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# 12.5: Axiom II

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Note that once the following claim is proved, Axiom II follows from Corollary 10.5.2.

Claim $$\PageIndex{1}$$

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

$$\iota(X)=\ln \dfrac{AX}{XB}.$$

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.