Distributive justice in taxation

A classical criterion for apportioning taxes is that all should sacrifice equally in loss of utility. Suppose that a method of apportioning taxes is continuous and has the following four properties: (i) the way that taxpayers split a given tax total depends only on their own taxable incomes; (ii) an increase in the tax total implies that everyone pays more; (iii) every incremental increase in tax is apportioned according to taxpayers' current after-tax incomes; (iv) the ordering of taxpayers by pre-tax income and after-tax income is the same. Then there exists a utility function relative to which all sacrifice equally.