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.