  From VR, VD, NC and STAT FUNC, we can derive FUNC as follows. Consider an arbitrary state of a system and an arbitrary observable Q.  By VD, Q possesses a value v(Q)=a. Thus, we can form the number f(v(Q))=b for an arbitrary function f. For this number, by STAT FUNC, prob[f(v(Q))=b] = prob[v(f(Q))=b].  Hence, we have, by transforming probabilities according to STAT FUNC, created a new self-adjoint operator f(Q), and associated it with the two real numbers b and prob[f(v(Q))=b]. Thus, by VR, there is an observable corresponding to f(Q) with value b, hence f(v(Q))=v(f(Q)). By NC, that observable is unique, hence FUNC follows.