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 Th59:
  seq is bounded_below implies (inferior_realsequence seq).n = - (
  superior_realsequence(-seq)).n
proof
  assume
A1: seq is bounded_below;
  (inferior_realsequence seq).n = lower_bound (seq ^\n) by Th36
    .= - upper_bound -(seq ^\n) by A1,Th14,SEQM_3:28
    .= - upper_bound ((-seq) ^\n) by SEQM_3:16;
  hence thesis by Th37;
end;
