Summary: We enrich the traditional model of bargaining by adding to the disagreement point and the feasible set a point representing claims (or expectations) that agents may have when they come to the bargaining table. We assume this point to be outside of the feasible set: the claims are incompatible. We look for solutions to the class of problems so defined, using the axiomatic method. We follow the various approaches that have been found useful in the classical axiomatic theory of bargaining. We successively consider how solutions respond to certain changes in (i) the feasible set, (ii) the disagreement point and the claims point, and (iii) the number of agents. In each case we formulate appropriate axioms and study their implications. Surprisingly, each of these approaches leads to the same solution: it is the solution that associates with each problem the maximal point of the feasible set on the line segment connecting the disagreement point to the claims point. We name this solution the proportional solution.