The aim of this appendix is to prove that a metric space which satisfies the incidence axiom and the reflection axiom is isometric to the Euclidean plane or (after rescaling) the hyperbolic plane. For a precise statement see the first page of the introduction. The need for a proof comes from the fact that our axioms are somewhat untraditional. The proof is presented as a systematic series of deductions from the axioms. This will be carried out to the point where we have proved the “Pasch's axiom”, see the discussion at the end of the appendix.

Let us agree that a straight geodesic curve in a metric space X is a distance preserving map γ:ℝ-> X and recall that a line in X is the image of a straight geodesic curve, compare II.1. In the following we let X be a plane i.e. a metric space X which satisfies the following two axioms

INCIDENCE AXIOM Through two distinct points of X there passes a unique line. The space X has at least one point.

REFLECTION AXIOM The complement of a given line in X has two connected components. There exists an isometry σ of X which fixes the points of the line, but interchanges the two connected components of its complement.