let f be PartFunc of REAL,REAL; :: thesis: ( f is convergent_in+infty & lim_in+infty f = 0 & ( for r being Real ex g being Real st
( r < g & g in dom f & f . g <> 0 ) ) & ex r being Real st
for g being Real st g in (dom f) /\ (right_open_halfline r) holds
0 <= f . g implies f ^ is divergent_in+infty_to+infty )
assume that
A1: ( f is convergent_in+infty & lim_in+infty f = 0 ) and
A2: for r being Real ex g being Real st
( r < g & g in dom f & f . g <> 0 ) ; :: thesis: ( for r being Real ex g being Real st
( g in (dom f) /\ (right_open_halfline r) & not 0 <= f . g ) 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
0 <= f . g ; :: 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 7 :: 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
A5: g1 in dom f and
A6: f . g1 <> 0 by A2;
take g1 ; :: thesis: ( r1 < g1 & g1 in dom (f ^) )
thus r1 < g1 by A4; :: thesis: g1 in dom (f ^)
not f . g1 in {0} by A6, TARSKI:def 1;
then not g1 in f " {0} by FUNCT_1:def 7;
then g1 in (dom f) \ (f " {0}) by A5, XBOOLE_0:def 5;
hence g1 in dom (f ^) by RFUNCT_1:def 2; :: 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
A7: s is divergent_to+infty and
A8: rng s c= dom (f ^) ; :: thesis: (f ^) /* s is divergent_to+infty
consider k being Nat such that
A9: for n being Nat st k <= n holds
r < s . n by A7;
A10: rng (s ^\ k) c= rng s by VALUED_0:21;
dom (f ^) = (dom f) \ (f " {0}) by RFUNCT_1:def 2;
then A11: dom (f ^) c= dom f by XBOOLE_1:36;
then A12: rng s c= dom f by A8;
then A13: rng (s ^\ k) c= dom f by A10;
A14: f /* (s ^\ k) is non-zero by A8, A10, RFUNCT_2:11, XBOOLE_1:1;

now :: thesis: for n being Nat holds 0 < (f /* (s ^\ k)) . n

let n be Nat; :: thesis: 0 < (f /* (s ^\ k)) . n
A15: n in NAT by ORDINAL1:def 12;
r < s . (n + k) by A9, 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 A13, XBOOLE_0:def 4;
then A16: 0 <= f . ((s ^\ k) . n) by A3;
(f /* (s ^\ k)) . n <> 0 by A14, SEQ_1:5;
hence 0 < (f /* (s ^\ k)) . n by A12, A10, A16, FUNCT_2:108, XBOOLE_1:1, A15; :: thesis: verum

end;

then A17: for n being Nat st 0 <= n holds
0 < (f /* (s ^\ k)) . n ;
s ^\ k is divergent_to+infty by A7, Th26;
then ( f /* (s ^\ k) is convergent & lim (f /* (s ^\ k)) = 0 ) by A1, A13, Def12;
then A18: (f /* (s ^\ k)) " is divergent_to+infty by A17, Th35;
(f /* (s ^\ k)) " = ((f /* s) ^\ k) " by A8, A11, VALUED_0:27, XBOOLE_1:1
.= ((f /* s) ") ^\ k by SEQM_3:18
.= ((f ^) /* s) ^\ k by A8, RFUNCT_2:12 ;
hence (f ^) /* s is divergent_to+infty by A18, Th7; :: thesis: verum