let x0 be Real; for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st
( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds
f1 . r <> lim (f1,x0) ) ) holds
f2 * f1 is_divergent_to+infty_in x0
let f1, f2 be PartFunc of REAL,REAL; ( f1 is_convergent_in x0 & f2 is_divergent_to+infty_in lim (f1,x0) & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) ) ) & ex g being Real st
( 0 < g & ( for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds
f1 . r <> lim (f1,x0) ) ) implies f2 * f1 is_divergent_to+infty_in x0 )
assume that
A1:
f1 is_convergent_in x0
and
A2:
f2 is_divergent_to+infty_in lim (f1,x0)
and
A3:
for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f2 * f1) & g2 < r2 & x0 < g2 & g2 in dom (f2 * f1) )
; ( for g being Real holds
( not 0 < g or ex r being Real st
( r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) & not f1 . r <> lim (f1,x0) ) ) or f2 * f1 is_divergent_to+infty_in x0 )
given g being Real such that A4:
0 < g
and
A5:
for r being Real st r in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[) holds
f1 . r <> lim (f1,x0)
; f2 * f1 is_divergent_to+infty_in x0
now for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} holds
(f2 * f1) /* s is divergent_to+infty let s be
Real_Sequence;
( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) \ {x0} implies (f2 * f1) /* s is divergent_to+infty )assume that A6:
(
s is
convergent &
lim s = x0 )
and A7:
rng s c= (dom (f2 * f1)) \ {x0}
;
(f2 * f1) /* s is divergent_to+infty consider k being
Element of
NAT such that A8:
for
n being
Element of
NAT st
k <= n holds
(
x0 - g < s . n &
s . n < x0 + g )
by A4, A6, LIMFUNC3:7;
set q =
(f1 /* s) ^\ k;
A9:
rng s c= dom (f2 * f1)
by A7, Th2;
rng (f1 /* s) c= dom f2
by A7, Th2;
then A10:
f2 /* ((f1 /* s) ^\ k) =
(f2 /* (f1 /* s)) ^\ k
by VALUED_0:27
.=
((f2 * f1) /* s) ^\ k
by A9, VALUED_0:31
;
A11:
rng s c= (dom f1) \ {x0}
by A7, Th2;
then A12:
f1 /* s is
convergent
by A1, A2, A6, LIMFUNC3:def 4;
A13:
rng s c= dom f1
by A7, Th2;
now for x being object st x in rng ((f1 /* s) ^\ k) holds
x in (dom f2) \ {(lim (f1,x0))}let x be
object ;
( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) \ {(lim (f1,x0))} )assume
x in rng ((f1 /* s) ^\ k)
;
x in (dom f2) \ {(lim (f1,x0))}then consider n being
Element of
NAT such that A14:
((f1 /* s) ^\ k) . n = x
by FUNCT_2:113;
A15:
f1 . (s . (n + k)) =
(f1 /* s) . (n + k)
by A13, FUNCT_2:108
.=
x
by A14, NAT_1:def 3
;
k <= n + k
by NAT_1:12;
then
(
x0 - g < s . (n + k) &
s . (n + k) < x0 + g )
by A8;
then
s . (n + k) in { g1 where g1 is Real : ( x0 - g < g1 & g1 < x0 + g ) }
;
then A16:
s . (n + k) in ].(x0 - g),(x0 + g).[
by RCOMP_1:def 2;
A17:
s . (n + k) in rng s
by VALUED_0:28;
then
not
s . (n + k) in {x0}
by A7, XBOOLE_0:def 5;
then
s . (n + k) in ].(x0 - g),(x0 + g).[ \ {x0}
by A16, XBOOLE_0:def 5;
then
s . (n + k) in ].(x0 - g),x0.[ \/ ].x0,(x0 + g).[
by A4, LIMFUNC3:4;
then
s . (n + k) in (dom f1) /\ (].(x0 - g),x0.[ \/ ].x0,(x0 + g).[)
by A13, A17, XBOOLE_0:def 4;
then
f1 . (s . (n + k)) <> lim (
f1,
x0)
by A5;
then A18:
not
f1 . (s . (n + k)) in {(lim (f1,x0))}
by TARSKI:def 1;
f1 . (s . (n + k)) in dom f2
by A9, A17, FUNCT_1:11;
hence
x in (dom f2) \ {(lim (f1,x0))}
by A18, A15, XBOOLE_0:def 5;
verum end; then A19:
rng ((f1 /* s) ^\ k) c= (dom f2) \ {(lim (f1,x0))}
;
lim (f1 /* s) = lim (
f1,
x0)
by A1, A6, A11, LIMFUNC3:def 4;
then
lim ((f1 /* s) ^\ k) = lim (
f1,
x0)
by A12, SEQ_4:20;
then
f2 /* ((f1 /* s) ^\ k) is
divergent_to+infty
by A2, A12, A19, LIMFUNC3:def 2;
hence
(f2 * f1) /* s is
divergent_to+infty
by A10, LIMFUNC1:7;
verum end;
hence
f2 * f1 is_divergent_to+infty_in x0
by A3, LIMFUNC3:def 2; verum