we let $${{{\mathbb P}^\theta_\lambda}}=\bigcup\limits_{\alpha<\lambda} P^{\theta,\lambda}_\alpha\quad\mbox{
and }\quad{}\leq_{\rm pr}{}={}\bigcup\limits_{\alpha<\lambda}{\leq^\alpha_{
\rm pr}}\quad\mbox{ and }\quad{}\leq{}={}\bigcup\limits_{\alpha<\lambda}
{\leq^\alpha}.$$ One easily checks that $\leq_{\rm pr}$ is a partial order on ${{{\mathbb P}^\theta_\lambda}}$ and that the relation $\leq$ is transitive and reflexive, and that ${\leq_{\rm pr}} 
\subseteq {\leq}$.

\[3.1\] Let $p,q\in{{{\mathbb P}^\theta_\lambda}}$.

1.  If $p\leq q$ then ${{\rm ht}}(p)\leq{{\rm ht}}(q)$, $u^p\subseteq u^q$ and $F^p=\{
    f{\restriction}u^p: f\in F^q\}$ (so ${{\mathbb B}}^p$ is a subalgebra of ${{\mathbb B}}^q$). If $p\leq q$ and ${{\rm ht}}(p)={{\rm ht}}(q)$, then $q\leq p$.

2.  For each $j\in u^p$, the set $\{\beta<{{\rm ht}}(p): h^p(j,\beta)<\theta\}$ is finite.

3.  If $p\leq_{\rm pr} q$ and $i\in u^p$, then $h^q(i,\beta)\geq\theta$ for all $\beta$ such that ${{\rm ht}}(p)\leq\beta<{{\rm ht}}(q)$.

4.  If $i,j\in u^p$ are distinct, then there is $\beta<{{\rm ht}}(p)$ such that $\theta\neq h^p(i,\beta)\neq h^p(j,\beta)\neq\theta$.

5.  For each finite set $X\subseteq{{\rm ht}}(p)$ there is $i\in u^p$ such that $$\{\beta<{{\rm ht}}(p):h^p(i,\beta)<\theta\}=X.$$

6.  If $p\leq_{\rm pr} q$ then there is a $\leq_{\rm pr}$–increasing sequence $\langle p_\xi:\xi\leq{{\rm ht}}(p)\rangle\subseteq{{{\mathbb P}^\theta_\lambda}}$ such that $p_{{{\rm ht}}(p)}=p$, $p_{{{\rm ht}}(q)}=q$ and ${{\rm ht}}(p_\xi)=\xi$ (for $\xi\leq{{\rm ht}}(p)$). (In particular, if $p\leq_{\rm pr} q$ and ${{\rm ht}}(p)={{\rm ht}}(q)$ then $p=q$.)

7.  If ${{\rm ht}}(p)=\gamma$ is a limit ordinal, $p=\langle p_\xi:\xi<\gamma
    \rangle$, then for each $i\in u^p$ and $\xi<\gamma$: $$i\in u^{p_\xi}\quad\mbox{ 