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## Derivation of Sum Rule and Product Rule from FUNC

The three principles, in full detail, are:
FUNC: Let A be a self-adjoint operator associated with observable A, let f: R R be an arbitrary function, such that f(A) is self-adjoint operator, and let | f > be an arbitrary state; then f(A) is associated uniquely with an observable f(A) such that:
v(f(A)) = f(v(A)) Sum Rule: If A and B are commuting self-adjoint operators corresponding to observables A and B, respectively, then A + B is the unique observable corresponding to the self-adjoint operator A + B and

v(A + B) = v(A) + v(B) Product Rule: If A and B are commuting self-adjoint operators corresponding to observables A and B, respectively, then if A B is the unique observable corresponding to the self-adjoint operator A B and

v(AB) = v(A)  v(B) In order to derive Sum Rule and Product Rule from FUNC, we use the following mathematical fact: Let A and B be commuting operators, then there is a maximal operator C and there are functions f, g such that A = f(C) and B = g(C).

So, for two commuting operators A, B:

Since A = f(C) and B = g(C), there is a function h = f+g, such that A + B = h(C).
Therefore:
 v(A + B) = h(v(C) ) (by FUNC) = f(v(C) ) + g(v(C) ) = v(f(C)) + v(g(C)) (by FUNC) = v(A) + v(B) (Sum Rule)
Similarly:
Since A = f(C) and B = g(C), there is a function k = f g, such that A B = k(C).
Therefore:
 v(A B) = k(v(C) ) (by FUNC) = f(v(C) ) g(v(C) ) = v(f(C))  v(g(C)) (by FUNC) = v(A)  v(B) (Product Rule)