let Omega be non empty finite set ; :: thesis: for P being Probability of Trivial-SigmaField Omega
for X being Real-Valued-Random-Variable of (Trivial-SigmaField Omega) ex F being FinSequence of REAL ex s being FinSequence of Omega st
( len F = card Omega & s is one-to-one & rng s = Omega & len s = card Omega & ( for n being Nat st n in dom F holds
F . n = (X . (s . n)) * (P . {(s . n)}) ) & expect (X,P) = Sum F )
let P be Probability of Trivial-SigmaField Omega; :: thesis: for X being Real-Valued-Random-Variable of (Trivial-SigmaField Omega) ex F being FinSequence of REAL ex s being FinSequence of Omega st
( len F = card Omega & s is one-to-one & rng s = Omega & len s = card Omega & ( for n being Nat st n in dom F holds
F . n = (X . (s . n)) * (P . {(s . n)}) ) & expect (X,P) = Sum F )
let X be Real-Valued-Random-Variable of (Trivial-SigmaField Omega); :: thesis: ex F being FinSequence of REAL ex s being FinSequence of Omega st
( len F = card Omega & s is one-to-one & rng s = Omega & len s = card Omega & ( for n being Nat st n in dom F holds
F . n = (X . (s . n)) * (P . {(s . n)}) ) & expect (X,P) = Sum F )
( X is_integrable_on P & ex F being FinSequence of REAL ex s being FinSequence of Omega st
( len F = card Omega & s is one-to-one & rng s = Omega & len s = card Omega & ( for n being Nat st n in dom F holds
F . n = (X . (s . n)) * (P . {(s . n)}) ) & Integral ((P2M P),X) = Sum F ) ) by Th13, Th30;
hence ex F being FinSequence of REAL ex s being FinSequence of Omega st
( len F = card Omega & s is one-to-one & rng s = Omega & len s = card Omega & ( for n being Nat st n in dom F holds
F . n = (X . (s . n)) * (P . {(s . n)}) ) & expect (X,P) = Sum F ) by Def4; :: thesis: verum