
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 d be Nat holds
  RVPortfolioValueFutExt(phi,F,G,d) 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 d be Nat;
  defpred J[Nat] means RVPortfolioValueFutExt(phi,F,G,$1)
   is random_variable of F,Borel_Sets;
   ElementsOfPortfolioValueProb_fut(F,G.0) is
    random_variable of F,Borel_Sets by FINANCE2:28; then
   A1: (phi.0(#)ElementsOfPortfolioValueProb_fut(F,G.0)) is
    random_variable of F,Borel_Sets by FINANCE2:26;
  (RVPortfolioValueFutExt(phi,F,G,0)) is random_variable of F,Borel_Sets
  proof
   for w being Element of Omega holds
    (RVPortfolioValueFutExt(phi,F,G,0)).w =
     (phi.0(#)ElementsOfPortfolioValueProb_fut(F,G.0)).w
   proof
    let w be Element of Omega;
    (RVPortfolioValueFutExt(phi,F,G,0)).w = PortfolioValueFutExt(0,phi,F,G,w)
     by Def2; then
    (RVPortfolioValueFutExt(phi,F,G,0)).w =
     (RVPortfolioValueFutExt_El(phi,F,G,w)).0 by SERIES_1:def 1
     .= (RVElementsOfPortfolioValue_fut(phi,F,G,0)).w by FINANCE2:def 6
     .= phi.0 * ElementsOfPortfolioValueProb_fut(F,G.0).w by FINANCE2:def 5;
   hence thesis by VALUED_1:6;
   end;
  hence thesis by A1,FUNCT_2:63;
  end; then
  J0: J[0];
  J1: for n being Nat st J[n] holds J[n+1]
  proof
   let n be Nat;
   assume B1: J[n];
     C0: for w being Element of Omega holds
          (RVPortfolioValueFutExt(phi,F,G,(n+1))).w =
           (RVPortfolioValueFutExt(phi,F,G,n)).w +
            (RVElementsOfPortfolioValue_fut(phi,F,G,(n+1))).w
     proof
      let w be Element of Omega;
      (RVPortfolioValueFutExt(phi,F,G,(n+1))).w =
       PortfolioValueFutExt((n+1),phi,F,G,w) by Def2; then
      D1: (RVPortfolioValueFutExt(phi,F,G,(n+1))).w =
       Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))).n +
        (RVPortfolioValueFutExt_El(phi,F,G,w)).(n+1) by SERIES_1:def 1;
      (RVPortfolioValueFutExt(phi,F,G,n)).w = PortfolioValueFutExt(n,phi,F,G,w)
       by Def2;
     hence thesis by FINANCE2:def 6,D1;
     end;
K2:   RVElementsOfPortfolioValue_fut(phi,F,G,n+1)
      is random_variable of F,Borel_Sets by FINANCE2:30;
C2: dom (RVPortfolioValueFutExt(phi,F,G,n)) = Omega by FUNCT_2:def 1;
      dom (RVElementsOfPortfolioValue_fut(phi,F,G,(n+1))) = Omega
       by FUNCT_2:def 1; then
      dom (RVPortfolioValueFutExt(phi,F,G,(n+1))) =
      dom (RVPortfolioValueFutExt(phi,F,G,n)) /\
       dom (RVElementsOfPortfolioValue_fut(phi,F,G,(n+1))) &
     for c being object st c in dom (RVPortfolioValueFutExt(phi,F,G,(n+1)))
      holds (RVPortfolioValueFutExt(phi,F,G,(n+1))).c =
       (RVPortfolioValueFutExt(phi,F,G,n)).c +
        (RVElementsOfPortfolioValue_fut(phi,F,G,(n+1))).c
        by C0, FUNCT_2:def 1,C2;
     then (RVPortfolioValueFutExt(phi,F,G,(n+1))) =
      (RVPortfolioValueFutExt(phi,F,G,n)) +
       (RVElementsOfPortfolioValue_fut(phi,F,G,(n+1))) by VALUED_1:def 1;
     hence thesis by B1,K2,FINANCE2:23;
    end;
  for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
 hence thesis;
 end;
