My question is about some proof details in Lemma 6.4 in Gitik's chapter from the Handbook of Set Theory.
On the sequel suppose that,
- $j:V\longrightarrow M$ stands for the elementary embedding derived from a normal measure $U$ over $\kappa$ in $V$.
- $\mathbb{P}_\kappa$ is a Magidor iteration of Prikry type forcings with the following property: $$\text{$\forall\alpha<\kappa\;\forall p,q,r\in\mathbb{Q}_\alpha\; (p,q\leq^*_\alpha r\;\longrightarrow \;\exists s\in\mathbb{Q}_\alpha\;( s\leq^*_\alpha p,q))$}.$$ (For instance, any Magidor iteration of Prikry forcings enjoy this property.)
- If $\mathbb{P}_{j(\kappa)}$ is the image by $j$ of the iteration, the forcing $(\mathbb{P}_{j(\kappa)}\setminus\kappa,\leq^*)$ is $\kappa^+$-closed.
Now define the following ultrafilter over $\kappa$ in the generic extension $V[G]$:
$$X\in U^*\;\longleftrightarrow\; \exists p\in G\cap M\,\exists q\leq^* j(p)\setminus\kappa\; (p\frown q\Vdash_{j(\mathbb{P}_\kappa)} \kappa\in j(\dot{X})). $$
What I would like to know is the proof that this ultrafilter $U^*$ is also normal. Gitik gives some indications but I am not able to manage a proof. Any help will be wellcome.