reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem
  (for n being Nat holds S3.n = S1.n \/ S2.n)
implies lim_sup S3 = lim_sup S1 \/
  lim_sup S2
proof
A1: (for n being Nat holds A3.n = B.n \/ A2.n)
implies Intersection
superior_setsequence(B) \/ Intersection superior_setsequence(A2) = Intersection
  superior_setsequence(A3)
  proof
A2: lim_sup B = Intersection superior_setsequence(B) & lim_sup A2 =
    Intersection superior_setsequence(A2);
A3: lim_sup A3 = Intersection superior_setsequence(A3);
    assume for n being Nat holds A3.n = B.n \/ A2.n;
    hence thesis by A2,A3,KURATO_0:11;
  end;
  assume for n being Nat holds S3.n = S1.n \/ S2.n;
  hence thesis by A1;
end;
