Theorem 5.3: Arc length is preserved by invertible isometries.

Proof: Let   A   be an arc, and let   {P0, P1, P2, . . . , Pn}   be a partition of the arc   A.   Let O be the center of the circle which contains   A,   and let   M   be an invertible isometry. Then,   Pi   is in the angle between   OPi-1   and   OPi+1   which is to say that   Pi is on the same side of   OPi-1   as   Pi+1   and on the same side of   OPi+1 as   Pi-1. By Theorem 4.3   M(Pi)   will be on the same side of the line determined by   M(O)   and   M(Pi-1)   as   M(Pi+1)   and on the same side of the line determined by   M(O)   and   M(Pi+1)   as   M(Pi-1),   so   {M(P0), M(P1), M(P2), . . . , M(Pn)}   is a partition of   M(A).   Conversely, given a partition of   M(A),   since   M-1   is also an invertible isometry, the image of this partition under   M-1   will be a partition of   A. Thus invertible isometries will set up a one to one correspondence between partitions of the original arc and partitions of the arc to which is has been moved.

Since the lengths of all the partitions will be preserved, the limits will be the same.

next theorem (5.4)