p_\zeta)\big\},\\
P^{\theta,\lambda}_\gamma=\bigcup\limits_{\alpha<\gamma}P^{\theta,
\lambda}_\alpha\cup P^*_\gamma,
  \end{array}$$ and for $p=\langle p_\xi:\xi<\gamma\rangle\in P^*_\gamma$ we let $$u^p=\bigcup\limits_{\xi<\gamma} u^{p_\xi},\quad F^p=\{f\in 2^{\textstyle
u^p}:(\forall\xi<\gamma)(f{\restriction}u^{p_\xi}\in F^{p_\xi})\},\quad {{\rm ht}}(p)=
\gamma$$ and $h^p=\bigcup\limits_{\xi<\gamma} h^{p_\xi}$ and $g^p=\bigcup\limits_{\xi<
\gamma} g^{p_\xi}$. We define $\leq^\gamma$ and $\leq^\gamma_{\rm pr}$ by:

$p\leq^\gamma_{\rm pr}q$if and only if

[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$, $\alpha<\gamma$ and $p\leq^\alpha_{\rm pr}q$,

[*or*]{} $q=\langle q_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $p\in
P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha_{\rm pr}q_\alpha$ for some $\alpha<\gamma$,

[*or*]{} $p=q$;

$p\leq^\gamma q$if and only if

[*either*]{} $p,q\in P^{\theta,\lambda}_\alpha$, $\alpha<\gamma$ and $p\leq^\alpha q$,

[*or*]{} $q=\langle q_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $p\in
P^{\theta,\lambda}_\alpha$ and $p\leq^\alpha q_\alpha$ for some $\alpha<
\gamma$,

[*or*]{} $p=\langle p_\xi:\xi<\gamma\rangle\in P^*_\gamma$, $q=\langle
q_\xi:\xi<\gamma\rangle\in P^*_\gamma$ and $$(\exists\delta<\gamma)(\forall\xi<\gamma)(\delta\leq\xi\ \Rightarrow\
p_\xi\leq^\xi q_\xi).$$

\[It is straightforward to show that clauses (i)$_\gamma$–(iv)$_\gamma$ hold true.\]

Suppose now that $\alpha<\lambda$. Let $P^*_{\alpha+1}$ consist of all tuples $$\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<\theta\rangle\rangle$$ such that for each $\xi_0<\xi_1<\theta$:

1.  $\zeta^*<\theta$, $n^*<\omega$, $\tau^*=\tau^*(y_1,\ldots,y_{
    n^*})$ is a Boolean term, $u^*\in [\lambda^+]^{\textstyle<\lambda}$,

2.  $p_{\xi_0}\in P^{\theta,\lambda}_\alpha$, ${{\rm ht}}(p)=\alpha$, $v_{\xi_0}\in [u^{p_{\xi_0}}]^{\textstyle n^*}$,

3.  the family $\{u^{p_\xi}:\xi<\theta\}$ forms a $\Delta$–system with heart $u^*$ and $u^{p_{\xi_0}}\setminus u^*\neq
    \emptyset$ and $$\sup(u^*)<\min(u^{p_{\xi_0}}\setminus u^*)\leq\sup (u^{p_{\xi_0}}\setminus
    u^*)<\min(u^{p_{\xi_1}}\setminus u^*),$$

4.  ${{\rm otp}}(u^{p_{\xi_0}})={{\rm otp}}(u^{p_{\xi_1}})$ and if $H:u^{p_{
    \xi_0}}\longrightarrow u^{p_{\xi_1}}$ is the order isomorphism then $H{\restriction}u^*$ is the identity on $u^*$, $F^{p_{\xi_0}}=\{f{\circ}H:f\in F^{p_{\xi_1}}
    \}$, $H[v_{\xi_0}]=v_{\xi_1}$ and $$(\forall j\in u^{p_{\xi_0}})(\forall\beta<\alpha)(h^{p_{\xi_0}}(j,\beta)=
    h^{p_{\xi_1}}(H(j),\beta)\ \&\ g^{p_{\xi_0}}(j,\beta)=g^{p_{\xi_1}}(H(j),
    \beta)).$$

We put $P^{\theta,\lambda}_{\alpha+1}=P^{\theta,\lambda}_\alpha\cup P^*_{
