
theorem
  ex Omega, Omega2 be non empty set,
     Sigma be SigmaField of Omega,
     Sigma2 be SigmaField of Omega2,
     I be non empty real-membered set,
     Q be Filtration of I,Sigma st
  ex rv being Function of Omega,Omega2 st
    for i being Element of I holds
      rv is random_variable of El_Filtration(i,Q),Sigma2
proof
  set Omega={1,2,3,4};
  set Omega2={1};
  consider Sigma being SigmaField of Omega such that A3: Sigma=bool(Omega);
  reconsider Sigma2=bool(Omega2) as non empty set;
  reconsider Sigma2 as SigmaField of Omega2;
  consider I being non empty real-membered set such that A5: I={1,2,3};
  consider MyFunc being Filtration of I,Sigma such that
  A6: MyFunc.1=Special_SigmaField1 &
      MyFunc.2=Special_SigmaField2 &
      MyFunc.3=Trivial-SigmaField {1,2,3,4} by A3,A5,MyNeed;
  consider X1 being Function of Omega,{1} such that
  A7:X1 is random_variable of Special_SigmaField1,Sigma2 &
     X1 is random_variable of Special_SigmaField2,Sigma2 &
     X1 is random_variable of Trivial-SigmaField {1,2,3,4},Sigma2 by THJ1;
  F2: for i being Element of I holds
   X1 is random_variable of El_Filtration(i,MyFunc),Sigma2
   proof
    let i be Element of I;
    i = 1 or i = 2 or i = 3 by A5,ENUMSET1:def 1;
    hence thesis by A6,A7;
   end;
  take Omega, Omega2, Sigma, Sigma2;
  thus thesis by F2;
 end;
