Proof of The Interior-Exterior Theorem

The Interior-Exterior Theorem.
If L is an i-line and tεI(2), then t maps the sides of L
to the sides of t(L).

 Proof Observe first, that if A and B lie on opposite sides of L, then all i-lines through A and B must meet L. Thus, if there is an i-line through A and B which does not meet L, then A and B must lie on the same side of L. We proceed by stages: If a point P lies outside a circle C , then there is an extended line through P which does not meet C. Let Q be the centre of C. We may choose the line through P perpendicular to PQ. If a point P lies inside a circle C , then there is a circle through P which does not meet C. Let Q be the centre of C. We may choose the circle on QP as diameter - all of its points are less than |QP| from Q. If C is a circle, and tεI(2), then t maps all points in the exterior of C to points on one side of t(C), and all points in the interior of C to points on the other side of t(C). If P lies in the exterior of C, then, by (1), there is an i-line (an extended line) M through P which does not meet C. Then t(M) is an i-line through t(P) and t(∞) which does not meet t(C). Thus t(P) lies on the same side of t(C) as t(∞). Similarly, using (2), any for any Q in the interior of C, t(Q) lies on the same side of t(C) as t(C), where C is the centre of C. Since t maps C to t(C), it maps the complement of C to the complement of t(C). Thus, the images of the interior and exterior must cover both sides of t(C), so the interior and exterior must map to opposite sides. Finally, suppose that L is an extended line. We can choose an inversion h which maps L to a circle C, so h-1(C) = L. Applying (3) to C and h-1, we see that h-1 maps the interior of C to one side of L. Thus, h maps this side of L to the interior of C, and the other side to the exterior. Now apply (3) to C and toh-1. This maps the interior of C to one side of (toh-1)(C) = t(h-1(C)) = t(L), and the exteriorto the other side. Now, applying h then toh-1, we see that t = (toh-1)oh maps the sides of L to the sides of t(L), as required.