\begin{definition}[Definition:Euclidean Plane]
For any real number $a$ let:
:$L_a = \set {\tuple {x, y} \in \R^2: x = a}$
Furthermore, define:
:$L_A = \set {L_a: a \in \R}$
For any two real numbers $m$ and $b$ let:
:$L_{m, b} = \set {\tuple {x, y} \in \R^2: y = m x + b}$
Furthermore, define:
:$L_{M, B} = \set {L_{m, b}: m, b \in \R}$
Finally let:
:$L_E = L_A \cup L_{M, B}$
The abstract geometry $\struct {\R^2, L_E}$ is called the '''Euclidean plane'''.
\end{definition}