I must prove that the map g from point to linear surface is well defined.
Consider a linear surface b through point p that f maps to coordinates c0, c1, ... cn-1. Since b goes through p, one coordinate ci depends on the other coordinates. If every coordinate except ci and one other cj is held fixed, then by similar triangles,
Implications of this algorithm for abstract linear spaces include that every linear space of the same number of boundaries has the same number of sections, and that the notion of sidedness of n-tuples of boundaries wrt boundaries is well defined.
Section Space TheoremIf space S0n is linear, then there exists linear space S1n, unique up to isomorphism, with boundaries corresponding to n-tuples of boundaries from S0n and regions corresponding to sections of S0n.
The proof follows from the algorithm above.