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;
