... A or C, and the values of the distance at the points A, C are greater than the values of the distance at the points E, H, I, we get the desiredstrict inequality and this part of the proof is ... completes the proof.It should be noted that the above theorem does not have an empty scope. There exists a class ofincome distribution functions fulfilling all the conditions in the theorem. Take the ... R+, hence the supremum over the empty set is 0 and the infimum over the empty set is ∞.(i), (ii) The proofs can b e left to the reader since they are immediate consequences of the definitions...