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_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in
  x0) & (for r st r<x0 ex g st r<g & g<x0 & g in dom f & f.g<>0) implies f^
  is_left_convergent_in x0 & lim_left(f^,x0)=0
proof
  assume
A1: f is_left_divergent_to+infty_in x0 or f is_left_divergent_to-infty_in x0;
A2: now
    dom(f^)=dom f\f"{0} by RFUNCT_1:def 2;
    then
A3: dom(f^)c=dom f by XBOOLE_1:36;
    let seq such that
A4: seq is convergent and
A5: lim seq=x0 and
A6: rng seq c=dom(f^)/\left_open_halfline(x0);
    dom(f^)/\left_open_halfline(x0)c=left_open_halfline(x0) by XBOOLE_1:17;
    then
A7: rng seq c=left_open_halfline(x0) by A6,XBOOLE_1:1;
A8: dom(f^)/\left_open_halfline(x0)c=dom(f^) by XBOOLE_1:17;
    then rng seq c=dom(f^) by A6,XBOOLE_1:1;
    then rng seq c=dom f by A3,XBOOLE_1:1;
    then
A9: rng seq c=dom f/\left_open_halfline(x0) by A7,XBOOLE_1:19;
    now
      per cases by A1;
      suppose
        f is_left_divergent_to+infty_in x0;
        then
A10:    f/*seq is divergent_to+infty by A4,A5,A9;
        then
A11:    lim((f/*seq)")=0 by LIMFUNC1:34;
        (f/*seq)" is convergent by A10,LIMFUNC1:34;
        hence (f^)/*seq is convergent & lim((f^)/*seq)=0 by A6,A8,A11,
RFUNCT_2:12,XBOOLE_1:1;
      end;
      suppose
        f is_left_divergent_to-infty_in x0;
        then
A12:    f/*seq is divergent_to-infty by A4,A5,A9;
        then
A13:    lim((f/*seq)")=0 by LIMFUNC1:34;
        (f/*seq)" is convergent by A12,LIMFUNC1:34;
        hence (f^)/*seq is convergent & lim((f^)/*seq)=0 by A6,A8,A13,
RFUNCT_2:12,XBOOLE_1:1;
      end;
    end;
    hence (f^)/*seq is convergent & lim((f^)/*seq)=0;
  end;
  assume
A14: for r st r<x0 ex g st r<g & g<x0 & g in dom f & f.g<>0;
  now
    let r;
    assume r<x0;
    then consider g such that
A15: r<g and
A16: g<x0 and
A17: g in dom f and
A18: f.g<>0 by A14;
    take g;
    thus r<g & g<x0 by A15,A16;
    not f.g in {0} by A18,TARSKI:def 1;
    then not g in f"{0} by FUNCT_1:def 7;
    then g in dom f\f"{0} by A17,XBOOLE_0:def 5;
    hence g in dom(f^) by RFUNCT_1:def 2;
  end;
  hence f^ is_left_convergent_in x0 by A2;
  hence thesis by A2,Def7;
end;
