\quad h^p(j_0,\beta)=h^p(j_1,\beta).$$ Moreover, for each $i\in u^p$ there is a unique $\alpha$–component $q$ of $p$ such that $i\in u^q$ and $$(\forall j\in u^q)(\forall\beta\in [\alpha,{{\rm ht}}(p)))(h^p(i,\beta)\geq
    \theta\ \Rightarrow\ h^p(j,\beta)\geq\theta).$$

2.  If $H$ is an isomorphism from $p$ onto $p'\in{{{\mathbb P}^\theta_\lambda}}$, and $q$ is an $\alpha$–component of $p$, then $H(q)$ is an $\alpha$–component of $p'$. If $q_0,q_1$ are $\alpha$–components of $p$ then $q_0,q_1$ are isomorphic.

3.  There is a unique $\alpha$–component $q$ of $p$ such that $q\leq_{\rm
    pr} p$.

Easy inductions on ${{\rm ht}}(p)$.

\[closed\] By induction on ${{\rm ht}}(p)$ we define when a set $Z\subseteq \lambda$ is $p$–closed for a condition $p\in{{{\mathbb P}^\theta_\lambda}}$.

-   If ${{\rm ht}}(p)=0$ then every $Z\subseteq\lambda$ is $p$–closed;

-   if ${{\rm ht}}(p)$ is limit, $p=\langle p_\xi:\xi<{{\rm ht}}(p)\rangle$, then $Z$ is $p$–closed provided it is $p_\xi$–closed for each $\xi<{{\rm ht}}(p)$;

-   if ${{\rm ht}}(p)=\alpha+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle
    p_\xi,v_\xi:\xi<\theta\rangle\rangle$ and $\alpha\notin Z$, then $Z$ is $p$–closed whenever it is $p_{\zeta^*}$–closed;

-   if ${{\rm ht}}(p)=\alpha+1$, $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle
    p_\xi,v_\xi:\xi<\theta\rangle\rangle$ and $\alpha\in Z$, then $Z$ is $p$–closed provided it is $p_{\zeta^*}$–closed and $$\{\beta<\alpha:(\exists j\in v_{\zeta^*}\cup\{\min(u^{p_{\zeta^*}}
    \setminus u^*)\})(h^{p_{\zeta^*}}(j,\beta)<\theta)\}\subseteq Z.$$

\[3.5.1\]

1.  If $p\in{{{\mathbb P}^\theta_\lambda}}$ and $w\in [{{\rm ht}}(p)]^{\textstyle<\omega}$, then there is a finite $p$–closed set $Z\subseteq{{\rm ht}}(p)$ such that $w\subseteq Z$.

2.  If $p,q\in{{{\mathbb P}^\theta_\lambda}}$ are 