Suppose $ E $ is an extender on the sequence of some premouse $\mathcal{N}$ such that $ \mathcal{P}^{\mathcal{N}|lh(E)}(\kappa) = \mathcal{P}^{\mathcal{N}}(\kappa) = \mathcal{P}^{\mathcal{M}}(\kappa),$ where $crit(E) = \kappa$.
- If $\mathcal{M}$ and $\mathcal{N}$ are premice and $ \mathcal{P}^{\mathcal{N}}(\kappa) = \mathcal{P}^{\mathcal{M}}(\kappa)$, then $\kappa^{+\mathcal{M}} = \kappa^{+ \mathcal{N}} =: \theta $ and $ \mathcal{M}|\theta = \mathcal{N}|\theta$.
This implies the following:
- $\pi^{\mathcal{M}}_{E}(\kappa) = \pi^{\mathcal{N}}_{E}(\kappa)$, $\pi^{\mathcal{M}}_{E}(\kappa^{+}) = \pi^{\mathcal{N}}_{E}(\kappa^{+})$ and $Ult_{0}(\mathcal{M},E)|\pi^{\mathcal{M}}_{E}(\kappa^{+}) = Ult_{0}(\mathcal{N},E)|\pi^{\mathcal{N}}_{E}(\kappa^{+})$
We also have from the coherence of $E$ and the fact that $\kappa$ is a cardinal in $\mathcal{N}$ that
- $ Ult_{0}(\mathcal{N},E)|\pi^{\mathcal{N}}_{E}(\kappa) \models \kappa \ \text{is a cardinal}$
Thus by 2. we have
- $ Ult_{0}(\mathcal{M},E)|\pi^{\mathcal{M}}_{E}(\kappa) \models \kappa \ \text{is a cardinal}.$
Since $Ult_{0}(\mathcal{M},E) \models \pi(\kappa) \ \text{is a cardinal} $, it follows from 4. and acceptability that
- $Ult_{0}(\mathcal{M},E) \models \ \kappa \ \text{is a cardinal}$
From 5. and the fact that $\mathcal{M} \models (\kappa \ \text{is a regular caridnal} )$ it follows that
- $\mathcal{M} \models \ \kappa \ \text{is a inaccessible cardinal}$
We have $Ult(\mathcal{M},E) \models h:\pi^{\mathcal{M}}_{E}(\mu) \rightarrow \bigcup_{n\in\omega}\mathcal{P}^{\mathcal{M}}([\pi^{\mathcal{M}}_{E}(\mu)]^{n}) $ and $h=[b,g]_{E}^{\mathcal{M}}$, by Los, we can assume that $g:\kappa^{|b|} \longrightarrow \bigcup_{n\in\omega}\mathcal{P}^{\mathcal{M}}([\mu]^{n})^{\mu}$. We are in the case $ \mu < \kappa$ so from 6. we have
$\mathcal{M} \models |\bigcup_{n\in\omega}\mathcal{P}^{\mathcal{M}}([\mu]^{n})|^{\mu} < \kappa $
So we can assume that $g$ is constant a.e. $E_{b}$.