This theorem is a simple result of the fact that U($(X+55))-U($X) must be greater than 55MU($(X+54)) (each dollar up from the Xth up to the (X+54)th must have marginal utility at least MU($(X+55))), while U($X)-U($(X-50)) must be less than 50MU($(X-100)) (each dollar from the (X-50)th up to (X-1)th must have marginal utility at most MU($(X-50))).

Are you sure you didn't intend to have (X+55) in the formula just after "greater than" instead of (X+54) and (X-50) in the last formula before the final parenthetical instead of (X-100)?

Also I think the formulas would be more readable if you omitted the dollar sign.