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 Th13:
  seq is bounded_above implies upper_bound seq = -lower_bound (-seq)
proof
  assume seq is bounded_above;
  then rng seq is non empty bounded_above by Th5,RELAT_1:41;
  then upper_bound rng seq = - lower_bound (--(rng seq)) by MEASURE6:44
    .= - lower_bound rng -seq by Th4;
  hence thesis;
end;
