reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve S for non empty Subset of REAL;
reserve r for Real;
reserve T for Nat;

theorem
for T be non zero Nat,
    MyFunc be Filtration of StoppingSet(T),Sigma holds
 not for k1,k2 be StoppingTime_Func of MyFunc holds
         k1+k2 is StoppingTime_Func of MyFunc
proof
  let T be non zero Nat;
  let MyFunc be Filtration of StoppingSet(T),Sigma;
  T in NAT by ORDINAL1:def 12;
  then T in StoppingSet(T) or T in {+infty}; then
  reconsider MyT = T as Element of StoppingSetExt(T) by XBOOLE_0:def 3;
  consider k1 being Function of Omega,StoppingSetExt(T) such that
A1: k1=Omega-->MyT & k1 is_StoppingTime_wrt MyFunc,T by FINANCE4:3;
  reconsider k1 as StoppingTime_Func of MyFunc by A1,Def1;
  consider k2 being Function of Omega,StoppingSetExt(T) such that
A2: k2=Omega-->MyT & k2 is_StoppingTime_wrt MyFunc,T by FINANCE4:3;
  reconsider k2 as StoppingTime_Func of MyFunc by A2,Def1;
  take k1,k2;
F1: not (T+T) in {+infty} by TARSKI:def 1;
  ex w being Element of dom(k1+k2) st
   w in dom(k1+k2) & not (k1+k2).w in StoppingSetExt(T)
   proof
    consider w2 being object such that
C1: w2 in dom(k1+k2) by XBOOLE_0:def 1;
    reconsider w2 as Element of Omega by C1;
e3: k1.w2=T & k2.w2=T by A1,FUNCOP_1:7,A2;
XX: T + T = k1.w2+(k2.w2) by e3,XXREAL_3:def 2
         .= (k1+k2).w2 by C1,Def888;
    not (k1+k2).w2 in StoppingSetExt(T)
    proof
     not (T+T) in StoppingSetExt(T)
     proof
      not (T+T) in {t where t is Element of NAT: 0<=t<=T}
       proof
        not ex t2 being Element of NAT st t2=T+T & 0<=t2<=T by NAT_1:16;
        hence thesis;
      end;
     hence thesis by F1,XBOOLE_0:def 3;
     end;
    hence thesis by XX;
    end;
   hence thesis by C1;
   end;
   hence thesis by FUNCT_2:5;
 end;
