12.5: Axiom II
Note that once the following claim is proved, Axiom II follows from Corollary 10.5.2 .
A subset of the h-plane is an h-line if and only if it forms a line for the h-distance in the sense of Definition 1.5.1 .
- Proof
-
Let \(\ell\) be an h-line. Applying the main observation ( Theorem 12.3.1 ) we can assume that \(\ell\) contains the center of the absolute. In this case, \(\ell\) is an intersection of a diameter of the absolute and the h-plane. Let \(A\) and \(B\) be the endpoints of the diameter.
Consider the map \(\iota : \ell \to \mathbb{R}\) defined as
Note that \(\iota :\ell\to \mathbb{R}\) is a bijection.
Further, if \(X,Y\in \ell\) and the points \(A\) , \(X\) , \(Y\) , and \(B\) appear on \([AB]\) in the same order, then
\(\iota(Y)-\iota(X)=\ln \dfrac{AY}{YB}-\ln \dfrac{AX}{XB}=\ln \dfrac{AY\cdot BX}{YB\cdot XB}=XY_h.\)
We proved that any h-line is a line for h-distance. The converse follows from Claim 12.4.3 .