theorem :: LIMFUNC4:47

for f1, f2 being PartFunc of REAL,REAL st f1 is convergent_in-infty & f2 is_divergent_to+infty_in lim_in-infty f1 & ( for r being Real ex g being Real st
( g < r & g in dom (f2 * f1) ) ) & ex r being Real st
for g being Real st g in (dom f1) /\ (left_open_halfline r) holds
f1 . g <> lim_in-infty f1 holds
f2 * f1 is divergent_in-infty_to+infty