let x0 be Real; for f1, f2 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 (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) & ex r being Real st
( 0 < r & f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below ) holds
f1 + f2 is_divergent_to+infty_in x0
let f1, f2 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 (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) & ex r being Real st
( 0 < r & f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below ) implies f1 + f2 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 (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) )
; ( for r being Real holds
( not 0 < r or not f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below ) or f1 + f2 is_divergent_to+infty_in x0 )
given r being Real such that A3:
0 < r
and
A4:
f2 | (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) is bounded_below
; f1 + f2 is_divergent_to+infty_in x0
now for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom (f1 + f2)) \ {x0} holds
(f1 + f2) /* s is divergent_to+infty let s be
Real_Sequence;
( s is convergent & lim s = x0 & rng s c= (dom (f1 + f2)) \ {x0} implies (f1 + f2) /* s is divergent_to+infty )assume that A5:
s is
convergent
and A6:
lim s = x0
and A7:
rng s c= (dom (f1 + f2)) \ {x0}
;
(f1 + f2) /* 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 - r < s . n &
s . n < x0 + r )
by A3, A5, A6, Th7;
rng (s ^\ k) c= rng s
by VALUED_0:21;
then A9:
rng (s ^\ k) c= (dom (f1 + f2)) \ {x0}
by A7;
then A10:
rng (s ^\ k) c= (dom f1) \ {x0}
by Lm4;
A11:
rng (s ^\ k) c= dom f2
by A9, Lm4;
now ex r2 being set st
for n being Nat holds r2 < (f2 /* (s ^\ k)) . nconsider r1 being
Real such that A12:
for
g being
object st
g in (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) /\ (dom f2) holds
r1 <= f2 . g
by A4, RFUNCT_1:71;
take r2 =
r1 - 1;
for n being Nat holds r2 < (f2 /* (s ^\ k)) . nlet n be
Nat;
r2 < (f2 /* (s ^\ k)) . nA13:
n in NAT
by ORDINAL1:def 12;
A14:
k <= n + k
by NAT_1:12;
then
s . (n + k) < x0 + r
by A8;
then A15:
(s ^\ k) . n < x0 + r
by NAT_1:def 3;
x0 - r < s . (n + k)
by A8, A14;
then
x0 - r < (s ^\ k) . n
by NAT_1:def 3;
then
(s ^\ k) . n in { g2 where g2 is Real : ( x0 - r < g2 & g2 < x0 + r ) }
by A15;
then A16:
(s ^\ k) . n in ].(x0 - r),(x0 + r).[
by RCOMP_1:def 2;
A17:
(s ^\ k) . n in rng (s ^\ k)
by VALUED_0:28;
then
not
(s ^\ k) . n in {x0}
by A9, XBOOLE_0:def 5;
then
(s ^\ k) . n in ].(x0 - r),(x0 + r).[ \ {x0}
by A16, XBOOLE_0:def 5;
then
(s ^\ k) . n in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[
by A3, Th4;
then
(s ^\ k) . n in (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) /\ (dom f2)
by A11, A17, XBOOLE_0:def 4;
then
r1 - 1
< (f2 . ((s ^\ k) . n)) - 0
by A12, XREAL_1:15;
hence
r2 < (f2 /* (s ^\ k)) . n
by A11, FUNCT_2:108, A13;
verum end; then A18:
f2 /* (s ^\ k) is
bounded_below
by SEQ_2:def 4;
lim (s ^\ k) = x0
by A5, A6, SEQ_4:20;
then
f1 /* (s ^\ k) is
divergent_to+infty
by A1, A5, A10;
then A19:
(f1 /* (s ^\ k)) + (f2 /* (s ^\ k)) is
divergent_to+infty
by A18, LIMFUNC1:9;
A20:
rng s c= dom (f1 + f2)
by A7, Lm4;
rng (s ^\ k) c= dom (f1 + f2)
by A9, Lm4;
then
rng (s ^\ k) c= (dom f1) /\ (dom f2)
by VALUED_1:def 1;
then (f1 /* (s ^\ k)) + (f2 /* (s ^\ k)) =
(f1 + f2) /* (s ^\ k)
by RFUNCT_2:8
.=
((f1 + f2) /* s) ^\ k
by A20, VALUED_0:27
;
hence
(f1 + f2) /* s is
divergent_to+infty
by A19, LIMFUNC1:7;
verum end;
hence
f1 + f2 is_divergent_to+infty_in x0
by A2; verum