Theorem distributions [in standard mathematics]

The picture below shows which of the possible theorems from page 812 hold for each of the numbered standard mathematical theories from page 805. The theorem close to the right-hand end valid in many cases is (p ∘ p) ∘ p ＝ p ∘ (p ∘ p). The lack of regularity in this picture can be viewed as a sign that it is difficult to tell which theorems hold, and thus in effect to do mathematics.