may find a set $A\in [\lambda^+]^{\textstyle \lambda^+}$ such that

-   conditions $\{p_\delta:\delta\in A\}$ are pairwise isomorphic,

-   the family $\{u^{p_\delta}:\delta\in A\}$ forms a $\Delta$–system with heart $u^*$,

-   if $\delta_0<\delta_1$ are from $A$ then $$\sup(u^*)<\min(u^{p_{\delta_0}}\setminus u^*)\leq \sup(u^{p_{\delta_0}}
    \setminus u^*)<\min(u^{p_{\delta_0}}\setminus u^*).$$

Take an increasing sequence $\langle\delta_\xi:\xi<\theta\rangle$ of elements of $A$, let $\tau^*={\bf 1}$, $v_\xi=\emptyset$ (for $\xi<\theta$), and look at $p=\langle 0,\tau^*,0,u^*,\langle p_{\delta_\xi},v_\xi:\xi<
\theta\rangle\rangle$. It is a condition in ${{{\mathbb P}^\theta_\lambda}}$ stronger than all $p_{\delta_\xi}$’s.

\[defcompo\] By induction on ${{\rm ht}}(p)$ we define [*$\alpha$–components of $p$*]{} (for $p\in{{{\mathbb P}^\theta_\lambda}}$, $\alpha\leq{{\rm ht}}(p)$).

-   First we declare that the only ${{\rm ht}}(p)$–component of $p$ is the $p$ itself.

-   If ${{\rm ht}}(p)=\beta+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,
    v_\xi:\xi<\theta\rangle\rangle$ and $\alpha=\beta$, then $\alpha$–components of $p$ are $p_\xi$ (for $\xi<\theta$); if $\alpha<\beta$, then $\alpha$–components of $p$ are those $q$ which are $\alpha$–components of $p_\xi$ for some $\xi<\theta$.

-   If ${{\rm ht}}(p)$ is a limit ordinal, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$ and $\alpha<{{\rm ht}}(p)$, then $\alpha$–components of $p$ are $\alpha$–components of $p_\xi$ for $\xi\in [\alpha,{{\rm ht}}(p))$.

\[3.4x\] Assume $p\in {{{\mathbb P}^\theta_\lambda}}$ and $\alpha<{{\rm ht}}(p)$.

1.  If $q$ is an $\alpha$–component of $p$ then $q\leq p$, ${{\rm ht}}(q)=
    \alpha$, and for all $j_0,j_1\in u^q$ and every $\beta\in [\alpha,{{\rm ht}}(p))$: $$h^p(j_0,\beta)\neq\theta\ \&\ h^p(j_1,\beta)\neq \theta\quad\Rightarrow
    