\alpha+1}$ and for $p=\langle\zeta^*,\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<
\theta\rangle\rangle\in P^*_{\alpha+1}$ we let $u^p=\bigcup\limits_{\xi<
\theta} u^{p_\xi}$ and $$\begin{array}{ll}
F^p=\{f\in 2^{\textstyle u^p}:& (\forall\xi<\theta)(f{\restriction}u^{p_\xi}\in
F^{p_{\xi}})\mbox{ and for all }\xi<\zeta<\theta\\
\ &f(\sigma_{\rm maj}(\tau_{3\cdot\xi},\tau_{3\cdot\xi+1},\tau_{3\cdot\xi+
2}))\leq f(\sigma_{\rm maj}(\tau_{3\cdot\zeta},\tau_{3\cdot\zeta+1},\tau_{3
\cdot\zeta+2}))\},
  \end{array}$$ where $\tau_\xi=\tau^*(x_i:i\in v_\xi)$ for $\xi<\theta$ (so $\tau_\xi$ is an element of the algebra ${{\mathbb B}}^{p_\xi}={{\mathbb B}}_{(u^{p_\xi},F^{p_\xi})}$), and $\sigma_{\rm maj}(y_0,y_1,y_2)=(y_0\wedge y_1)\vee (y_0\wedge y_2)\vee
(y_1\wedge y_2)$. Next we let ${{\rm ht}}(p)=\alpha+1$ and we define functions $h^p,g^p$ on $u^p\times (\alpha+1)$ by $$h^p(j,\beta)=\left\{\begin{array}{lll}
h^{p_{\xi}}(j,\beta)&\mbox{if}&j\in u^{p_\xi},\ \xi<\theta,\ \beta<\alpha,\\
\theta                 &\mbox{if}&j\in u^*,\ \beta=\alpha,\\
\theta+1               &\mbox{if}&j\in u^{p_{\zeta^*}}\setminus u^*,\
\beta=\alpha,\\ 
\xi                    &\mbox{if}&j\in u^{p_\xi}\setminus u^*,\ \xi<\theta,\
\xi\neq\zeta^*,\ \beta=\alpha, 
             \end{array}\right.$$ $$g^p(j,\beta)=\left\{\begin{array}{lll}
g^{p_{\xi}}(j,\beta)&\mbox{if}&j\in u^{p_\xi},\ \xi<\theta,\ \beta<\alpha,\\
(1,\tau^*)          &\mbox{if}&j\in v_\xi,\ \xi<\theta,\ \beta=\alpha,\\
(0,\tau^*)          &\mbox{if}&j\in u^{p_\xi}\setminus v_\xi,\ \xi<\theta,\
\beta=\alpha.
             \end{array}\right.$$ Next we define the relations $\leq^{\alpha+1}_{\rm pr}$ and $\leq^{\alpha+1}$ by:

$p\leq^{\alpha+1}_{\rm pr}q$if and only if

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

[*or*]{} $q=\langle \zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:\xi<\theta
\rangle\rangle\in P^*_{\alpha+1}$, $p\in P^{\theta,\lambda}_\alpha$, and $p\leq^\alpha_{\rm pr} q_{\zeta^*}$,

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

$p\leq^{\alpha+1} q$if and only if

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

[*or*]{} $q=\langle\zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:\xi<\theta
\rangle\rangle\in P^*_{\alpha+1}$, $p\in P^{\theta,\lambda}_\alpha$, and $p\leq^\alpha q_\xi$ for some $\xi<\theta$,

[*or*]{} $p=\langle\zeta^{**},\tau^*,n^*,u^*,\langle p_\xi,v_\xi:\xi<
\theta\rangle\rangle$, $q=\langle\zeta^*,\tau^*,n^*,u^*,\langle q_\xi,v_\xi:
\xi<\theta\rangle\rangle$ are from $P^*_{\alpha+1}$ and $$(\forall\xi<\theta)(p_\xi\leq^\alpha q_\xi\ \&\ u^{p_\xi}=u^{q_\xi}).$$

\[Again, it is easy to show that clauses (i)$_{\alpha+1}$–(iv)$_{
\alpha+1}$ are satisfied.\]

After the construction is carried out 