
theorem
  for Omega, Omega2 being non empty set,
      Sigma being SigmaField of Omega,
      Sigma2 being SigmaField of Omega2,
      I being non empty real-membered set,
      Q being ManySortedSigmaField of I,Sigma holds
  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
  let Omega, Omega2 be non empty set;
  let Sigma be SigmaField of Omega;
  let Sigma2 be SigmaField of Omega2;
  let I be non empty real-membered set;
  let Q be ManySortedSigmaField of I,Sigma;
   consider w being object such that A00: w in Omega2 by XBOOLE_0:def 1;
   reconsider w as Element of Omega2 by A00;
   set myset=w;
   deffunc U(Element of Omega)=myset;
   consider myfunc being Function of Omega, Omega2 such that
    B3: myfunc=Omega-->myset;
    take myfunc;
    let i be Element of I;
    myfunc is (El_Filtration(i,Q),Sigma2)-random_variable-like
    proof
     for x being set st x in Sigma2 holds myfunc"x in El_Filtration(i,Q)
     proof
      let x be set;
      assume x in Sigma2;
      per cases;
      suppose CAS0: myset in x;
       H1: myfunc"x=Omega
       proof
        for y being object holds y in myfunc"x iff y in Omega
        proof
         let y be object;
         y in Omega implies y in myfunc"x
         proof
          assume I0: y in Omega; then
          I1: y in dom myfunc by FUNCT_2:def 1;
          myfunc.y in x by I0,B3,FUNCOP_1:7,CAS0;
          hence thesis by I1,FUNCT_1:def 7;
         end;
        hence thesis;
        end;
       hence thesis by TARSKI:def 3;
       end;
      myfunc"x in Q.i
      proof
       Q.i is SigmaField of Omega by KOLMOG01:def 2;
       hence thesis by H1,PROB_1:5;
       end;
      hence thesis;
      end;
      suppose CAS1: not myset in x;
       myfunc"x c= {}
       proof
        let z be object;
         assume z in myfunc"x; then
         z in dom myfunc & myfunc.z in x & myfunc.z=myset
          by FUNCT_1:def 7, B3,FUNCOP_1:7;
         hence thesis by CAS1;
       end; then
      myfunc"x={};
      hence thesis by PROB_1:4;
      end;
     end;
    hence thesis;
    end;
   hence thesis;
 end;
