Let p be another straight line which is not parallel to l, m or n, but is inclined to the first line l at an angle x. Furthermore, let p intersect all three straight lines.
p intersects l and m as a transversal. Since I and m are parallel, p meets l and m at the same angle x (corresponding angles). x is also the measure of the angle directly opposite it formed by the transversal on l, so we have equal (congruent) alternate angles, another property of transversals between parallels.
So we've established the transversal angles on m, and the same properties apply between m and n.
l and n are parallel because the rules applicable to l and m also apply to m and n because they're parallel, too. We conclude that the angles formed by the transversal on l match those on n, and therefore l is also parallel to n because of the corresponding and alternate angles.