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
  f1 is_left_convergent_in x0 & f2 is_left_convergent_in x0 & lim_left(
f2,x0 ) <> 0 & (for r st r<x0 ex g st r<g & g<x0 & g in dom(f1/f2)) implies f1/
f2 is_left_convergent_in x0 & lim_left(f1/f2,x0)=(lim_left(f1,x0))/(lim_left(f2
  ,x0))
proof
  assume that
A1: f1 is_left_convergent_in x0 and
A2: f2 is_left_convergent_in x0 and
A3: lim_left(f2,x0)<>0 and
A4: for r st r<x0 ex g st r<g & g<x0 & g in dom(f1/f2);
A5: now
    let r;
    assume r<x0;
    then consider g such that
A6: r<g and
A7: g<x0 and
A8: g in dom(f1/f2) by A4;
    take g;
    thus r<g & g<x0 by A6,A7;
    dom(f1/f2)=dom f1/\(dom f2\f2"{0}) by RFUNCT_1:def 1;
    then
A9: g in dom f2\f2"{0} by A8,XBOOLE_0:def 4;
    then
A10: not g in f2"{0} by XBOOLE_0:def 5;
    g in dom f2 by A9,XBOOLE_0:def 5;
    then not f2.g in {0} by A10,FUNCT_1:def 7;
    hence g in dom f2 & f2.g<>0 by A9,TARSKI:def 1,XBOOLE_0:def 5;
  end;
  then
A11: f2^ is_left_convergent_in x0 by A2,A3,Th49;
A12: f1/f2=f1(#)(f2^) by RFUNCT_1:31;
  hence f1/f2 is_left_convergent_in x0 by A1,A4,A11,Th50;
  lim_left(f2^,x0)=(lim_left(f2,x0))" by A2,A3,A5,Th49;
  hence
  lim_left(f1/f2,x0)=lim_left(f1,x0)*((lim_left(f2,x0))") by A1,A4,A12,A11,Th50
    .=lim_left(f1,x0)/(lim_left(f2,x0)) by XCMPLX_0:def 9;
end;
