
theorem
  for Omega being non empty set st Omega={1,2,3,4} holds
  for Sigma being SigmaField of Omega,
      I being non empty real-membered set
    st I={1,2,3} & Sigma = bool {1,2,3,4} holds
    for MyFunc being ManySortedSigmaField of I,Sigma st
      MyFunc.1=Special_SigmaField1 &
      MyFunc.2=Special_SigmaField2 &
      MyFunc.3=Trivial-SigmaField {1,2,3,4} holds
       for Prob being Function of Sigma, REAL holds
         for i being Element of I holds
          ex RV being Function of Omega,REAL st
           RV is (El_Filtration(i,MyFunc),Borel_Sets)-random_variable-like
 proof
  let Omega be non empty set;
  assume A0: Omega={1,2,3,4};
  let Sigma be SigmaField of Omega;
  let I be non empty real-membered set;
  assume A1: I={1,2,3} & Sigma=bool{1,2,3,4};
  let MyFunc be ManySortedSigmaField of I,Sigma;
  assume A2: MyFunc.1=Special_SigmaField1 & MyFunc.2=Special_SigmaField2 &
             MyFunc.3=Trivial-SigmaField {1,2,3,4};
  let Prob be Function of Sigma, REAL;
  let i be Element of I;
     per cases by A1,ENUMSET1:def 1;
     suppose a1: i=1;
     100 in REAL by NUMBERS:19; then
      ex f being Function of Omega,REAL st
      f.1=100 & f.2=100 & f.3=100 & f.4=100 &
      f is (El_Filtration(i,MyFunc),Borel_Sets)-random_variable-like
      by A0,A2,MyFunc7,a1;
     hence thesis;
     end;
     suppose a1: i=2;
     80 in REAL & 120 in REAL by NUMBERS:19; then
      ex f being Function of Omega,REAL st
       f.1=80 & f.2=80 & f.3=120 & f.4=120 &
       f is (El_Filtration(i,MyFunc),Borel_Sets)-random_variable-like
      by A0,A2,MyFunc6,a1;
     hence thesis;
     end;
     suppose a1: i=3;
     60 in REAL & 80 in REAL & 100 in REAL & 120 in REAL by NUMBERS:19; then
      ex f being Function of Omega,REAL st
      f.1=60 & f.2=80 & f.3=100 & f.4=120 &
       f is (El_Filtration(i,MyFunc),Borel_Sets)-random_variable-like
      by A0,A2,MyFunc5,a1;
     hence thesis;
     end;
 end;
