reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th90:
  seq is bounded implies lim_sup -seq = - lim_inf seq & lim_inf -
  seq = - lim_sup seq
proof
  assume
A1: seq is bounded;
  then inferior_realsequence seq is bounded by Th56;
  then
A2: - lim_inf seq = - - lower_bound - inferior_realsequence seq by Th13
    .= lower_bound - -superior_realsequence(-seq) by A1,Th61
    .= lim_sup -seq;
  superior_realsequence seq is bounded by A1,Th56;
  then - lim_sup seq = - - upper_bound - superior_realsequence seq by Th14
    .= upper_bound - -inferior_realsequence(-seq) by A1,Th62
    .= lim_inf -seq;
  hence thesis by A2;
end;
