let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL ,REAL st f1 is_left_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ) holds
( f2 * f1 is_left_convergent_in x0 & lim_left (f2 * f1),x0 = lim_in-infty f2 )
let f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( f1 is_left_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ) implies ( f2 * f1 is_left_convergent_in x0 & lim_left (f2 * f1),x0 = lim_in-infty f2 ) )
assume A1: ( f1 is_left_divergent_to-infty_in x0 & f2 is convergent_in-infty & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ) ) ; :: thesis: ( f2 * f1 is_left_convergent_in x0 & lim_left (f2 * f1),x0 = lim_in-infty f2 )

A2: now

let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) )
A3: lim_in-infty f2 = lim_in-infty f2 ;
assume A4: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ) ; :: thesis: ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 )
then A5: ( rng s c= dom (f2 * f1) & rng s c= (dom f1) /\ (left_open_halfline x0) & rng (f1 /* s) c= dom f2 ) by Th1;
then f1 /* s is divergent_to-infty by A1, A4, LIMFUNC2:def 3;
then ( f2 /* (f1 /* s) is convergent & lim (f2 /* (f1 /* s)) = lim_in-infty f2 ) by A1, A3, A5, LIMFUNC1:def 13;
hence ( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 ) by A5, VALUED_0:31; :: thesis: verum

end;

hence f2 * f1 is_left_convergent_in x0 by A1, LIMFUNC2:def 1; :: thesis: lim_left (f2 * f1),x0 = lim_in-infty f2
hence lim_left (f2 * f1),x0 = lim_in-infty f2 by A2, LIMFUNC2:def 7; :: thesis: verum