[contravariant]

Bit of an overcomplicated proof that DMU implies that gains add less utility than an equivalent loss removes. A simpler proof goes as follows:

Let U be a concave utility function, and let 'w' and 'e' be some amount of 'wealth'. Concavity implies:

U(w+e)/2 + U(w-e)/2 <= U(w)

hence

U(w+e) + U(w-e) <= 2*U(w)

rearranging the terms we find

(U(w+e) - U(w)) + (U(w-e) - U(w)) <= 0

and therefore

U(w+e) - U(w) <= U(w) - U(w-e).

So gains add less utility than losses remove.