a ${{\mathbb P}}$–name $\dot{X}$ and a ${{\mathbb P}}$–generic filter $G$ over ${{\bf V}}$, the interpretation of the name $\dot{X}$ by $G$ is denoted by $\dot{X}^G$.

3.  $i,j,\alpha,\beta,\gamma,\delta,\ldots$ will denote ordinals.

4.  For a set $X$ and a cardinal $\lambda$, $[X]^{\textstyle<\lambda}$ stands for the family of all subsets of $X$ of size less than $\lambda$. The family of all functions from $Y$ to $X$ is called $X^{\textstyle Y}$. If $X$ is a set of ordinals then its order type is denoted by ${{\rm otp}}(X)$.

5.  In Boolean algebras we use $\vee$ (and $\bigvee$), $\wedge$ (and $\bigwedge$) and $-$ for the Boolean operations. If ${{\mathbb B}}$ is a Boolean algebra, $x\in{{\mathbb B}}$ then $x^0=x$, $x^1=-x$.

6.  For a subset $Y$ of an algebra ${{\mathbb B}}$, the subalgebra of ${{\mathbb B}}$ generated by $Y$ is denoted by $\langle Y\rangle_{{{\mathbb B}}}$.

[**Acknowledgements:**]{}We would like to thank the referee for valuable comments and suggestions.

The forcing and its basic properties
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Let us start with the definition of the forcing notion ${{{\mathbb P}^\theta_\lambda}}$. By induction on $\alpha<\lambda$ we will define sets of conditions $P^{\theta,
\lambda}_\alpha$, and for each $p\in P^{\theta,\lambda}_\alpha$ we will define $u^p,F^p,{{\rm ht}}(p),h^p$ and $g^p$. Also we will define relations $\leq^\alpha$ and $\leq^\alpha_{\rm 