  This is the full proof of Theorem 3 on the existence of the largest known
degree 8 circulant graph for all diameters stated in the paper "The
degree-diameter problem for circulant graphs of degree 8 and 9" by the author.
To avoid the paper being unduly long the exceptions for only one case were
included in the statement of the proof. In the statement of the proof presented
in this paper the exceptions for all cases are included in full. This proof
closely follows the approach taken by Dougherty and Faber for the existence of
the largest known degree 6 circulant graph for all diameters.
