
theorem MyFunc7:
  for k1 being Element of REAL,
      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,
      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 eli being Element of I st eli=1 holds
    ex f being Function of Omega,REAL st
     f.1=k1 & f.2=k1 & f.3=k1 & f.4=k1 &
     f is (El_Filtration(eli,MyFunc),Borel_Sets)-random_variable-like
 proof
   let k1 be Element of REAL;
   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;
   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 eli be Element of I;
   assume A4: eli=1;
    consider f being Function of Omega,REAL such that
A3: f.1=k1 & f.2=k1 & f.3=k1 & f.4=k1 by A0,MYF30;
   take f;
    set i=eli;
     for x being set holds f"x in El_Filtration(i,MyFunc)
     proof
      let x be set;
       f"x in MyFunc.i
       proof
        f"x in {{},{1,2,3,4}}
        proof
          f"x={} or f"x={1,2,3,4}
          proof
           per cases;
           suppose JSUPP1: k1 in x;
             for z being object holds z in {1,2,3,4} iff
               ex y being object st [z,y] in f & y in x
             proof
              let z be object;
          I1: z in {1,2,3,4} implies
                   ex y being object st [z,y] in f & y in x
              proof
               assume ASSJ0: z in {1,2,3,4};
               [z,k1] in f
               proof
                z in dom f & k1 = f.z by ENUMSET1:def 2,A3,
                                         ASSJ0,FUNCT_2:def 1,A0;
               hence thesis by FUNCT_1:1;
               end;
              hence thesis by JSUPP1;
              end;
              (ex y being object st [z,y] in f & y in x) implies
               z in {1,2,3,4}
              proof
               given y being object such that ASSJ0: [z,y] in f & y in x;
               z in dom f & y=f.z by ASSJ0, FUNCT_1:1;
               hence thesis by A0;
              end;
             hence thesis by I1;
             end;
            hence thesis by RELAT_1:def 14;
           end;
           suppose JSUPP2: not k1 in x;
            for z being object holds z in {} iff
              ex y being object st [z,y] in f & y in x
            proof
             let z be object;
             (ex y being object st [z,y] in f & y in x) implies z in {}
              proof
              given y being object such that
               M1: [z,y] in f & y in x;
               z in dom f & y =f.z & not y=k1 by M1,FUNCT_1:1, JSUPP2;
               hence thesis by A3,A0,ENUMSET1:def 2;
              end;
            hence thesis;
            end;
            hence thesis by RELAT_1:def 14;
            end;
          end;
         hence thesis by TARSKI:def 2;
         end;
        hence thesis by A4,A2;
        end;
     hence thesis;
     end;
  hence thesis by A3;
 end;
