betweenness in rays

Let $S$ be a linear ordered geometry.Fix a point $p$ and consider the pencil $\mathrm{\Pi }(p)$ of all raysemanating from it. Let $\alpha \ne \beta \in \mathrm{\Pi }(p)$. A point $q$ issaid to be an interior point of $\alpha$ and $\beta$ if thereare points $s\in \alpha$ and $t\in \beta$ such that

1. $q$ and $s$ are on the same side of line $\overleftrightarrow{pt}$, and

2. $q$ and $t$ are on the same side of line $\overleftrightarrow{ps}$.

A point $q$ is said to be between $\alpha$ and $\beta$ ifthere are points $s\in \alpha$ and $t\in \beta$ such that $q$ isbetween $s$ and $t$. A point that is between two rays is aninterior point of these rays, but not vice versa in general. A ray$\rho \in \mathrm{\Pi }(p)$ is said to be between rays $\alpha$ and$\beta$ if there is an interior point of $\alpha$ and $\beta$ lyingon $\rho$.

Properties

1. Suppose $\alpha ,\beta ,\rho \in \mathrm{\Pi }(p)$ and $\rho$ is between$\alpha$ and $\beta$. Then

   1. any point on $\rho$ is an interior point of $\alpha$ and$\beta$;

   2. any point on the line containing $\rho$ that is an interiorpoint of $\alpha$ and $\beta$ must be a point on $\rho$;

   3. there is a point $q$ on $\rho$ that is between $\alpha$ and$\beta$. This is known as the Crossbar Theorem, the two “crossbars” being $\rho$ and a line segment joining a point on $\alpha$ and a point on $\beta$;

   4. if $q$ is defined as above, then any point between $p$ and$q$ is between $\alpha$ and $\beta$.

2. There are no rays between two opposite rays.

3. If $\rho$ is between $\alpha$ and $\beta$, then $\beta$ is notbetween $\alpha$ and $\rho$.

4. If $\alpha ,\beta \in \mathrm{\Pi }(p)$ has a ray $\rho$ between them, then$\alpha$ and $\beta$ must lie on the same half plane of some line.

5. The converse of the above statement is true too: if$\alpha ,\beta \in \mathrm{\Pi }(p)$ are distinct rays that are not opposite ofone another, then there exist a ray $\rho \in \mathrm{\Pi }(p)$ such that $\rho$is between $\alpha$ and $\beta$.

6. Given $\alpha ,\beta \in \mathrm{\Pi }(p)$ with $\alpha \ne \beta$ and$\alpha \ne -\beta$. We can write $\mathrm{\Pi }(p)$ as a disjoint union oftwo subsets:

   1. $A=\{\rho \in \mathrm{\Pi }(p)\mid \rho \text{ is between }\alpha \text{ and }\beta \}$,

   2. $B=\mathrm{\Pi }(p)-A$.

   Let $\rho ,\sigma \in \mathrm{\Pi }(p)$ be two rays distinct from both $\alpha$and $\beta$. Suppose $x\in \rho$ and $y\in \sigma$. Then$\rho ,\sigma$ belong to the same subset if and only if$\overline{xy}$ does not intersect either $\alpha$ or $\beta$.