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.

next theorem (5.5)