Making an S_x^y-block.

``Making an S<sub>x</sub><sup>y</sup>-block'' is the process which is applied to eliminate all deficiencies in x w.r.t. y, when $y \in C_x^B$ for some B. Eliminating a deficiency in x w.r.t. y is nothing but providing an S<sub>x</sub>-successor of y that suits the job. If we want to respect invariant 3 we must ensure that every S<sub>x</sub>-successor of y is B-critical as well. It turns out to be possible to define a finite construction of S<sub>x</sub>-successors of y such that no deficiency remains in x w.r.t. y and w.r.t. the whole S<sub>x</sub><sup>y</sup>-block, and such that moreover this whole construction lies B-critically above x. This is our so-called process of ``making an S_x^y-block'', and it proceeds as follows.

• We first set the S_x^y-block to be empty.

• Now for every deficiency in x w.r.t. y we put a y' in the S_x^y-block which repairs the deficiency in a sufficient way. If $C \rhd D \in x$is such a deficiency, with $C \in y$, lemma 4.8 guarantees the existence of a world y' such that $x \prec_{\mbox{\footnotesize {B}}} y' , D \in y' , y \subset_{\Box} y'$, and y' being maximal in this $\subset_{\Box}$-ordering. Note that there is no ambiguity of the B since invariant 4 guarantees its uniqueness. If we define y S_x y', this y' has repaired the deficiency $C \rhd D \in x$. We have to add also x R y', etcetera. In other words we have to close under the $\mathit{ILM}$ frame conditions in a minimal way. After all this no deficiencies in x exist any more w.r.t. y. There might be very well a whole lot of deficiencies in x w.r.t. the S_x^y-block constucted so far. They are taken care of in the following loop.

Repeat the following steps as long as deficiencies of x w.r.t. the S_x^y-block still do exist.

• Fix a y' in the S_x^y-block such that there is a deficiency of xw.r.t. y'. Say $C \rhd D$ is a deficiency of x w.r.t. y'. If there is a y'' in the S_x^y-block with $D \in y''$ and $\forall E ( \Box E \in y' \rightarrow \Box E \in y'' )$, then define y' S_x y''. If there is no such y'' then use lemma 4.8 to obtain a B-critical successor y'' of x with $D \in y''$ such that $\forall E ( \Box E \in y' \rightarrow \Box E \in y'' )$ and moreover such that y'' is maximal with respect to this $\Box$-inclusion. Again define y' S_x y''.

• Close off under the frame conditions of $\mathit{ILM}$ in a minimal way.

It is evident that the process of making an S_x^y-block will terminate. There is only a finite number of possible deficiencies of x w.r.t. any specific y'. This implies that every world in the S_x^y-block needs only a finite number of S_x-successors. An arbitrary chain (without loops) of S_y-successors is limited in length due to the maximality of all worlds w.r.t. the $\Box$-inclusion. By the clause ``If there is a y'' in the S_x^y-block with $\ldots$, we are forced to use the same y'' again if it is approppiate. These two ingredients ensure the finiteness of the S_x^y-block.