let f be PartFunc of REAL ,REAL ; :: thesis: ( f is convergent_in+infty & lim_in+infty f = 0 & ex r being Real st
for g being Real st g in (dom f) /\ (right_open_halfline r) holds
f . g < 0 implies f ^ is divergent_in+infty_to-infty )
assume that
A1: f is convergent_in+infty and
A2: lim_in+infty f = 0 ; :: thesis: ( for r being Real ex g being Real st
( g in (dom f) /\ (right_open_halfline r) & not f . g < 0 ) or f ^ is divergent_in+infty_to-infty )
given r being Real such that A3: for g being Real st g in (dom f) /\ (right_open_halfline r) holds
f . g < 0 ; :: thesis: f ^ is divergent_in+infty_to-infty
thus for r1 being Real ex g1 being Real st
( r1 < g1 & g1 in dom (f ^ ) ) :: according to LIMFUNC1:def 8 :: thesis: for seq being Real_Sequence st seq is divergent_to+infty & rng seq c= dom (f ^ ) holds
(f ^ ) /* seq is divergent_to-infty

proof

let r1 be Real; :: thesis: ex g1 being Real st
( r1 < g1 & g1 in dom (f ^ ) )
consider g1 being Real such that
A4: r1 < g1 and
g1 in dom f by A1, Def6;

now

per cases ( g1 <= r or r <= g1 ) ;

suppose A5: g1 <= r ; :: thesis: ex g2 being Real st
( r1 < g2 & g2 in dom (f ^ ) )

consider g2 being Real such that
A6: r < g2 and
A7: g2 in dom f by A1, Def6;
take g2 = g2; :: thesis: ( r1 < g2 & g2 in dom (f ^ ) )
g1 < g2 by A5, A6, XXREAL_0:2;
hence r1 < g2 by A4, XXREAL_0:2; :: thesis: g2 in dom (f ^ )
g2 in { r2 where r2 is Real : r < r2 } by A6;
then g2 in right_open_halfline r by XXREAL_1:230;
then g2 in (dom f) /\ (right_open_halfline r) by A7, XBOOLE_0:def 4;
then 0 <> f . g2 by A3;
then not f . g2 in {0 } by TARSKI:def 1;
then not g2 in f " {0 } by FUNCT_1:def 13;
then g2 in (dom f) \ (f " {0 }) by A7, XBOOLE_0:def 5;
hence g2 in dom (f ^ ) by RFUNCT_1:def 8; :: thesis: verum

end;

suppose A8: r <= g1 ; :: thesis: ex g2 being Real st
( r1 < g2 & g2 in dom (f ^ ) )

consider g2 being Real such that
A9: g1 < g2 and
A10: g2 in dom f by A1, Def6;
take g2 = g2; :: thesis: ( r1 < g2 & g2 in dom (f ^ ) )
thus r1 < g2 by A4, A9, XXREAL_0:2; :: thesis: g2 in dom (f ^ )
r < g2 by A8, A9, XXREAL_0:2;
then g2 in { r2 where r2 is Real : r < r2 } ;
then g2 in right_open_halfline r by XXREAL_1:230;
then g2 in (dom f) /\ (right_open_halfline r) by A10, XBOOLE_0:def 4;
then 0 <> f . g2 by A3;
then not f . g2 in {0 } by TARSKI:def 1;
then not g2 in f " {0 } by FUNCT_1:def 13;
then g2 in (dom f) \ (f " {0 }) by A10, XBOOLE_0:def 5;
hence g2 in dom (f ^ ) by RFUNCT_1:def 8; :: thesis: verum

end;

end;

end;

hence ex g1 being Real st
( r1 < g1 & g1 in dom (f ^ ) ) ; :: thesis: verum

end;

let s be Real_Sequence; :: thesis: ( s is divergent_to+infty & rng s c= dom (f ^ ) implies (f ^ ) /* s is divergent_to-infty )
assume that
A11: s is divergent_to+infty and
A12: rng s c= dom (f ^ ) ; :: thesis: (f ^ ) /* s is divergent_to-infty
consider k being Element of NAT such that
A13: for n being Element of NAT st k <= n holds
r < s . n by A11, Def4;
A14: rng (s ^\ k) c= rng s by VALUED_0:21;
dom (f ^ ) = (dom f) \ (f " {0 }) by RFUNCT_1:def 8;
then A15: dom (f ^ ) c= dom f by XBOOLE_1:36;
then A16: rng s c= dom f by A12, XBOOLE_1:1;
then A17: rng (s ^\ k) c= dom f by A14, XBOOLE_1:1;

now

let n be Element of NAT ; :: thesis: (f /* (s ^\ k)) . n < 0
r < s . (n + k) by A13, NAT_1:12;
then r < (s ^\ k) . n by NAT_1:def 3;
then (s ^\ k) . n in { g2 where g2 is Real : r < g2 } ;
then ( (s ^\ k) . n in rng (s ^\ k) & (s ^\ k) . n in right_open_halfline r ) by VALUED_0:28, XXREAL_1:230;
then (s ^\ k) . n in (dom f) /\ (right_open_halfline r) by A17, XBOOLE_0:def 4;
then f . ((s ^\ k) . n) < 0 by A3;
hence (f /* (s ^\ k)) . n < 0 by A16, A14, FUNCT_2:185, XBOOLE_1:1; :: thesis: verum

end;

then A18: for n being Element of NAT st 0 <= n holds
(f /* (s ^\ k)) . n < 0 ;
s ^\ k is divergent_to+infty by A11, Th53;
then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A2, A17, Def12;
then A19: (f /* (s ^\ k)) " is divergent_to-infty by A18, Th63;
(f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A12, A15, VALUED_0:27, XBOOLE_1:1
.= ((f /* s) " ) ^\ k by SEQM_3:41
.= ((f ^ ) /* s) ^\ k by A12, RFUNCT_2:27 ;
hence (f ^ ) /* s is divergent_to-infty by A19, Th34; :: thesis: verum