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 Th34:
  for seq be ExtREAL_sequence st seq is non-increasing & -infty =
  inf seq holds seq is convergent_to_-infty & lim seq = -infty
proof
  let seq be ExtREAL_sequence;
  assume that
A1: seq is non-increasing and
A2: -infty = inf seq;
A3: seq is convergent_to_-infty
  proof
    assume not seq is convergent_to_-infty;
    then consider g be Real such that
    g < 0 and
A4: for n be Nat ex m be Nat st n<=m & g < seq.m by MESFUNC5:def 10;
    for y be ExtReal st y in rng seq holds g <= y
    proof
      let y be ExtReal;
      assume y in rng seq;
      then consider n be object such that
A5:   n in NAT and
A6:   y=seq.n by FUNCT_2:11;
      reconsider n as Element of NAT by A5;
      consider m be Nat such that
A7:   n<=m and
A8:   g < seq.m by A4;
      seq.m <= seq.n by A1,A7,Th7;
      hence thesis by A6,A8,XXREAL_0:2;
    end;
    then g is LowerBound of rng seq by XXREAL_2:def 2;
    then
A9: g <= inf rng seq by XXREAL_2:def 4;
    g in REAL by XREAL_0:def 1;
    hence contradiction by A2,A9,XXREAL_0:12;
  end;
  then seq is convergent by MESFUNC5:def 11;
  hence thesis by A3,MESFUNC5:def 12;
end;
