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

theorem
  f1 is divergent_in-infty_to-infty & f2 is divergent_in-infty_to+infty
  & (for r ex g st g<r & g in dom(f2*f1)) implies f2*f1 is
  divergent_in-infty_to+infty
proof
  assume that
A1: f1 is divergent_in-infty_to-infty and
A2: f2 is divergent_in-infty_to+infty and
A3: for r ex g st g<r & g in dom(f2*f1);
  now
    let s be Real_Sequence;
    assume that
A4: s is divergent_to-infty and
A5: rng s c=dom(f2*f1);
    rng s c=dom f1 by A5,Lm2;
    then
A6: f1/*s is divergent_to-infty by A1,A4;
    rng(f1/*s)c=dom f2 by A5,Lm2;
    then f2/*(f1/*s) is divergent_to+infty by A2,A6;
    hence (f2*f1)/*s is divergent_to+infty by A5,VALUED_0:31;
  end;
  hence thesis by A3;
end;
