
theorem
  for Omega being non empty set,
      F being SigmaField of Omega,
      X being non empty set,
      G being sequence of X,
      phi being Real_Sequence,
      d being Nat holds
  RVPortfolioValueFutExt(phi,F,G,d+1) = RVPortfolioValueFut(phi,F,G,d) +
   RVElementsOfPortfolioValue_fut(phi,F,G,0)
 proof
  let Omega be non empty set;
  let F be SigmaField of Omega;
  let X be non empty set;
  let G be sequence of X;
  let phi be Real_Sequence;
  let d be Nat;
  C0: for w being Element of Omega holds
       RVPortfolioValueFutExt(phi,F,G,d+1).w =
        RVPortfolioValueFut(phi,F,G,d).w +
         (RVElementsOfPortfolioValue_fut(phi,F,G,0)).w
  proof
   let w be Element of Omega;
   A01: RVPortfolioValueFut(phi,F,G,d).w =
    PortfolioValueFut((d+1),phi,F,G,w) by Def4;
   for d being Nat holds
    Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).d =
     (Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))).(d+1) -
      (RVPortfolioValueFutExt_El(phi,F,G,w)).0)
   proof
    defpred J[Nat] means
     Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).$1 =
      (Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))).($1+1) -
       (RVPortfolioValueFutExt_El(phi,F,G,w)).0);
     B1: Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).0 =
      ((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).0 by SERIES_1:def 1;
    Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).0 =
     (Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))).(0+1) -
      (RVPortfolioValueFutExt_El(phi,F,G,w)).0)
    proof
     (Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w)))).(0+1) =
      (Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w)))).0 +
       (RVPortfolioValueFutExt_El(phi,F,G,w)).(0+1) by SERIES_1:def 1;
     then
     Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).0 =
      (Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w)))).(0+1) -
       Partial_Sums(RVPortfolioValueFutExt_El(phi,F,G,w)).0 by NAT_1:def 3,B1;
    hence thesis by SERIES_1:def 1;
    end;
    then
    J0: J[0];
    J1: for n being Nat st J[n] holds J[n+1]
    proof
     let n be Nat;
     assume Q0: J[n];
     Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).(n+1) =
           Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).n +
            ((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).(n+1)
       by SERIES_1:def 1
       .= ((Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w)))).(n+1) +
        ((RVPortfolioValueFutExt_El(phi,F,G,w))^\1).(n+1) -
         (RVPortfolioValueFutExt_El(phi,F,G,w)).0) by Q0
       .= ((Partial_Sums((RVPortfolioValueFutExt_El(phi,F,G,w)))).(n+1) +
        ((RVPortfolioValueFutExt_El(phi,F,G,w))).((n+1)+1)) -
         (RVPortfolioValueFutExt_El(phi,F,G,w)).0 by NAT_1:def 3;
     hence thesis by SERIES_1:def 1;
    end;
    for n being Nat holds J[n] from NAT_1:sch 2(J0,J1);
   hence thesis;
   end;
   then
   RVPortfolioValueFut(phi,F,G,d).w =
    PortfolioValueFutExt(d+1,phi,F,G,w) -
      (RVPortfolioValueFutExt_El(phi,F,G,w)).0 by A01
    .= PortfolioValueFutExt(d+1,phi,F,G,w) -
     (RVElementsOfPortfolioValue_fut(phi,F,G,0)).w by FINANCE2:def 6;
  hence thesis by Def2;
  end;
   C2: dom (RVPortfolioValueFut(phi,F,G,d)) = Omega by FUNCT_2:def 1;
   dom (RVElementsOfPortfolioValue_fut(phi,F,G,0)) = Omega
    by FUNCT_2:def 1; then
  dom RVPortfolioValueFutExt(phi,F,G,d+1) =
   dom (RVPortfolioValueFut(phi,F,G,d)) /\
    dom (RVElementsOfPortfolioValue_fut(phi,F,G,0)) &
  for c being object st c in dom RVPortfolioValueFutExt(phi,F,G,d+1)
   holds (RVPortfolioValueFutExt(phi,F,G,d+1)).c =
    RVPortfolioValueFut(phi,F,G,d).c +
     (RVElementsOfPortfolioValue_fut(phi,F,G,0)).c
     by C0, FUNCT_2:def 1,C2;
 hence thesis by VALUED_1:def 1;
 end;
