Theorem 1.13: A line is parallel to one of two parallel lines if and only if it is parallel to the other.

Proof: If a line is parallel to one of two parallel lines, then they all have the same slope, and are all parallel to each other.

If it is not parallel to one of two parallel lines then it has a different slope from the one parallel line and so has a different slope than the other of the two parallel lines, and is not parallel to the other line either.

next theorem (1.14)