 Positive affine uniqueness of affine preference representations

Two real affine functionals on a convex set of lotteries that represent the same strict preferences differ by $V=aU+b$ with $a>0$. Equal utility values must first be shown to give equal values of the other functional. The induced function on the real interval of utility values preserves mixtures; anchor interpolation, including exterior points represented through an anchor, makes it linear throughout. Constant representations are handled separately and still admit a positive slope.