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.
next theorem (4.4)