 reserve Omega for non empty set;
 reserve F for SigmaField of Omega;
 reserve phi for Real_Sequence;
 reserve jpi for pricefunction;
 reserve Prob for Probability of Special_SigmaField2;

theorem
  for r being Real st r>0 holds
  for jpi being pricefunction holds
  ex G being sequence of
      set_of_random_variables_on(Special_SigmaField2,Borel_Sets) st
  (for d being Nat holds
  G.d = ({1,2,3,4}-->(jpi.d*(1+r))) &
      Change_Element_to_Func(G,d) is_integrable_on P2M(Prob) &
      Change_Element_to_Func(G,d) is_simple_func_in Special_SigmaField2)
proof
 let r be Real;
 assume A1: r>0;
 let jpi be pricefunction;
 deffunc U(Nat) = RVfourth(jpi,r,$1);
 consider g being sequence of
  set_of_random_variables_on(Special_SigmaField2,Borel_Sets) such that
   A2: for d being Element of NAT holds g.d=U(d) from FUNCT_2:sch 4;
   take g;
 let d be Nat;
 d in NAT by ORDINAL1:def 12; then
b1: Change_Element_to_Func(g,d)=RVfourth(jpi,r,d) by A2;
 ex RV being Real-Valued-Random-Variable of Special_SigmaField2 st
 RV=({1,2,3,4}-->In((jpi.d*(1+r)),REAL)) &
     RV is_integrable_on P2M(Prob) &
     RV is_simple_func_in Special_SigmaField2 by A1,ThArbPrel;
 hence thesis by b1;
end;
