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 Th38:
  seq is bounded_below implies (inferior_realsequence seq).0 = lower_bound seq
proof
  reconsider Y1 = {seq.k : 0 <= k} as Subset of REAL by Th29;
  (inferior_realsequence seq).0 = lower_bound Y1 by Def4
    .= lower_bound seq by SETLIM_1:5;
  hence thesis;
end;
