#### Supplement to The Kochen-Specker Theorem

## Derivation of Sum Rule and Product Rule from FUNC

The three principles, in full detail, are:

FUNC:Letbe a self-adjoint operator associated with observableAA, letf:R→Rbe an arbitrary function, such thatis a self-adjoint operator, and let | φ> be an arbitrary state; thenf(A)is associated uniquely with an observablef(A)f(A) such that:v(f(A))^{|φ>}=f(v(A))^{|φ>}

Sum Rule:IfandAare commuting self-adjoint operators corresponding to observablesBAandB, respectively, thenA+Bis the unique observable corresponding to the self-adjoint operatorandA + Bv(A+B)^{|φ>}=v(A)^{|φ>}+v(B)^{|φ>}

Product Rule:IfandAare commuting self-adjoint operators corresponding to observablesBAandB, respectively, then ifA·Bis the unique observable corresponding to the self-adjoint operator·AandBv(AB)^{|φ>}=v(A)^{|φ>}·v(B)^{|φ>}

In order to derive Sum Rule and Product Rule from FUNC, we use the
following mathematical fact: Let __ A__ and

__be commuting operators, then there is a maximal operator__

**B**__and there are functions__

**C***f*,

*g*such that

__=__

**A***f*(

__) and__

**C**__=__

**B***g*(

__).__

**C**
So, for two commuting operators __ A__,

__:__

**B**Since=Af() andC=Bg(), there is a functionCh=f+g, such that=A + Bh().C

Therefore:

v(A+B)^{|φ>}= h(v(C)^{|φ>})(by FUNC) = f(v(C)^{|φ>}) +g(v(C)^{|φ>})= v(f(C))^{|φ>}+v(g(C))^{|φ>}(by FUNC) = v(A)^{|φ>}+v(B)^{|φ>}(Sum Rule)

Similarly:

Since=Af() andC=Bg(), there is a functionCk=f·g, such that·A=Bk().C

Therefore:

v(A·B)^{|φ>}= k(v(C)^{|φ>})(by FUNC) = f(v(C)^{|φ>}) ·g(v(C)^{|φ>})= v(f(C))^{|φ>}·v(g(C))^{|φ>}(by FUNC) = v(A)^{|φ>}·v(B)^{|φ>}(Product Rule)