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_right_convergent_in x0 & f2 is_right_convergent_in x0 &
  lim_right(f2,x0)<>0 & (for r st x0<r ex g st g<r & x0<g & g in dom(f1/f2))
implies f1/f2 is_right_convergent_in x0 & lim_right(f1/f2,x0)=(lim_right(f1,x0)
  )/(lim_right(f2,x0))
proof
  assume that
A1: f1 is_right_convergent_in x0 and
A2: f2 is_right_convergent_in x0 and
A3: lim_right(f2,x0)<>0 and
A4: for r st x0<r ex g st g<r & x0<g & g in dom(f1/f2);
A5: now
    let r;
    assume x0<r;
    then consider g such that
A6: g<r and
A7: x0<g and
A8: g in dom(f1/f2) by A4;
    take g;
    thus g<r & x0<g 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_right_convergent_in x0 by A2,A3,Th58;
A12: f1/f2=f1(#)(f2^) by RFUNCT_1:31;
  hence f1/f2 is_right_convergent_in x0 by A1,A4,A11,Th59;
  lim_right(f2^,x0)=(lim_right(f2,x0)) " by A2,A3,A5,Th58;
  hence lim_right(f1/f2,x0)=lim_right(f1,x0)*((lim_right(f2,x0))") by A1,A4,A12
,A11,Th59
    .=lim_right(f1,x0)/(lim_right(f2,x0)) by XCMPLX_0:def 9;
end;
