In chapter 8 of Mitchell's and Steel's FSIT, they prove a central fine structural result, which basically states that if $\mathcal{M}$ is 1-small, $k$-sound, $k$-iterable premouse then the $k+1$-strandard parameter is $k+1$-solid/universal over $\mathcal M$.

The proof is not too complicated but on page 76 they state that if $\rho_{k+1}^{\mathcal M}> lh(E)$ where $E$ is some extender on the $\mathcal M$-sequence then $\mathcal H= \mathcal H_{k+1}^{\mathcal M}(\alpha_s)$ (I omitted the rest of the parameter and $u$ ($u$ is the tuple having the solidity witnesses and the appropriate projectums whenever defined)from the hull so that it is easier to read, this is well written in details page 74) then $\mathcal H$ is an initial segment of $\mathcal M$. I can't see why this is true. Could anyone help me figure out the reason behind this? Thx.