Parallelism is Equivalence Relation/Transitivity

Theorem
Let $S$ be the set of straight lines in the plane.

For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$.

Then $\parallel$ is a transitive relation on $S$.

Proof
From Parallelism is Transitive Relation:


 * $l_1 \parallel l_2$ and $l_2 \parallel l_3$ implies $l_1 \parallel l_3$.

Thus $\parallel$ is seen to be transitive.