$\leq_{\rm pr}$ represents “the official line in history”, and sometimes we will have to rewrite that official history, see Definition \[defptran\] and Lemma \[3y1\] (on changing history see also Orwell [@Or49]).

The forcing notion ${{{\mathbb P}^\theta_\lambda}}$ has some other interesting features. (For example, conditions are very much like fractals, they contain many self-similar pieces (see Definition \[defcompo\] and Lemma \[3.4x\]).) The method of historic forcing notions could be applicable to more problems, and this is why in our presentation we separated several observations of general character (presented in the first section) from the problem specific arguments (section 2)

[**Notation:**]{}Our notation is standard and compatible with that of classical textbooks on set theory (like Jech [@J]) and Boolean algebras (like Monk [@M1], [@M2]). However in forcing considerations we keep the older tradition that

[*the stronger condition is the greater one.* ]{}

Let us list some of our notation and conventions.

1.  Throughout the paper, $\theta,\lambda$ are fixed regular infinite cardinals, $\theta<\lambda$.

2.  A name for an object in a forcing extension is denoted with a dot above (like $\dot{X}$) with one exception: the canonical name for a generic filter in a forcing notion ${{\mathbb P}}$ will be called $\Gamma_{{\mathbb P}}$. For 