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 Th14:
  seq is bounded_below implies lower_bound seq = -upper_bound (-seq)
proof
  assume seq is bounded_below;
  then rng seq is non empty bounded_below by Th6,RELAT_1:41;
  then lower_bound rng seq = - upper_bound --(rng seq) by MEASURE6:43
    .= - upper_bound rng -seq by Th4;
  hence thesis;
end;
