reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve T for Nat;
 reserve TFix for Element of StoppingSetExt(T);
 reserve MyFunc for Filtration of StoppingSet(T),Sigma;
 reserve k,k1,k2 for Function of Omega,StoppingSetExt(T);

theorem
  Omega = {1,2,3,4} implies
  for MyFunc being Filtration of StoppingSet(2),Sigma st
      MyFunc.0 = Special_SigmaField1 &
      MyFunc.1 = Special_SigmaField2 &
      MyFunc.2 = Trivial-SigmaField (Omega)
    ex ST being Function of Omega,StoppingSetExt(2) st
      ST is_StoppingTime_wrt MyFunc,2 &
       ST.1=1 & ST.2=1 & ST.3=2 & ST.4=2 &
       {w where w is Element of Omega: ST.w=0} = {} &
       {w where w is Element of Omega: ST.w=1} = {1,2} &
       {w where w is Element of Omega: ST.w=2} = {3,4}
proof
  assume
ASS0: Omega={1,2,3,4};
  let MyFunc be Filtration of StoppingSet(2),Sigma;
  assume
ASS2: MyFunc.0=Special_SigmaField1 &
      MyFunc.1=Special_SigmaField2 &
      MyFunc.2=Trivial-SigmaField(Omega);
  deffunc U(Element of Omega) = Special_StoppingSet($1);
  consider f being Function of Omega,StoppingSetExt(2) such that
A1: for d be Element of Omega holds f.d = U(d) from FUNCT_2:sch 4;
B1: 1 in {1,2} & 2 in {1,2} & not 3 in {1,2} & not 4 in {1,2}
    by TARSKI:def 2;
  take f;
A2: f.1=1 & f.2=1 & f.3=2 & f.4=2
 proof
   set O1=1, O2=2, O3=3, O4=4;
   reconsider O1,O2,O3,O4 as Element of Omega by ASS0,ENUMSET1:def 2;
   f.1=U(O1) & f.2=U(O2) & f.3=U(O3) & f.4=U(O4) by A1;
   hence thesis by B1,MATRIX_7:def 1;
 end;
 f is_StoppingTime_wrt MyFunc,2 &
  {w where w is Element of Omega: f.w=0}={} &
  {w where w is Element of Omega: f.w=1}={1,2} &
  {w where w is Element of Omega: f.w=2}={3,4}
 proof
G1: for t being Element of StoppingSet(2) holds
   {w where w is Element of Omega: f.w=t} in MyFunc.t &
   (t=0 implies {w where w is Element of Omega: f.w=0}={}) &
   (t=1 implies {w where w is Element of Omega: f.w=1}={1,2}) &
   (t=2 implies {w where w is Element of Omega: f.w=2}={3,4})
  proof
   let t be Element of StoppingSet(2);
   t in StoppingSet 2; then
   consider t1 being Element of NAT such that H1: t=t1 & 0<=t1 & t1<=2;
   t<=1 or t=1+1 by NAT_1:9,H1; then
g2:t<=0 or t=0+1 or t=2 by NAT_1:9,H1;
   {w where w is Element of Omega: f.w=t} in MyFunc.t &
   (t=0 implies {w where w is Element of Omega: f.w=0}={}) &
   (t=1 implies {w where w is Element of Omega: f.w=1}={1,2}) &
   (t=2 implies {w where w is Element of Omega: f.w=2}={3,4})
   proof
    reconsider M = MyFunc.t as SigmaField of Omega by KOLMOG01:def 2;
    per cases by g2,H1;
    suppose S1: t=0;
     {w where w is Element of Omega: f.w=0} in M &
     (t=0 implies {w where w is Element of Omega: f.w=0}={})
     proof
      {w where w is Element of Omega: f.w=0} c= {}
      proof
        let y be object;
        assume y in {w where w is Element of Omega: f.w=0};
        then ex y1 being Element of Omega st y=y1 & f.y1=0;
        hence thesis by A2,ASS0,ENUMSET1:def 2;
      end;
     then {w where w is Element of Omega: f.w=0} = {};
     hence thesis by PROB_1:4;
     end;
    hence thesis by S1;
    end;
    suppose S1: t=1;
    {w where w is Element of Omega: f.w=1} = {1,2}
    proof
     for x being object holds
      x in {w where w is Element of Omega: f.w=1} iff x in {1,2}
     proof
      let x be object;
      thus x in {w where w is Element of Omega: f.w=1} implies x in {1,2}
      proof
       assume x in {w where w is Element of Omega: f.w=1};
       then consider w2 being Element of Omega such that K2: x=w2 & f.w2=1;
        not w2 in {1,2} implies f.w2>1
        proof
         assume ASSJ: not w2 in {1,2};
         w2=1 or w2=2 or w2=3 or w2=4 by ASS0,ENUMSET1:def 2;
        hence thesis by A2,ASSJ,TARSKI:def 2;
       end;
      hence thesis by K2;
      end;
       assume ASSJ: x in {1,2}; then
       x=1 or x=2 or x=3 or x=4 by TARSKI:def 2; then
S:     x in Omega by ASS0,ENUMSET1:def 2;
       x=1 or x=2 by ASSJ,TARSKI:def 2;
       hence thesis by S,A2;
     end;
    hence thesis by TARSKI:2;
    end;
    hence thesis by S1,ENUMSET1:def 2,ASS2;
    end;
    suppose S1: t=2;
    S2: {w where w is Element of Omega: f.w=t} = {3,4}
    proof
     for x being object holds
      x in {w where w is Element of Omega: f.w=t} iff x in {3,4}
     proof
      let x be object;
      thus x in {w where w is Element of Omega: f.w=t} implies x in {3,4}
      proof
       assume x in {w where w is Element of Omega: f.w=t};
       then consider w2 being Element of Omega such that
   K2: x=w2 & f.w2=2 by S1;
       assume ASSJ: not x in {3,4};
       w2=1 or w2=2 or w2=3 or w2=4 by ASS0,ENUMSET1:def 2;
       hence thesis by A2,ASSJ,TARSKI:def 2,K2;
      end;
      assume x in {3,4}; then
T:    x=3 or x=4 by TARSKI:def 2; then
      x in Omega by ASS0,ENUMSET1:def 2;
      hence thesis by A2,S1,T;
    end;
   hence thesis by TARSKI:2;
   end;
   {3,4} in Special_SigmaField2 &
   Special_SigmaField2 c= Trivial-SigmaField({1,2,3,4}) by FINANCE3:24;
   hence thesis by S2,S1,ASS2;
   end;
   end;
   hence thesis;
   end;
j1:0 in StoppingSet(2);
j2:1 in StoppingSet(2);
   2 in StoppingSet(2);
  hence thesis by j1,j2,G1;
 end;
hence thesis by A2;
end;
