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 Th9:
  seq = rseq & rseq is bounded implies superior_realsequence seq =
  superior_realsequence rseq & lim_sup seq = lim_sup rseq
proof
  assume that
A1: seq=rseq and
A2: rseq is bounded;
A3: NAT = dom superior_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;
    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;
A5: Y2 is bounded_above by A2,RINFSUP1:31;
    ex Y1 being non empty Subset of ExtREAL st Y1
     = {seq.k: n <= k} & (superior_realsequence seq).n = sup Y1 by Def7;
    then (superior_realsequence seq).x =upper_bound Y2 by A1,A5,Th1;
    hence (superior_realsequence seq).x =(superior_realsequence rseq).x by
RINFSUP1:def 5;
  end;
  superior_realsequence rseq is bounded by A2,RINFSUP1:56;
  then
A6: rng superior_realsequence rseq is bounded_below by RINFSUP1:6;
  NAT = dom superior_realsequence seq by FUNCT_2:def 1;
  then superior_realsequence seq = superior_realsequence rseq by A4,A3,
FUNCT_1:2;
  hence thesis by A6,Th3;
end;
