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

theorem
  f1 is_divergent_to+infty_in x0 & f2 is divergent_in+infty_to+infty & (
for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom(f2*f1) & g2<r2
  & x0<g2 & g2 in dom(f2*f1)) implies f2*f1 is_divergent_to+infty_in x0
proof
  assume that
A1: f1 is_divergent_to+infty_in x0 and
A2: f2 is divergent_in+infty_to+infty and
A3: for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom(f2*
  f1) & g2<r2 & x0<g2 & g2 in dom(f2*f1);
  now
    let s be Real_Sequence;
    assume that
A4: s is convergent & lim s=x0 and
A5: rng s c=dom(f2*f1)\{x0};
A6: rng s c=dom(f2*f1) by A5,Th2;
    rng s c=dom f1\{x0} by A5,Th2;
    then
A7: f1/*s is divergent_to+infty by A1,A4,LIMFUNC3:def 2;
    rng(f1/*s)c=dom f2 by A5,Th2;
    then f2/*(f1/*s) is divergent_to+infty by A2,A7;
    hence (f2*f1)/*s is divergent_to+infty by A6,VALUED_0:31;
  end;
  hence thesis by A3,LIMFUNC3:def 2;
end;
