It always works

It now remains to demonstrate that the polygons generated by RPA are representative of the class of all polygons – i.e., that any polygon may be generated via this method. This follows from the following observations:

Suppose we have an n-gon P. Let the N vertices of the n-gon be partitioned into the concave and convex sets, V and X
respectively where |N|=|V|+|X| . Clearly |X| > 2 . If we can demonstrate that there are always three adjacent
vertices m, n and o in N where mn and no are edges in the polygon in which m and o are mutually visible, then we are
certain that P can be derived from a polygon of |N| - 1 sides, since the two edges E(mn) and E(no) could have been derived
from a single edge E(mo).

If |V| = 0 then the polygon is convex and any three consecutive nodes are mutually visible. If |V| = 1 then the three vertices of the single concavity are necessarily all mutually visible and we are done.

If |V|>1, choose any concave angle. If its three points are mutually visible then its two line segments may be replaced by
one showing the n-gon is derivable from an n-1 gon by RPA. If they are not, then the line of site between the two endpoints of the angle must be interrupted by a part of the n-gon. This part of the n-gon must, in turn, contain a concavity. Repeat this process obtaining a finite sequence of nested concavities. That finite sequence terminates with a last internal concavity which itself, must have no inner blocking concavities. The three points of a concave angle within that unblocked concavity must all be
mutually visible, implying that the n-gon could have been derived from an n-1-gon