predicate for finitely presentable Distributive structures a, where for each point p in Point a there is an associated FinitePresentation s i a such that its finitePoint is equal to p.

the finite presentation of a given point and according to a finitaly presentable structure.

predicate for splitable slices.

Propterties Let d be Oriented and Sliced ( i k) d for all k, then holds:

1. Let Splitable splt be in Splitable From i d then holds: For all k and s in Slice From (i k) d holds: end s' == end s for all s' in splt s.

splitting of a slice of dimension i k.

finitely presentable Points within a Distributive structure.

Property 1 Let GeneratorTo d k' k'' coker ker lft be in FinitePresentation From i a and let DiagramChainTo p (g:|g':|Nil) = d

         g           g'
  p <<------- p' <------< p''

then holds:

1. coker is the cokernel of g' with g as the shell factor of its universal cone.
2. ker is the kernel of g with g' as the shell factor of its universal cone.
3. KernelSliceFromSomeFreeTip k'' k' ker is valid.

4. For all h in Slice From (i k) a with end h == p holds:

   1. lft h is valid.
   2. orientation (lft h) == start h :> start g.
   3. g *> lft h == h.

             p'
           ^ |
          /  |
  lft h  /   | g
        /    |
       /     v
      * ---> p
         h

Property 2 Let EmbeddingFrom d k' k'' ker coker lft be in -- FinitePresentation To i a then holds the dual of the property 1 above.

the underlying finite point given by the finite presentation.

the generator

the embedding of the finite point.

some slice.

random variable of factors in a having a free start and as end-point the given one.

the underlying random variable for some free slice.

random variable of lift-able free slice froms.

Property Let a be in instance of XStandardSomeFreeSliceFromLiftable then holds: For all p in Point a and SomeFreeSlice (SliceFrom _ h) in the range of xStandardSomeFreeSliceFromLiftable p holds: end h == p.