Plane/Parallel lines/Equivalence classes/Example

On the set of all lines in the plane, we can consider the property of being parallel as an equivalence relation. A line is parallel to itself, the relation is obviously symmetric, and if $G_{1}$ and $G_{2}$ are parallel, and $G_{2}$ and $G_{3}$ are parallel, then also $G_{1}$ and $G_{3}$ are parallel. The equivalence class of a line $G$ consists of all lines that are parallel to $G$ ; these lines form a parallel linenschar. We fix a point $M$ in the plane. Then there exists, for every line $G$ , a parallel line $G^{'}$ running through $M$ . Therefore, every equivalence class can be uniquely represented by a line through the point $M$ . The set of lines through $M$ form a system of representatives for this equivalence relation.