reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k for Nat;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  f is_left_convergent_in x0 & lim_left(f,x0)=0 & (ex r st 0<r & for g
  st g in dom f /\ ].x0-r,x0.[ holds 0<f.g) implies f^
  is_left_divergent_to+infty_in x0
proof
  assume that
A1: f is_left_convergent_in x0 and
A2: lim_left(f,x0)=0;
  given r such that
A3: 0<r and
A4: for g st g in dom f/\].x0-r,x0.[ holds 0<f.g;
  thus for r1 st r1<x0 ex g1 st r1<g1 & g1<x0 & g1 in dom(f^)
  proof
    let r1;
    assume r1<x0;
    then consider g1 such that
A5: r1<g1 and
A6: g1<x0 and
    g1 in dom f by A1;
    now
      per cases;
      suppose
A7:     g1<=x0-r;
        x0-r<x0 by A3,Lm1;
        then consider g2 such that
A8:     x0-r<g2 and
A9:     g2<x0 and
A10:    g2 in dom f by A1;
        take g2;
        g1<g2 by A7,A8,XXREAL_0:2;
        hence r1<g2 & g2<x0 by A5,A9,XXREAL_0:2;
        g2 in {r2: x0-r<r2 & r2<x0} by A8,A9;
        then g2 in ].x0-r,x0.[ by RCOMP_1:def 2;
        then g2 in dom f/\].x0-r,x0.[ by A10,XBOOLE_0:def 4;
        then 0<>f.g2 by A4;
        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;
      suppose
A11:    x0-r<=g1;
        consider g2 such that
A12:    g1<g2 and
A13:    g2<x0 and
A14:    g2 in dom f by A1,A6;
        take g2;
        thus r1<g2 & g2<x0 by A5,A12,A13,XXREAL_0:2;
        x0-r<g2 by A11,A12,XXREAL_0:2;
        then g2 in {r2: x0-r<r2 & r2<x0} by A13;
        then g2 in ].x0-r,x0.[ by RCOMP_1:def 2;
        then g2 in dom f/\].x0-r,x0.[ by A14,XBOOLE_0:def 4;
        then 0<>f.g2 by A4;
        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 A14,XBOOLE_0:def 5;
        hence g2 in dom(f^) by RFUNCT_1:def 2;
      end;
    end;
    hence thesis;
  end;
  let s be Real_Sequence such that
A15: s is convergent and
A16: lim s=x0 and
A17: rng s c=dom(f^)/\left_open_halfline(x0);
  x0-r<x0 by A3,Lm1;
  then consider k such that
A18: for n st k<=n holds x0-r<s.n by A15,A16,Th1;
A19: lim(s^\k)=x0 by A15,A16,SEQ_4:20;
  dom(f^)=dom f\f"{0} by RFUNCT_1:def 2;
  then
A20: dom(f^)c=dom f by XBOOLE_1:36;
A21: rng(s^\k)c=rng s by VALUED_0:21;
  dom(f^)/\left_open_halfline(x0)c=left_open_halfline(x0) by XBOOLE_1:17;
  then rng s c=left_open_halfline(x0) by A17,XBOOLE_1:1;
  then
A22: rng(s^\k)c=left_open_halfline(x0) by A21,XBOOLE_1:1;
A23: dom(f^)/\left_open_halfline(x0)c=dom(f^) by XBOOLE_1:17;
  then
A24: rng s c=dom(f^) by A17,XBOOLE_1:1;
  then
A25: rng s c=dom f by A20,XBOOLE_1:1;
  then
A26: rng(s^\k)c=dom f by A21,XBOOLE_1:1;
  then
A27: rng(s^\k)c=dom f/\left_open_halfline(x0) by A22,XBOOLE_1:19;
  then
A28: lim(f/*(s^\k))=0 by A1,A2,A15,A19,Def7;
  now
    let n;
A29: n in NAT by ORDINAL1:def 12;
    x0-r<s.(n+k) by A18,NAT_1:12;
    then
A30: x0-r<(s^\k).n by NAT_1:def 3;
A31: (s^\k).n in rng(s^\k) by VALUED_0:28;
    then (s^\k).n in left_open_halfline(x0) by A22;
    then (s^\k).n in {g1: g1<x0} by XXREAL_1:229;
    then ex g1 st g1=(s^\k).n & g1<x0;
    then (s^\k).n in {g2: x0-r<g2 & g2<x0} by A30;
    then (s^\k).n in ].x0-r,x0.[ by RCOMP_1:def 2;
    then (s^\k).n in dom f/\].x0-r,x0.[ by A26,A31,XBOOLE_0:def 4;
    then 0<f.((s^\k).n) by A4;
    hence 0<(f/*(s^\k)).n by A25,A21,FUNCT_2:108,XBOOLE_1:1,A29;
  end;
  then
A32: for n holds 0<=n implies 0<(f/*(s^\k)).n;
  f/*(s^\k) is convergent by A1,A15,A19,A27;
  then
A33: (f/*(s^\k))" is divergent_to+infty by A28,A32,LIMFUNC1:35;
  (f/*(s^\k))"=((f/*s)^\k)" by A24,A20,VALUED_0:27,XBOOLE_1:1
    .=((f/*s)")^\k by SEQM_3:18
    .=((f^)/*s)^\k by A17,A23,RFUNCT_2:12,XBOOLE_1:1;
  hence thesis by A33,LIMFUNC1:7;
end;
