That makes sense to me. I'm assuming that another way to state this proposition is simply this: a dollar is of greater value to a poor person than it is to a rich person.
This is, I think, true, for a couple of reasons. First, a dollar would increase a poor person's wealth by a greater percent than it would increase a rich person's wealth. Therefore it would be of greater value. Second, the things a poor person would be likely to buy with an extra dollar are likely to be more important (more closely related to simple survival) than the things a rich person would be likely to buy (luxuries).
Therefore, it seems to me that the statement is true. As far as I can tell, though, this is simply an opinion question.
A dollar taken away from a rich person and given to a rich person creates more utility for the poor person more than the reduction of utility for the rich person. This happens because of the law of diminishing marginal utility. This law of economics holds that marginal (or additional) utility of a dollar spent on a good reduces with the total quantity of that goods consumed increases.
Implication of the law of diminishing marginal utility is that a person will spend the first dollar available on the good that gives maximum utility per dollar spent. As more and more dollars are spent on that good, its marginal utility will decrease to a level below that of the marginal utility of some other good. In this situation the next dollar will be spent on this second good. In this way, as more and more money is available with a person, more and more goods will be added to the purchase basket, In this purchase basket, the last unit of the last item added will have the lowest utility. Further, the utility of the last unit of last item spent in terms of utility per dollar will decrease as the total spending decreases. As per this reasoning the utility of the last dollar spent by a rich person will be less than utility of last dollar spent by a poor person because the total spending of a rich person is more than that of a poor person.
The logic given above assumes that both rich person derive same kind of utilities from same goods, or that their preferences among different goods are same. Let us assume a situation where the rich person is faced with a life threatening health disorder so that he has to cut down expenses on many other items to be able to pay the medical bills. In this case the utility of the last dollar spent by a rich person in such a situation may be more than the last dollar spent by a comparatively poorer person.