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

theorem
  (f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0) & (
for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f & g2<r2 & x0
<g2 & g2 in dom f & f.g1<>0 & f.g2<>0) implies f^ is_convergent_in x0 & lim(f^,
  x0)=0
proof
A1: dom f\f"{0}=dom(f^) by RFUNCT_1:def 2;
  assume
A2: f is_divergent_to+infty_in x0 or f is_divergent_to-infty_in x0;
A3: now
    let seq;
    assume that
A4: seq is convergent and
A5: lim seq=x0 and
A6: rng seq c=dom(f^)\{x0};
    rng seq c=dom(f^) by A6,XBOOLE_1:1;
    then
A7: rng seq c=dom f by A1,XBOOLE_1:1;
A8: rng seq c=dom f\{x0}
    proof
      let x be object;
      assume
A9:   x in rng seq;
      then not x in {x0} by A6,XBOOLE_0:def 5;
      hence thesis by A7,A9,XBOOLE_0:def 5;
    end;
    now
      per cases by A2;
      suppose
        f is_divergent_to+infty_in x0;
        then
A10:    f/*seq is divergent_to+infty by A4,A5,A8;
        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,A11,RFUNCT_2:12
,XBOOLE_1:1;
      end;
      suppose
        f is_divergent_to-infty_in x0;
        then
A12:    f/*seq is divergent_to-infty by A4,A5,A8;
        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,A13,RFUNCT_2:12
,XBOOLE_1:1;
      end;
    end;
    hence (f^)/*seq is convergent & lim((f^)/*seq)=0;
  end;
  assume
A14: for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f &
  g2<r2 & x0<g2 & g2 in dom f & f.g1<>0 & f.g2<>0;
  now
    let r1,r2;
    assume that
A15: r1<x0 and
A16: x0<r2;
    consider g1,g2 such that
A17: r1<g1 and
A18: g1<x0 and
A19: g1 in dom f and
A20: g2<r2 and
A21: x0<g2 and
A22: g2 in dom f and
A23: f.g1<>0 and
A24: f.g2<>0 by A14,A15,A16;
    take g1,g2;
    not f.g2 in {0} by A24,TARSKI:def 1;
    then
A25: not g2 in f"{0} by FUNCT_1:def 7;
    not f.g1 in {0} by A23,TARSKI:def 1;
    then not g1 in f"{0} by FUNCT_1:def 7;
    hence r1<g1 & g1<x0 & g1 in dom(f^) & g2<r2 & x0<g2 & g2 in dom(f^) by A1
,A17,A18,A19,A20,A21,A22,A25,XBOOLE_0:def 5;
  end;
  hence f^ is_convergent_in x0 by A3;
  hence thesis by A3,Def4;
end;
