 reserve Omega for non empty set;
 reserve F for SigmaField of Omega;
 reserve phi for Real_Sequence;
 reserve jpi for pricefunction;

theorem JB3:
  for d,d2 being Nat
  for r being Real
  for G being sequence of set_of_random_variables_on (F,Borel_Sets) holds
   (RVPortfolioValueFut(phi,F,G,d2)-
    (Omega-->(1+r)*BuyPortfolio(phi,jpi,d)))"[.0,+infty.[ is Event of F
proof
 let d,d2 be Nat;
 let r be Real;
 let G be sequence of set_of_random_variables_on (F,Borel_Sets);
 set RV= RVPortfolioValueFut(phi,F,G,d2);
 reconsider RV as random_variable of F,Borel_Sets by FINANCE3:7;
 set myr=(1+r)*BuyPortfolio(phi,jpi,d);
 reconsider myr as Element of REAL by XREAL_0:def 1;
 reconsider constRV = Omega-->myr as random_variable of F,Borel_Sets
   by FINANCE3:10;
B5: [.0,+infty.[ is Element of Borel_Sets by FINANCE1:3;
 RV-constRV is (F,Borel_Sets)-random_variable-like by FINANCE2:24;
 hence thesis by B5;
end;
