Euclidean Plane is Abstract Geometry

Theorem
The Euclidean plane $\left({\mathbb{R}, L_E}\right)$ is an abstract geometry.

Proof
We will show that the axioms for an abstract geometry hold.

Axiom 1
Let $P=\left({x_1,y_1}\right)$ and $Q=\left({x_2,y_2}\right)$ be two points in $\left({\mathbb{R}, L_E}\right)$.

If $x_1 = x_2 = a$ then $P,Q \in L_a$.

If $x_1 \neq x_2$ then let:


 * $m = \frac{y_2-y_1}{x_2-x_1}$


 * $b = y_2 - mx_2$

By definition then $P,Q \in L_{m,b}$.

So any two points in $\R^2$ lie on a line in $L_E$.

Axiom 2
For any $a \in \R$:


 * $\left({a,0}\right),\left({a,1}\right) \in L_a$

Also for any $m, b \in \R$:


 * $\left({0,b}\right),\left({1,m+b}\right) \in L_{m,b}$

So every line in $L_E$ has at least two points.

Hence $\left({\mathbb{R}, L_E}\right)$ is an abstract geometry.