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

theorem Th11:
  for phi being Real_Sequence, jpi being pricefunction,
      d being Nat st d>0 holds
  BuyPortfolioExt(phi,jpi,d) = phi.0 + BuyPortfolio(phi,jpi,d)
proof
 let phi be Real_Sequence, jpi be pricefunction,
     d be Nat;
 assume d>0; then
 A1: d-1 is Element of NAT by NAT_1:20;
 defpred J[Nat] means
      Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).($1+1) =
       phi.0 + Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).$1;
 Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).(0+1) =
   phi.0 + Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).0
 proof
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).(0+1) =
     Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).0+
     ElementsOfBuyPortfolio(phi,jpi).1 by SERIES_1:def 1; then
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).(0+1) =
     ElementsOfBuyPortfolio(phi,jpi).0+
     ElementsOfBuyPortfolio(phi,jpi).1 by SERIES_1:def 1; then
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).(0+1) =
     ElementsOfBuyPortfolio(phi,jpi).0+
     (ElementsOfBuyPortfolio(phi,jpi)^\1).0 by NAT_1:def 3; then
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).(0+1) =
     ElementsOfBuyPortfolio(phi,jpi).0+
     Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).0
       by SERIES_1:def 1; then
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).(0+1) =
     (phi.0 * jpi.0)+
     Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).0 by VALUED_1:5; then
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).(0+1) =
     (phi.0 * 1)+
     Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).0 by Def2;
   hence thesis;
 end; then
A2: J[0];
A3: for k being Nat st J[k] holds J[k+1]
 proof
   let k be Nat;
   assume A4: Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).(k+1) =
   phi.0 + Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).k;
   reconsider k as Element of NAT by ORDINAL1:def 12;
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).((k+1)+1) =
   (phi.0 + Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).k) +
   ElementsOfBuyPortfolio(phi,jpi).((k+1)+1) by A4,SERIES_1:def 1; then
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).((k+1)+1) =
   (phi.0 + Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).k) +
   (ElementsOfBuyPortfolio(phi,jpi)^\1).(k+1) by NAT_1:def 3; then
   Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).((k+1)+1) =
     phi.0 + (Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).k +
     (ElementsOfBuyPortfolio(phi,jpi)^\1).(k+1));
   hence thesis by SERIES_1:def 1;
 end;
 for k being Nat holds J[k] from NAT_1:sch 2(A2,A3); then
 Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).((d-1)+1) =
  phi.0 + BuyPortfolio(phi,jpi,d) by A1;
 hence thesis;
end;
