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 convergent_in-infty & (for r
  ex g st g<r & g in dom(f2*f1)) implies f2*f1 is convergent_in-infty &
  lim_in-infty(f2*f1)=lim_in-infty f2
proof
  assume that
A1: f1 is divergent_in-infty_to-infty and
A2: f2 is convergent_in-infty and
A3: for r ex g st g<r & g in dom(f2*f1);
A4: now
    let s be Real_Sequence;
    assume that
A5: s is divergent_to-infty and
A6: rng s c=dom(f2*f1);
    rng s c=dom f1 by A6,Lm2;
    then
A7: f1/*s is divergent_to-infty by A1,A5;
A8: rng(f1/*s)c=dom f2 by A6,Lm2;
    then
A9: lim(f2/*(f1/*s))=lim_in-infty f2 by A2,A7,LIMFUNC1:def 13;
    f2/*(f1/*s) is convergent by A2,A8,A7;
    hence (f2*f1)/*s is convergent & lim((f2*f1)/*s)=lim_in-infty f2 by A6,A9,
VALUED_0:31;
  end;
  hence f2*f1 is convergent_in-infty by A3;
  hence thesis by A4,LIMFUNC1:def 13;
end;
