let x0 be Real; 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; ( 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 that
A1:
f1 is_left_divergent_to-infty_in x0
and
A2:
f2 is convergent_in-infty
and
A3:
for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) )
; ( f2 * f1 is_left_convergent_in x0 & lim_left ((f2 * f1),x0) = lim_in-infty f2 )
A4:
now for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) holds
( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 )let s be
Real_Sequence;
( 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 ) )assume that A5:
(
s is
convergent &
lim s = x0 )
and A6:
rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0)
;
( (f2 * f1) /* s is convergent & lim ((f2 * f1) /* s) = lim_in-infty f2 )
rng s c= (dom f1) /\ (left_open_halfline x0)
by A6, Th1;
then A7:
f1 /* s is
divergent_to-infty
by A1, A5, LIMFUNC2:def 3;
A8:
rng (f1 /* s) c= dom f2
by A6, Th1;
then A9:
lim (f2 /* (f1 /* s)) = lim_in-infty f2
by A2, A7, LIMFUNC1:def 13;
A10:
rng s c= dom (f2 * f1)
by A6, Th1;
f2 /* (f1 /* s) is
convergent
by A2, A8, A7;
hence
(
(f2 * f1) /* s is
convergent &
lim ((f2 * f1) /* s) = lim_in-infty f2 )
by A10, A9, VALUED_0:31;
verum end;
hence
f2 * f1 is_left_convergent_in x0
by A3, LIMFUNC2:def 1; lim_left ((f2 * f1),x0) = lim_in-infty f2
hence
lim_left ((f2 * f1),x0) = lim_in-infty f2
by A4, LIMFUNC2:def 7; verum