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)