 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
  for 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 holds
  (Risk_neutral_measure_exists_wrt G,jpi,r &
  for s being Nat holds
    jpi.s = expect(Real_RV(r,Change_Element_to_Func(G,s)),Prob))
proof
 let r be Real;
 assume A0: r>0;
 let jpi be pricefunction;
 let G be sequence of
    set_of_random_variables_on(Special_SigmaField2,Borel_Sets);
 assume A01: 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;
 for s being Nat holds jpi.s=
  expect(Real_RV(r,Change_Element_to_Func(G,s)),Prob)
 proof
  let s be Nat;
  set RV=Change_Element_to_Func(G,s);
  RV=(({1,2,3,4}-->In((jpi.s*(1+r)),REAL))) &
  RV is_integrable_on P2M(Prob) &
  RV is_simple_func_in Special_SigmaField2 by A01;
  hence thesis by A0,ThArb;
 end;
 hence thesis;
end;
