**Theorem 4.3**: Let *M *be an
invertible isometry. If *A *and *B *are two points which are on the
same side of a
line *L*, then *M*(*A*) and *M*(*B*) are on the same
side of *M*(*L*).

**Proof**: By Theorem 4.2, *M*(*L*) will be a line. Suppose *M*(*A*) and *M*(*B*) are on opposite
sides of *M*(*L*), then by Theorem 2.8, there is
a point *X *which is on the line segment
between* M*(*A*)* *and* **M*(*B*)* *which is on* **M*(*L*).* *Then* **C* = *M*^{-1}(*X*)* *is
a point which is on the
line segment
between* **A *and* **B *and is on the line* L*.* *Then by Theorem 2.4,* **A *and* **B *are on opposite
sides of* **C*,* *so by Theorem 2.5, they are
on opposite
sides of* **L *which contradicts the hypothesis.