let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( f1 is divergent_in-infty_to+infty & f2 is convergent_in+infty & ( for r being Real ex g being Real st
( g < r & g in dom (f2 * f1) ) ) implies ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in+infty f2 ) )
assume that
A1: f1 is divergent_in-infty_to+infty and
A2: f2 is convergent_in+infty and
A3: for r being Real ex g being Real st
( g < r & g in dom (f2 * f1) ) ; :: thesis: ( f2 * f1 is convergent_in-infty & lim_in-infty (f2 * f1) = lim_in+infty f2 )

A4: now :: thesis: for s being Real_Sequence st s is divergent_to-infty & rng s c= dom (f2 * f1) holds
( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 )

let s be Real_Sequence; :: thesis: ( s is divergent_to-infty & rng s c= dom (f2 * f1) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 ) )
assume that
A5: s is divergent_to-infty and
A6: rng s c= dom (f2 * f1) ; :: thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in+infty f2 )
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 12;
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; :: thesis: verum

end;

hence f2 * f1 is convergent_in-infty by A3; :: thesis: lim_in-infty (f2 * f1) = lim_in+infty f2
hence lim_in-infty (f2 * f1) = lim_in+infty f2 by A4, LIMFUNC1:def 13; :: thesis: verum