let Omega be non empty set ; :: thesis: for F being SigmaField of Omega
for d being Nat st d > 0 holds
for r being Real st r > - 1 holds
for phi being Real_Sequence
for jpi being pricefunction
for G being sequence of st Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & BuyPortfolioExt (phi,jpi,d) <= 0 holds
( { 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)) } )

let F be SigmaField of Omega; :: thesis: for d being Nat st d > 0 holds
for r being Real st r > - 1 holds
for phi being Real_Sequence
for jpi being pricefunction
for G being sequence of st Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & BuyPortfolioExt (phi,jpi,d) <= 0 holds
( { 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)) } )

let d be Nat; :: thesis: ( d > 0 implies for r being Real st r > - 1 holds
for phi being Real_Sequence
for jpi being pricefunction
for G being sequence of st Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & BuyPortfolioExt (phi,jpi,d) <= 0 holds
( { 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)) } ) )

assume A1: d > 0 ; :: thesis: for r being Real st r > - 1 holds
for phi being Real_Sequence
for jpi being pricefunction
for G being sequence of st Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & BuyPortfolioExt (phi,jpi,d) <= 0 holds
( { 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)) } )

let r be Real; :: thesis: ( r > - 1 implies for phi being Real_Sequence
for jpi being pricefunction
for G being sequence of st Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & BuyPortfolioExt (phi,jpi,d) <= 0 holds
( { 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)) } ) )

assume A2: r > - 1 ; :: thesis: for phi being Real_Sequence
for jpi being pricefunction
for G being sequence of st Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & BuyPortfolioExt (phi,jpi,d) <= 0 holds
( { 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)) } )

let phi be Real_Sequence; :: thesis: for jpi being pricefunction
for G being sequence of st Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & BuyPortfolioExt (phi,jpi,d) <= 0 holds
( { 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)) } )

let jpi be pricefunction ; :: thesis: for G being sequence of st Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & BuyPortfolioExt (phi,jpi,d) <= 0 holds
( { 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)) } )

set X = set_of_random_variables_on (F,Borel_Sets);
let G be sequence of ; :: thesis: ( Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) & 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)) } ) )
assume A3: Element_Of (F,Borel_Sets,G,0) = Omega --> (1 + r) ; :: thesis: ( not BuyPortfolioExt (phi,jpi,d) <= 0 or ( { 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)) } ) )
assume A4: BuyPortfolioExt (phi,jpi,d) <= 0 ; :: thesis: ( { 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)) } )
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 ; :: according to TARSKI:def 3 :: thesis: ( not x in { w where w is Element of Omega : PortfolioValueFutExt (d,phi,F,G,w) >= 0 } or x in { w where w is Element of Omega : PortfolioValueFut (d,phi,F,G,w) >= (1 + r) * (BuyPortfolio (phi,jpi,d)) } )
assume x in { w where w is Element of Omega : PortfolioValueFutExt (d,phi,F,G,w) >= 0 } ; :: thesis: x in { w where w is Element of Omega : PortfolioValueFut (d,phi,F,G,w) >= (1 + r) * (BuyPortfolio (phi,jpi,d)) }
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 x in { w where w is Element of Omega : PortfolioValueFut (d,phi,F,G,w) >= (1 + r) * (BuyPortfolio (phi,jpi,d)) } by A6; :: thesis: verum
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 ; :: according to TARSKI:def 3 :: thesis: ( not x in { w where w is Element of Omega : PortfolioValueFutExt (d,phi,F,G,w) > 0 } or x in { w where w is Element of Omega : PortfolioValueFut (d,phi,F,G,w) > (1 + r) * (BuyPortfolio (phi,jpi,d)) } )
assume x in { w where w is Element of Omega : PortfolioValueFutExt (d,phi,F,G,w) > 0 } ; :: thesis: x in { w where w is Element of Omega : PortfolioValueFut (d,phi,F,G,w) > (1 + r) * (BuyPortfolio (phi,jpi,d)) }
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 ;
hence x in { w where w is Element of Omega : PortfolioValueFut (d,phi,F,G,w) > (1 + r) * (BuyPortfolio (phi,jpi,d)) } by A7; :: thesis: verum
end;
hence ( { 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)) } ) by A5; :: thesis: verum