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

theorem Th13:
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
for w being Element of Omega holds
 BuyPortfolioExt(phi,jpi,d)<=0 implies
(PortfolioValueFutExt(d,phi,F,G,w) <=
 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;
 let w be Element of Omega;
 assume A4: BuyPortfolioExt(phi,jpi,d)<=0;
A5: (1+r)*BuyPortfolioExt(phi,jpi,d) <= 0
 proof
   1+r>0 by A2,XREAL_1:62;
   hence thesis by A4;
 end;
 ((1+r)*BuyPortfolioExt(phi,jpi,d)) + (PortfolioValueFut(d,phi,F,G,w) -
((1+r)*BuyPortfolioExt(phi,jpi,d))) <=
 (PortfolioValueFut(d,phi,F,G,w) -
((1+r)*BuyPortfolioExt(phi,jpi,d))) by A5,XREAL_1:32; then
 PortfolioValueFut(d,phi,F,G,w) <= (PortfolioValueFut(d,phi,F,G,w) -
(1+r)*(phi.0 + BuyPortfolio(phi,jpi,d))) by A1,Th11; then
 PortfolioValueFut(d,phi,F,G,w) <= ((PortfolioValueFut(d,phi,F,G,w) -
 (1+r)*BuyPortfolio(phi,jpi,d))) - (1+r)*phi.0; then
 PortfolioValueFut(d,phi,F,G,w) + (1+r)*phi.0 <=
 ((PortfolioValueFut(d,phi,F,G,w) -
 (1+r)*BuyPortfolio(phi,jpi,d))) by XREAL_1:19;
hence thesis by A1,A3,Th12;
end;
