
theorem MyNeed:
  for Omega being non empty set,
      Sigma being SigmaField of Omega,
      I being non empty real-membered set st
      I={1,2,3} & Sigma=bool {1,2,3,4} & Omega = {1,2,3,4} holds
  ex MyFunc being Filtration of I,Sigma st
     MyFunc.1=Special_SigmaField1 &
     MyFunc.2=Special_SigmaField2 &
     MyFunc.3=Trivial-SigmaField {1,2,3,4}
 proof
  let Omega be non empty set;
  let Sigma be SigmaField of Omega;
  let I be non empty real-membered set;
  assume A0: I={1,2,3} & Sigma=bool {1,2,3,4} & Omega={1,2,3,4};
  consider MyFunc be ManySortedSigmaField of I,Sigma such that A1:
     MyFunc.1=Special_SigmaField1 &
     MyFunc.2=Special_SigmaField2 &
     MyFunc.3=Trivial-SigmaField {1,2,3,4} by A0,Th3;
  A3: for s,t being Element of I st s<=t holds MyFunc.s is Subset of MyFunc.t
  proof
   let s,t be Element of I;
   assume B1: s<=t;
   per cases by A0,ENUMSET1:def 1;
   suppose SUPP1: s=1;
    per cases by A0,ENUMSET1:def 1;
    suppose t=1; then
    MyFunc.t=MyFunc.1 & MyFunc.s=MyFunc.1 by SUPP1;
    hence thesis;
    end;
    suppose t=2 or t = 3;
    hence thesis by SUPP1,A1,XX1;
    end;
   end;
   suppose SUPP1: s=2;
    per cases by A0,ENUMSET1:def 1;
    suppose t=1;
    hence thesis by SUPP1,B1;
    end;
    suppose t=2; then
    MyFunc.t=MyFunc.2 & MyFunc.s=MyFunc.2 by SUPP1;
    hence thesis;
    end;
    suppose t=3;
    hence thesis by SUPP1,A1;
    end;
   end;
   suppose SUPP1: s=3;
    per cases by A0,ENUMSET1:def 1;
    suppose t=1 or t = 2;
      hence thesis by SUPP1,B1;
    end;
    suppose t=3; then
      MyFunc.t=MyFunc.3 & MyFunc.s=MyFunc.3 by SUPP1;
      hence thesis;
    end;
   end;
  end;
  MyFunc is Filtration of I,Sigma by A3, Def2000;
  hence thesis by A1;
 end;
