, where $f$ is quantifier free?
Yes. Quite generally, the conjunction of two $\forall\exists$ sentences is equivalent to another $\forall\exists$ sentence. Specifically, $(\forall x\exists y\,\alpha)\land(\forall u\exists v\,\beta)$ is equivalent to $\forall x\forall u\exists y\exists v\,(\alpha\land\beta)$.