reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;
reserve seq for ExtREAL_sequence;
reserve e1,e2 for ExtReal;
reserve rseq for Real_Sequence;

theorem Th10:
  seq = rseq & rseq is bounded implies inferior_realsequence seq =
  inferior_realsequence rseq & lim_inf seq=lim_inf rseq
proof
  assume that
A1: seq=rseq and
A2: rseq is bounded;
A3: NAT = dom (inferior_realsequence rseq) by FUNCT_2:def 1;
A4: now
    let x be object;
    assume x in NAT;
    then reconsider n = x as Element of NAT;
    consider Y1 being non empty Subset of ExtREAL such that
A5: Y1 = {seq.k: n <= k} and
A6: (inferior_realsequence seq).n = inf Y1 by Def6;
    now
      let x be object;
      assume x in {rseq.k: n <= k};
      then ex k st x=rseq.k & n<= k;
      hence x in REAL by XREAL_0:def 1;
    end;
    then reconsider
    Y2={rseq.k: n <= k} as Subset of REAL by TARSKI:def 3;
    Y2 is bounded_below by A2,RINFSUP1:32;
    then inf Y1= lower_bound Y2 by A1,A5,Th3;
    hence
    (inferior_realsequence seq).x = (inferior_realsequence rseq).x by A6,
RINFSUP1:def 4;
  end;
  inferior_realsequence rseq is bounded by A2,RINFSUP1:56;
  then
A7: rng inferior_realsequence rseq is bounded_above by RINFSUP1:5;
  NAT = dom (inferior_realsequence seq) by FUNCT_2:def 1;
  then inferior_realsequence seq = inferior_realsequence rseq by A4,A3,
FUNCT_1:2;
  hence thesis by A7,Th1;
end;
