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structures are decomposable
...A further simplification of the measurement function may be achieved by requiring a special kind of non interaction of the components which has become known as additive independence
...R,P >R,P <>[[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P where F is simply the addition function
...R,P >R,P <>[[Phi]]1 R [[Phi]]2 P [[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P [[Phi]]1 R [[Phi]]2 P It can be shown that starting at the other end given an additively independent representation the properties defined in 1 and 3,and the Archimedean property are necessary
...Here the term [[Phi]]1 [[Phi]]2 is referred to as the interaction term,its absence accounts for the non interaction in the previous condition
...We are now in a position to state the main representation theorem
...Theorem Suppose <R x P,>>is an additive conjoint structure,then there exist functions,[[Phi]]1 from R,and [[Phi]]2 from P into the real numbers such that,for all R,R [[propersubset]]R and P,P [[propersubset]]P:R,P >R,P <>[[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P If [[Phi]]i []are two other functions with the same property,then there exist constants [[Theta]]>0,[[gamma]]1,and [[gamma]]2 such that [[Phi]]1 [][[Theta]][[Phi]]1 [[gamma]]1 [[Phi]]2 [][[Theta]][[Phi]]2 [[gamma]]2 The proof of this theorem may be found in Krantz et al
...Let us stop and take stock of this situation
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