We have systems of distribution.
In S1 your probability of getting A is Ps1a and your probability of getting B is Ps1b.
In S2 your probability of getting A is Ps2a and your probability of getting B is Ps2b.
1) You either prefer S1 to S2 or S2 to S1 or you are indifferent
2) If for you S1>S2 and S2>S3 then S1>S3
3) You preference between S1 and S2 and S3 is continious
4) If you prefer S1 to S2 (S1>S2)
and you have a chance of p to get involved in S1 when you must inevitably engage in S3 with a chance of (1-p)
You would prefer this scenario compared with getting involved in S2 when you must inevitably engage in S3 with a chance of (1-p)
p.S1+(1-p).S3 > p.S2+(1-p).S3
They prove that
there exists a utility function U such that if you prefer S1 to S2
Your utility U under S1 will be U(S1)=U(Ps1a.A+Ps2b.B)
There is also U under which U(S2)=U(Ps2a.A+Ps2b.B)
This means that Ps1a.U(A)+Ps2b.U(B)>Ps2a.U(A)+Ps2b.U(B)
(c) Amir H. Ghaseminejad, 3 Jan 2014