 reserve Omega, Omega2 for non empty set;
 reserve Sigma, F for SigmaField of Omega;
 reserve Sigma2, F2 for SigmaField of Omega2;

theorem
for d being Nat st d>0 holds
for r being Real st r>-1 holds
for phi being Real_Sequence,
    jpi being pricefunction holds
for G being sequence of set_of_random_variables_on (F,Borel_Sets) st
 Element_Of(F,Borel_Sets,G,0) = Omega-->1+r holds
(BuyPortfolioExt(phi,jpi,d)<=0 implies
({w where w is Element of Omega:
      PortfolioValueFutExt(d,phi,F,G,w) >= 0} c=
     {w where w is Element of Omega:
      PortfolioValueFut(d,phi,F,G,w)
                    >= (1+r)*BuyPortfolio(phi,jpi,d)} &
{w where w is Element of Omega: PortfolioValueFutExt(d,phi,F,G,w) > 0} c=
{w where w is Element of Omega: PortfolioValueFut(d,phi,F,G,w)
                    > (1+r)*BuyPortfolio(phi,jpi,d)}))
proof
 let d be Nat;
 assume A1: d>0;
 let r be Real;
 assume A2:r>-1;
 let phi be Real_Sequence,
     jpi be pricefunction;
 set X = set_of_random_variables_on (F,Borel_Sets);
 let G be sequence of X;
 assume A3: Element_Of(F,Borel_Sets,G,0) = Omega-->1+r;
 assume A4: BuyPortfolioExt(phi,jpi,d)<=0;
 A5: {w where w is Element of Omega:
      PortfolioValueFutExt(d,phi,F,G,w) >= 0} c=
     {w where w is Element of Omega:
      PortfolioValueFut(d,phi,F,G,w)
                    >= (1+r)*BuyPortfolio(phi,jpi,d)}
 proof
   let x be object;
   assume x in {w where w is Element of Omega:
     PortfolioValueFutExt(d,phi,F,G,w) >= 0}; then
   consider w being Element of Omega such that
A6: x=w & PortfolioValueFutExt(d,phi,F,G,w) >= 0;
   0<= PortfolioValueFut(d,phi,F,G,w) -
   ((1+r)*BuyPortfolio(phi,jpi,d)) by A1,A2,A3,A4,Th13,A6; then
   0+ ((1+r)*BuyPortfolio(phi,jpi,d))
     <= PortfolioValueFut(d,phi,F,G,w) by XREAL_1:19;
   hence thesis by A6;
 end;
 {w where w is Element of Omega:
      PortfolioValueFutExt(d,phi,F,G,w) > 0} c=
   {w where w is Element of Omega:
   PortfolioValueFut(d,phi,F,G,w) > (1+r)*BuyPortfolio(phi,jpi,d)}
 proof
   let x be object;
   assume x in {w where w is Element of Omega:
      PortfolioValueFutExt(d,phi,F,G,w) > 0}; then
   consider w being Element of Omega such that
   A7: x=w & PortfolioValueFutExt(d,phi,F,G,w) > 0;
   PortfolioValueFutExt(d,phi,F,G,w) <= (PortfolioValueFut(d,phi,F,G,w) -
   (1+r)*BuyPortfolio(phi,jpi,d)) by A1,A2,A3,A4,Th13; then
   0+((1+r)*BuyPortfolio(phi,jpi,d))
      < (PortfolioValueFut(d,phi,F,G,w) - (1+r)*BuyPortfolio(phi,jpi,d))
     +((1+r)*BuyPortfolio(phi,jpi,d)) by A7,XREAL_1:6;
   hence thesis by A7;
 end;
 hence thesis by A5;
end;
