let Omega be non empty set ; :: thesis: for F being SigmaField of Omega
for G being sequence of (set_of_random_variables_on (F,Borel_Sets))
for phi being Real_Sequence
for d being Nat holds RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
let F be SigmaField of Omega; :: thesis: for G being sequence of (set_of_random_variables_on (F,Borel_Sets))
for phi being Real_Sequence
for d being Nat holds RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
let G be sequence of (set_of_random_variables_on (F,Borel_Sets)); :: thesis: for phi being Real_Sequence
for d being Nat holds RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
let phi be Real_Sequence; :: thesis: for d being Nat holds RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
let d be Nat; :: thesis: RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
defpred S₁[ 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; :: thesis: (RVPortfolioValueFutExt (phi,F,G,0)) . w = ((phi . 0) (#) (ElementsOfPortfolioValueProb_fut (F,(G . 0)))) . w
(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 (RVPortfolioValueFutExt (phi,F,G,0)) . w = ((phi . 0) (#) (ElementsOfPortfolioValueProb_fut (F,(G . 0)))) . w by VALUED_1:6; :: thesis: verum

end;

hence RVPortfolioValueFutExt (phi,F,G,0) is random_variable of F, Borel_Sets by A1, FUNCT_2:63; :: thesis: verum

end;

then J0: S₁[ 0 ] ;
J1: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume B1: S₁[n] ; :: thesis: S₁[n + 1]
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; :: thesis: (RVPortfolioValueFutExt (phi,F,G,(n + 1))) . w = ((RVPortfolioValueFutExt (phi,F,G,n)) . w) + ((RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1))) . w)
(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 (RVPortfolioValueFutExt (phi,F,G,(n + 1))) . w = ((RVPortfolioValueFutExt (phi,F,G,n)) . w) + ((RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1))) . w) by FINANCE2:def 6, D1; :: thesis: verum

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 S₁[n + 1] by B1, K2, FINANCE2:23; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(J0, J1);
hence RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets ; :: thesis: verum