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

theorem JB2:
  for d,d2 being Nat st d2=d-1 holds
  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: PortfolioValueFut(d,phi,F,G,w)
                               >= (1+r)*BuyPortfolio(phi,jpi,d)} =
      (RVPortfolioValueFut(phi,F,G,d2)-
       (Omega-->(1+r)*BuyPortfolio(phi,jpi,d)))"[.0,+infty.[
proof
 let d,d2 be Nat;
 assume a10: d2=d-1;
 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: PortfolioValueFut(d,phi,F,G,w)
                               >= (1+r)*BuyPortfolio(phi,jpi,d)};
 set Set2=(RVPortfolioValueFut(phi,F,G,d2)-
          (Omega-->(1+r)*BuyPortfolio(phi,jpi,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 & PortfolioValueFut(d,phi,F,G,w)
              >= (1+r)*BuyPortfolio(phi,jpi,d);
   reconsider x as Element of Omega by A1;
   set myel=(1+r)*BuyPortfolio(phi,jpi,d);
a2: PortfolioValueFut(d2+1,phi,F,G,w)=RVPortfolioValueFut(phi,F,G,d2).w
    by FINANCE3:def 3;
    RVPortfolioValueFut(phi,F,G,d2).x-(Omega-->myel).x >=0
      by XREAL_1:48,A1,a2,a10;
    then Q1: RVPortfolioValueFut(phi,F,G,d2).x+ (-1)*(Omega-->(myel)).x>=0;
     x in dom(Omega-->myel);
     then x in dom ((-1)(#)(Omega-->myel)) by VALUED_1:def 5;
    then RVPortfolioValueFut(phi,F,G,d2).x+ ((-1)(#)(Omega-->(myel))).x>=0
     by Q1,VALUED_1:def 5;
    then (RVPortfolioValueFut(phi,F,G,d2) + (- (Omega-->myel))).x >=0
     by VALUED_1:1;
    then T1: (RVPortfolioValueFut(phi,F,G,d2) - (Omega-->myel)).x
     in [.0,+infty.[ by XXREAL_1:3,XXREAL_0:9;
     dom (RVPortfolioValueFut(phi,F,G,d2) +(- (Omega-->myel)))=
      Omega/\Omega by FUNCT_2:def 1;
   hence thesis by T1,FUNCT_1:def 7;
  end;
   assume a1: x in Set2;
   then A1:x in dom (RVPortfolioValueFut(phi,F,G,d2)-
          (Omega-->(1+r)*BuyPortfolio(phi,jpi,d))) &
          (RVPortfolioValueFut(phi,F,G,d2)-
          (Omega-->(1+r)*BuyPortfolio(phi,jpi,d))).x in [.0,+infty.[
           by FUNCT_1:def 7;
   set my1= ((1+r)*BuyPortfolio(phi,jpi,d));
   set my2=RVPortfolioValueFut(phi,F,G,d2).x;
   reconsider x as Element of Omega by a1;
   set RVmyx=(RVPortfolioValueFut(phi,F,G,d2)-
            (Omega-->(1+r)*BuyPortfolio(phi,jpi,d))).x;
   RVmyx=RVPortfolioValueFut(phi,F,G,d2).x + ((-1)(#)
    (Omega-->(1+r)*BuyPortfolio(phi,jpi,d))).x by VALUED_1:1;
   then RVmyx=RVPortfolioValueFut(phi,F,G,d2).x + ((-1)*
    (Omega-->(1+r)*BuyPortfolio(phi,jpi,d)).x) by VALUED_1:6;
   then 0<=RVPortfolioValueFut(phi,F,G,d2).x -
           (Omega-->(1+r)*BuyPortfolio(phi,jpi,d)).x by A1,XXREAL_1:3;
   then 0<=my2-my1;
   then 0+my1<=my2 by XREAL_1:19;
   then ((1+r)*BuyPortfolio(phi,jpi,d))<=PortfolioValueFut(d2+1,phi,F,G,x)
    by FINANCE3:def 3;
  hence thesis by a10;
 end;
hence thesis by TARSKI:2;
end;
