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

theorem JB1:
  for n being Nat
  for r being Real
  for G being sequence of set_of_random_variables_on (F,Borel_Sets) holds
  {w where w is Element of Omega: PortfolioValueFutExt(n,phi,F,G,w) >= 0} =
      RVPortfolioValueFutExt(phi,F,G,n)"[.0,+infty.[
proof
 let d be Nat;
 let r be Real;
 let G be sequence of set_of_random_variables_on (F,Borel_Sets);
 set Set1={w where w is Element of Omega:
           PortfolioValueFutExt(d,phi,F,G,w) >= 0};
 set Set2=RVPortfolioValueFutExt(phi,F,G,d)"[.0,+infty.[;
 for x being object holds x in Set1 iff x in Set2
 proof
  let x be object;
  thus x in Set1 implies x in Set2
  proof
   assume x in Set1;
   then consider w being Element of Omega such that
    A1: w=x & PortfolioValueFutExt(d,phi,F,G,w) >= 0;
   PortfolioValueFutExt(d,phi,F,G,w)=RVPortfolioValueFutExt(phi,F,G,d).w
    by FINANCE3:def 1;
   then A2: RVPortfolioValueFutExt(phi,F,G,d).w in [.0,+infty.[
     by XXREAL_1:3,A1,XXREAL_0:9;
   dom RVPortfolioValueFutExt(phi,F,G,d)=Omega by FUNCT_2:def 1;
  hence thesis by A1,A2,FUNCT_1:def 7;
  end;
   assume a1: x in Set2;
   then A1: x in dom RVPortfolioValueFutExt(phi,F,G,d) &
        RVPortfolioValueFutExt(phi,F,G,d).x in [.0,+infty.[ by FUNCT_1:def 7;
   reconsider x as Element of Omega by a1;
   0<=RVPortfolioValueFutExt(phi,F,G,d).x &
      RVPortfolioValueFutExt(phi,F,G,d).x < +infty by A1,XXREAL_1:3;
   then 0<=PortfolioValueFutExt(d,phi,F,G,x) by FINANCE3:def 1;
  hence thesis;
 end;
hence thesis by TARSKI:2;
end;
