Let $G=A*B$. And let $N\unlhd A$ be a normal subgroup of A. Let $H\leq G$ be the kernel of the following map: $$\Psi:A*B\to A/N*1.$$ With Kurosh's Theorem there exists a splitting $H=(H\cap A)*(H\cap B)*F$, where $F\leq G$ is a free subgroup of $G$. Why can I choose $F\neq 1$? Or is $F$ always nontrivial? And if so, why?
Thanks for help.