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 & f"{0}={} & (lim_in-infty f)<>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: f"{0}={} and
A3: (lim_in-infty f)<>0;
A4: dom f= dom f\f"{0} by A2
    .=dom(f^) by RFUNCT_1:def 2;
A5: now
    let seq;
    assume that
A6: seq is divergent_to-infty and
A7: rng seq c=dom(f^);
A8: f/*seq is convergent & lim(f/*seq)=lim_in-infty f by A1,A4,A6,A7,Def13;
    then (f/*seq)" is convergent by A3,A7,RFUNCT_2:11,SEQ_2:21;
    hence (f^)/*seq is convergent by A7,RFUNCT_2:12;
    thus lim((f^)/*seq)=lim((f/*seq)") by A7,RFUNCT_2:12
      .=(lim_in-infty f)" by A3,A7,A8,RFUNCT_2:11,SEQ_2:22;
  end;
  for r ex g st g<r & g in dom(f^) by A1,A4;
  hence f^ is convergent_in-infty by A5;
  hence thesis by A5,Def13;
end;
