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

theorem
  f1 is_left_divergent_to-infty_in x0 & f2 is
divergent_in-infty_to+infty & (for r st r<x0 ex g st r<g & g<x0 & g in dom(f2*
  f1)) implies f2*f1 is_left_divergent_to+infty_in x0
proof
  assume that
A1: f1 is_left_divergent_to-infty_in x0 and
A2: f2 is divergent_in-infty_to+infty and
A3: for r st r<x0 ex g st r<g & g<x0 & g 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)/\left_open_halfline(x0);
A6: rng s c=dom(f2*f1) by A5,Th1;
    rng s c=dom f1/\left_open_halfline(x0) by A5,Th1;
    then
A7: f1/*s is divergent_to-infty by A1,A4,LIMFUNC2:def 3;
    rng(f1/*s)c=dom f2 by A5,Th1;
    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,LIMFUNC2:def 2;
end;
