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Parts I and II of this three-part paper provided the fundamental concepts underlying constraint theory whose goal is the systematic determination of whether a mathematical model and its computations are well posed. In addition to deriving results for the general relation, special relations defined as universal and regular were treated. This concluding part treats two more special relations: inequality and discrete. Employing the axiom of transitivity for inequalities, results relating to the consistency of a mathematical model of inequalities in terms of its model graph are derived. Rules for the simultaneous propagation of four types of constraint, over, point, interval, and slack, through a heterogeneous model graph are established. In contrast to other relation types, discrete relations point constrain every relevant variable, so that finding intrinsic constraint sources is trivial. A general procedure is provided to determine the allowability of requested computations on a discrete model.