reserve r,r1,r2,g,g1,g2,x0 for Real;
reserve f1,f2 for PartFunc of REAL,REAL;

theorem
  f1 is_right_divergent_to+infty_in x0 & f2 is convergent_in+infty & (
  for r st x0<r ex g st g<r & x0<g & g in dom(f2*f1)) implies f2*f1
  is_right_convergent_in x0 & lim_right(f2*f1,x0)=lim_in+infty f2
proof
  assume that
A1: f1 is_right_divergent_to+infty_in x0 and
A2: f2 is convergent_in+infty and
A3: for r st x0<r ex g st g<r & x0<g & g in dom(f2*f1);
A4: now
    let s be Real_Sequence;
    assume that
A5: s is convergent & lim s=x0 and
A6: rng s c=dom(f2*f1)/\right_open_halfline(x0);
    rng s c=dom f1/\right_open_halfline(x0) by A6,Th1;
    then
A7: f1/*s is divergent_to+infty by A1,A5,LIMFUNC2:def 5;
A8: rng(f1/*s)c=dom f2 by A6,Th1;
    then
A9: lim(f2/*(f1/*s))=lim_in+infty f2 by A2,A7,LIMFUNC1:def 12;
A10: rng s c=dom(f2*f1) by A6,Th1;
    f2/*(f1/*s) is convergent by A2,A8,A7;
    hence (f2*f1)/*s is convergent & lim((f2*f1)/*s)=lim_in+infty f2 by A10,A9,
VALUED_0:31;
  end;
  hence f2*f1 is_right_convergent_in x0 by A3,LIMFUNC2:def 4;
  hence thesis by A4,LIMFUNC2:def 8;
end;
