 reserve Exx for Real;
 reserve Omega,Omega2 for non empty set;
 reserve Sigma for SigmaField of Omega;
 reserve Sigma2 for SigmaField of Omega2;
 reserve X,Y,Z for Function of Omega,REAL;

theorem
  for Omega be non empty set
  for F be SigmaField of Omega
  for G be sequence of set_of_random_variables_on(F,Borel_Sets)
  for phi be Real_Sequence
  for n be Nat holds
  RVElementsOfPortfolioValue_fut(phi,F,G,n) is
    random_variable of F,Borel_Sets
proof
  let Omega be non empty set;
  let F be SigmaField of Omega;
  let G be sequence of set_of_random_variables_on(F,Borel_Sets);
  let phi be Real_Sequence;
  let n be Nat;
  ElementsOfPortfolioValueProb_fut(F,G.n)
  is random_variable of F,Borel_Sets by T1;
  then
A1: phi.n(#)ElementsOfPortfolioValueProb_fut(F,G.n)
   is random_variable of F,Borel_Sets by Th8;
  for w being Element of Omega holds
  (phi.n(#)ElementsOfPortfolioValueProb_fut(F,G.n)).w =
    (RVElementsOfPortfolioValue_fut(phi,F,G,n)).w
  proof
   let w be Element of Omega;
   phi.n * ElementsOfPortfolioValueProb_fut(F,G.n).w =
       (phi.n(#)ElementsOfPortfolioValueProb_fut(F,G.n)).w by VALUED_1:6;
   hence thesis by Def5000;
  end;
 hence thesis by A1,FUNCT_2:63;
end;
