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-->TFix is_StoppingTime_wrt MyFunc,T
 proof
 set const = Omega --> TFix;
  for t being Element of StoppingSet(T) holds
  {w where w is Element of Omega: const.w=t} in MyFunc.t
  proof
   let t be Element of StoppingSet(T);
   per cases;
   suppose S0: t=TFix;
    C1: {w where w is Element of Omega: const.w=t}=Omega
    proof
     for x being object holds
      x in {w where w is Element of Omega: const.w=t} iff x in Omega
     proof
      let x be object;
      thus x in {w where w is Element of Omega: const.w=t} implies x in Omega
      proof
       assume x in {w where w is Element of Omega: const.w=t};
       then consider s being Element of Omega such that E1: s=x & const.s=t;
       thus thesis by E1;
      end;
       assume x in Omega; then
       consider y being Element of Omega such that
       F10: y=x & y in Omega;
       y in Omega implies t=const.y by FUNCOP_1:7,S0;
       hence thesis by F10;
     end;
    hence thesis by TARSKI:2;
    end;
    MyFunc.t is SigmaField of Omega by KOLMOG01:def 2;
    hence thesis by C1,PROB_1:5;
   end;
   suppose S1: t<>TFix;
c1:  {w where w is Element of Omega: const.w=t} c= {}
     proof
      let x be object;
       assume x in {w where w is Element of Omega: const.w=t};
       then ex s being Element of Omega st s=x & const.s=t;
       then consider s being Element of Omega such that
       E1: s=x & const.s<>TFix by S1;
      thus thesis by E1,FUNCOP_1:7;
     end;
    MyFunc.t is SigmaField of Omega by KOLMOG01:def 2; then
    {} in MyFunc.t by PROB_1:4;
   hence thesis by c1;
  end;
  end;
 hence thesis;
 end;
