let x0 be Real; for f, f1 being PartFunc of REAL,REAL st f1 is_divergent_to-infty_in 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 f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st
( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f . g <= f1 . g ) ) holds
f is_divergent_to-infty_in x0
let f, f1 be PartFunc of REAL,REAL; ( f1 is_divergent_to-infty_in 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 f & g2 < r2 & x0 < g2 & g2 in dom f ) ) & ex r being Real st
( 0 < r & (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f . g <= f1 . g ) ) implies f is_divergent_to-infty_in x0 )
assume that
A1:
f1 is_divergent_to-infty_in x0
and
A2:
for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
; ( for r being Real holds
( not 0 < r or not (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) or ex g being Real st
( g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & not f . g <= f1 . g ) ) or f is_divergent_to-infty_in x0 )
given r being Real such that A3:
0 < r
and
A4:
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[)
and
A5:
for g being Real st g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f . g <= f1 . g
; f is_divergent_to-infty_in x0
thus
for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
by A2; LIMFUNC3:def 3 for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds
f /* seq is divergent_to-infty
let s be Real_Sequence; ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to-infty )
assume that
A6:
s is convergent
and
A7:
lim s = x0
and
A8:
rng s c= (dom f) \ {x0}
; f /* s is divergent_to-infty
consider k being Element of NAT such that
A9:
for n being Element of NAT st k <= n holds
( x0 - r < s . n & s . n < x0 + r )
by A3, A6, A7, Th7;
A10:
rng (s ^\ k) c= rng s
by VALUED_0:21;
then A11:
rng (s ^\ k) c= (dom f) \ {x0}
by A8;
now for x being object st x in rng (s ^\ k) holds
x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[let x be
object ;
( x in rng (s ^\ k) implies x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ )assume
x in rng (s ^\ k)
;
x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[then consider n being
Element of
NAT such that A12:
(s ^\ k) . n = x
by FUNCT_2:113;
A13:
k <= n + k
by NAT_1:12;
then
s . (n + k) < x0 + r
by A9;
then A14:
(s ^\ k) . n < x0 + r
by NAT_1:def 3;
x0 - r < s . (n + k)
by A9, A13;
then
x0 - r < (s ^\ k) . n
by NAT_1:def 3;
then
(s ^\ k) . n in { g1 where g1 is Real : ( x0 - r < g1 & g1 < x0 + r ) }
by A14;
then A15:
(s ^\ k) . n in ].(x0 - r),(x0 + r).[
by RCOMP_1:def 2;
(s ^\ k) . n in rng (s ^\ k)
by VALUED_0:28;
then
not
(s ^\ k) . n in {x0}
by A11, XBOOLE_0:def 5;
then
(s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0}
by A15, XBOOLE_0:def 5;
hence
x in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[
by A3, A12, Th4;
verum end;
then A16:
rng (s ^\ k) c= ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[
;
A17:
rng s c= dom f
by A8, XBOOLE_1:1;
then
rng (s ^\ k) c= dom f
by A10;
then A18:
rng (s ^\ k) c= (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[)
by A16, XBOOLE_1:19;
then A19:
rng (s ^\ k) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[)
by A4;
A22:
rng (s ^\ k) c= dom f1
by A19, XBOOLE_1:18;
then A24:
rng (s ^\ k) c= (dom f1) \ {x0}
;
lim (s ^\ k) = x0
by A6, A7, SEQ_4:20;
then
f1 /* (s ^\ k) is divergent_to-infty
by A1, A6, A24;
then
f /* (s ^\ k) is divergent_to-infty
by A20, LIMFUNC1:43;
then
(f /* s) ^\ k is divergent_to-infty
by A8, VALUED_0:27, XBOOLE_1:1;
hence
f /* s is divergent_to-infty
by LIMFUNC1:7; verum