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
  f1 is convergent_in-infty & f2 is convergent_in-infty & lim_in-infty
  f2<> 0 & (for r ex g st g<r & g in dom(f1/f2)) implies f1/f2 is
  convergent_in-infty & lim_in-infty(f1/f2)=(lim_in-infty f1)/(lim_in-infty f2)
proof
  assume that
A1: f1 is convergent_in-infty and
A2: f2 is convergent_in-infty & lim_in-infty f2<>0 and
A3: for r ex g st g<r & g in dom(f1/f2);
  dom(f1/f2)=dom f1/\(dom f2\f2"{0}) by RFUNCT_1:def 1;
  then
A4: dom(f1/f2)=dom f1/\dom(f2^) by RFUNCT_1:def 2;
A5: dom f1/\dom(f2^) c=dom(f2^) by XBOOLE_1:17;
A6: now
    let r;
    consider g such that
A7: g<r and
A8: g in dom(f1/f2) by A3;
    take g;
    g in dom(f2^) by A4,A5,A8;
    then
A9: g in dom f2\f2"{0} by RFUNCT_1:def 2;
    then g in dom f2 & not g in f2"{0} by XBOOLE_0:def 5;
    then not f2.g in {0} by FUNCT_1:def 7;
    hence g<r & g in dom f2 & f2.g<>0 by A7,A9,TARSKI:def 1,XBOOLE_0:def 5;
  end;
  then
A10: f2^ is convergent_in-infty by A2,Th95;
A11: lim_in-infty(f2^)=(lim_in-infty f2)" by A2,A6,Th95;
A12: now
    let r;
    consider g such that
A13: g<r & g in dom(f1/f2) by A3;
    take g;
    thus g<r & g in dom(f1(#)(f2^)) by A4,A13,VALUED_1:def 4;
  end;
  then f1(#)(f2^) is convergent_in-infty by A1,A10,Th96;
  hence f1/f2 is convergent_in-infty by RFUNCT_1:31;
  thus lim_in-infty(f1/f2)=lim_in-infty(f1(#)(f2^)) by RFUNCT_1:31
    .=(lim_in-infty f1)*((lim_in-infty f2)") by A1,A12,A10,A11,Th96
    .=(lim_in-infty f1)/(lim_in-infty f2) by XCMPLX_0:def 9;
end;
