pr}$ on $P^{\theta,\lambda}_\alpha$. Our inductive requirements are:

1.  for each $p\in P^{\theta,\lambda}_\alpha$:\
    $u^p\in [\lambda^+]^{\textstyle<\lambda}$, ${{\rm ht}}(p)\leq\alpha$, $F^p
    \subseteq 2^{\textstyle u^p}$ is a non-empty closed set, $g^p$ is a function with domain ${{\rm dom}}(g^p)=u^p\times{{\rm ht}}(p)$ and values of the form $(\ell,
    \tau)$, where $\ell<2$ and $\tau$ is a Boolean term, and $h^p:u^p\times
    {{\rm ht}}(p)\longrightarrow\theta+2$ is a function,

2.  $\leq^\alpha,\leq^\alpha_{\rm pr}$ are transitive and reflexive relations on $P^{\theta,\lambda}_\alpha$, and $\leq^\alpha$ extends $\leq^\alpha_{\rm pr}$,

3.  if $p,q\in P^{\theta,\lambda}_\alpha$, $p\leq^\alpha
    q$, then $u^p\subseteq u^q$, ${{\rm ht}}(p)\leq{{\rm ht}}(q)$, and $F^p=\{f{\restriction}u^p:
    f\in F^q\}$, and if $p\leq^\alpha_{\rm pr} q$, then for every $i\in u^p$ and $\xi<{{\rm ht}}(p)$ we have $h^p(i,\xi)=h^q(i,\xi)$ and $g^p(i,\xi)= g^q(i,\xi)$,

4.  if $\beta<\alpha$ then $P^{\theta,\lambda}_\beta
    \subseteq P^{\theta,\lambda}_\alpha$, and $\leq^\alpha_{\rm pr}$ extends $\leq^\beta_{\rm pr}$, and $\leq^\alpha$ extends $\leq^\beta$.

For a condition $p\in P^{\theta,\lambda}_\alpha$, we will also declare that ${{\mathbb B}}^p={{\mathbb B}}_{(u^p,F^p)}$ (the Boolean algebra defined in Definition \[0.C\]).

We define $P^{\theta,\lambda}_0=\{\langle\xi\rangle:\xi<\lambda^+\}$ and for $p=\langle\xi\rangle$ we let $F^p=2^{\textstyle\{\xi\}}$, ${{\rm ht}}(p)=0$ and $h^p=\emptyset=g^p$. The relations $\leq^0_{\rm pr}$ and $\leq^0$ both are the equality. \[Clearly these objects are as declared, i.e, clauses (i)$_0$–(iv)$_0$ hold true.\]

If $\gamma<\lambda$ is a limit ordinal, then we put $$\begin{array}{l}
P^*_\gamma=\big\{\langle p_\xi:\xi<\gamma\rangle: (\forall\xi<\zeta<
\gamma)(p_\xi\in P^{\theta,\lambda}_\xi\ \&\ {{\rm ht}}(p_\xi)=\xi \ \&\ p_\xi
\leq^\zeta_{\rm pr} 