Proof of this theorem?

Question

Does anyone know the proof of this theorem:
If line l is parallel to line m, and m is parallel to line n, then l is parallel to n or l is equal to n.

I would LOVE a paragraph proof, but a column proof would also help. Thanks!

Answer 1

Suppose the contrary, that l∦n ∧ l≠n. Then l and n meet at a single point P. Since l∥m, they have no points in common, so in particular P∉m. Then l and n are two distinct lines both parallel to m passing through a point P not on m - contradicting the parallel postulate, which states there is only one such line. Therefore our initial assumption was false, and l∥n ∨ l=n. Q.E.D.