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 Th58:
  B is constant & the_value_of B = A implies B is convergent & lim
  B = A & lim_inf B = A & lim_sup B = A
proof
  assume
A1: B is constant & the_value_of B = A;
  then for n holds (superior_setsequence(B)).n = A by Th39;
  then
A2: lim_sup B = A by Th11;
  for n holds (inferior_setsequence(B)).n = A by A1,Th38;
  then lim_inf B = A by Th10;
  hence thesis by A2,KURATO_0:def 5;
end;
