In this section we complete the proofs of the Theorems 3–11 by putting all the pieces together. First recall Theorem 11 was proved in Subsection 14.3.

Here we first use the result of Subsection 14.2 to reduce all assertions on phases 0–4 in the transition from types in E j−1 to E j to the corresponding scenario for finite multi-level systems with (j + 1)-levels of fitness and j-levels (or (j + 1)-levels) of spatial organization by Proposition 14.3. Then we conclude with the proposition from Subsection 14.1 which were based on the arguments in Section 13.2, the statement of the Theorems 3–10 immediately. (The basis for these results were statements in Section 10 up to Section 12 on phases 0–4).