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
  S is constant & the_value_of S = A implies S is convergent & lim S = A
  & lim_inf S = A & lim_sup S = A
proof
A1: B is constant & the_value_of B = A implies Union inferior_setsequence(B)
  = A & Intersection superior_setsequence(B) = A
  proof
A2: lim_inf B = Union inferior_setsequence(B) & lim_sup B = Intersection
    superior_setsequence(B);
    assume B is constant & the_value_of B = A;
    hence thesis by A2,Th58;
  end;
  assume S is constant & the_value_of S = A;
  then lim_inf S = A & lim_sup S = A by A1;
  hence thesis by KURATO_0:def 5;
end;
