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 Th86:
  f is convergent_in+infty & lim_in+infty f<>0 & (for r ex g st r
<g & g in dom f & f.g<>0) implies f^ is convergent_in+infty & lim_in+infty(f^)=
  (lim_in+infty f)"
proof
  assume that
A1: f is convergent_in+infty and
A2: lim_in+infty f<>0 and
A3: for r ex g st r<g & g in dom f & f.g<>0;
A4: now
    let seq such that
A5: seq is divergent_to+infty and
A6: rng seq c=dom(f^);
    dom(f^)=dom f\f"{0} & dom f\f"{0}c=dom f by RFUNCT_1:def 2,XBOOLE_1:36;
    then rng seq c=dom f by A6;
    then
A7: f/*seq is convergent & lim(f/*seq)=lim_in+infty f by A1,A5,Def12;
    then (f/*seq)" is convergent by A2,A6,RFUNCT_2:11,SEQ_2:21;
    hence (f^)/*seq is convergent by A6,RFUNCT_2:12;
    thus lim((f^)/*seq)=lim((f/*seq)") by A6,RFUNCT_2:12
      .=(lim_in+infty f)" by A2,A6,A7,RFUNCT_2:11,SEQ_2:22;
  end;
  now
    let r;
    consider g such that
A8: r<g and
A9: g in dom f and
A10: f.g<>0 by A3;
    take g;
    not f.g in {0} by A10,TARSKI:def 1;
    then not g in f"{0} by FUNCT_1:def 7;
    then g in dom f\f"{0} by A9,XBOOLE_0:def 5;
    hence r<g & g in dom(f^) by A8,RFUNCT_1:def 2;
  end;
  hence f^ is convergent_in+infty by A4;
  hence thesis by A4,Def12;
end;
