reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem
  f is convergent_in-infty & lim_in-infty f=0 & (ex r st for g st g in
  dom f /\ left_open_halfline(r) holds f.g<0) implies f^ is
  divergent_in-infty_to-infty
proof
  assume that
A1: f is convergent_in-infty and
A2: lim_in-infty f=0;
  given r such that
A3: for g st g in dom f/\left_open_halfline(r) holds f.g<0;
  thus for r1 ex g1 st g1<r1 & g1 in dom(f^)
  proof
    let r1;
    consider g1 such that
A4: g1<r1 and
    g1 in dom f by A1;
    now
      per cases;
      suppose
A5:     g1<=r;
        consider g2 such that
A6:     g2<g1 and
A7:     g2 in dom f by A1;
        take g2;
        thus g2<r1 by A4,A6,XXREAL_0:2;
        g2<r by A5,A6,XXREAL_0:2;
        then g2 in {r2: r2<r};
        then g2 in left_open_halfline(r) by XXREAL_1:229;
        then g2 in dom f/\left_open_halfline(r) by A7,XBOOLE_0:def 4;
        then 0<>f.g2 by A3;
        then not f.g2 in {0} by TARSKI:def 1;
        then not g2 in f"{0} by FUNCT_1:def 7;
        then g2 in dom f\f"{0} by A7,XBOOLE_0:def 5;
        hence g2 in dom(f^) by RFUNCT_1:def 2;
      end;
      suppose
A8:     r<=g1;
        consider g2 such that
A9:     g2<r and
A10:    g2 in dom f by A1;
        take g2;
        g2<g1 by A8,A9,XXREAL_0:2;
        hence g2<r1 by A4,XXREAL_0:2;
        g2 in {r2: r2<r} by A9;
        then g2 in left_open_halfline(r) by XXREAL_1:229;
        then g2 in dom f/\left_open_halfline(r) by A10,XBOOLE_0:def 4;
        then 0<>f.g2 by A3;
        then not f.g2 in {0} by TARSKI:def 1;
        then not g2 in f"{0} by FUNCT_1:def 7;
        then g2 in dom f\f"{0} by A10,XBOOLE_0:def 5;
        hence g2 in dom(f^) by RFUNCT_1:def 2;
      end;
    end;
    hence thesis;
  end;
  let s be Real_Sequence;
  assume that
A11: s is divergent_to-infty and
A12: rng s c=dom (f^);
  consider k such that
A13: for n st k<=n holds s.n<r by A11;
A14: rng(s^\k)c=rng s by VALUED_0:21;
  dom(f^)=dom f\f"{0} by RFUNCT_1:def 2;
  then
A15: dom(f^)c=dom f by XBOOLE_1:36;
  then
A16: rng s c=dom f by A12;
  then
A17: rng(s^\k)c=dom f by A14;
  now
    let n;
A18: n in NAT by ORDINAL1:def 12;
    s.(n+k)<r by A13,NAT_1:12;
    then (s^\k).n<r by NAT_1:def 3;
    then (s^\k).n in {g2: g2<r};
    then (s^\k).n in rng(s^\k) & (s^\k).n in left_open_halfline(r) by
VALUED_0:28,XXREAL_1:229;
    then (s^\k).n in dom f/\left_open_halfline(r) by A17,XBOOLE_0:def 4;
    then f.((s^\k).n)<0 by A3;
    hence (f/*(s^\k)).n<0 by A16,A14,FUNCT_2:108,XBOOLE_1:1,A18;
  end;
  then
A19: for n holds 0<=n implies (f/*(s^\k)).n<0;
  s^\k is divergent_to-infty by A11,Th27;
  then f/*(s^\k) is convergent & lim(f/*(s^\k))=0 by A1,A2,A17,Def13;
  then
A20: (f/*(s^\k))" is divergent_to-infty by A19,Th36;
  (f/*(s^\k))"=((f/*s)^\k)" by A12,A15,VALUED_0:27,XBOOLE_1:1
    .=((f/*s)")^\k by SEQM_3:18
    .=((f^)/*s)^\k by A12,RFUNCT_2:12;
  hence thesis by A20,Th7;
end;
