Theorem 5.4: (The angle addition axiom) Let C be a point on the arc between A and B. Then the length of the arc from A to B is the sum of the lengths of the arcs from A to C and from C to B.
Proof: If C is not in a partition, we will get a better approxmation by adding it to the partition. In all of the partitions which include C the lengths of the inside and outside approximations of the arc length from A to B will be equal to the lengths of the inside and outside approximations of the arc length from A to C plus the lengths of the inside and outside approximations of the arc length from C to B.