let f1, f be PartFunc of REAL,REAL; :: thesis: ( f1 is divergent_in-infty_to-infty & ( for r being Real ex g being Real st
( g < r & g in dom f ) ) & ex r being Real st
( (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) & ( for g being Real st g in (dom f) /\ (left_open_halfline r) holds
f . g <= f1 . g ) ) implies f is divergent_in-infty_to-infty )
assume that
A1: f1 is divergent_in-infty_to-infty and
A2: for r being Real ex g being Real st
( g < r & g in dom f ) ; :: thesis: ( for r being Real holds
( not (dom f) /\ (left_open_halfline r) c= (dom f1) /\ (left_open_halfline r) or ex g being Real st
( g in (dom f) /\ (left_open_halfline r) & not f . g <= f1 . g ) ) or f is divergent_in-infty_to-infty )
given r1 being Real such that A3: (dom f) /\ (left_open_halfline r1) c= (dom f1) /\ (left_open_halfline r1) and
A4: for g being Real st g in (dom f) /\ (left_open_halfline r1) holds
f . g <= f1 . g ; :: thesis: f is divergent_in-infty_to-infty

now

let seq be Real_Sequence; :: thesis: ( seq is divergent_to-infty & rng seq c= dom f implies f /* seq is divergent_to-infty )
assume that
A5: seq is divergent_to-infty and
A6: rng seq c= dom f ; :: thesis: f /* seq is divergent_to-infty
consider k being Element of NAT such that
A7: for n being Element of NAT st k <= n holds
seq . n < r1 by A5, Def5;

now

let x be set ; :: thesis: ( x in rng (seq ^\ k) implies x in left_open_halfline r1 )
assume x in rng (seq ^\ k) ; :: thesis: x in left_open_halfline r1
then consider n being Element of NAT such that
A8: (seq ^\ k) . n = x by FUNCT_2:113;
seq . (n + k) < r1 by A7, NAT_1:12;
then (seq ^\ k) . n < r1 by NAT_1:def 3;
then x in { g2 where g2 is Real : g2 < r1 } by A8;
hence x in left_open_halfline r1 by XXREAL_1:229; :: thesis: verum

end;

then A9: rng (seq ^\ k) c= left_open_halfline r1 by TARSKI:def 3;
A10: rng (seq ^\ k) c= rng seq by VALUED_0:21;
then rng (seq ^\ k) c= dom f by A6, XBOOLE_1:1;
then A11: rng (seq ^\ k) c= (dom f) /\ (left_open_halfline r1) by A9, XBOOLE_1:19;
then A12: rng (seq ^\ k) c= (dom f1) /\ (left_open_halfline r1) by A3, XBOOLE_1:1;
A13: (dom f1) /\ (left_open_halfline r1) c= dom f1 by XBOOLE_1:17;

A14: now

let n be Element of NAT ; :: thesis: (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n
(seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28;
then f . ((seq ^\ k) . n) <= f1 . ((seq ^\ k) . n) by A4, A11;
then (f /* (seq ^\ k)) . n <= f1 . ((seq ^\ k) . n) by A6, A10, FUNCT_2:108, XBOOLE_1:1;
hence (f /* (seq ^\ k)) . n <= (f1 /* (seq ^\ k)) . n by A12, A13, FUNCT_2:108, XBOOLE_1:1; :: thesis: verum

end;

A15: seq ^\ k is divergent_to-infty by A5, Th54;
rng (seq ^\ k) c= dom f1 by A12, A13, XBOOLE_1:1;
then f1 /* (seq ^\ k) is divergent_to-infty by A1, A15, Def11;
then A16: f /* (seq ^\ k) is divergent_to-infty by A14, Th70;
f /* (seq ^\ k) = (f /* seq) ^\ k by A6, VALUED_0:27;
hence f /* seq is divergent_to-infty by A16, Th34; :: thesis: verum

end;

hence f is divergent_in-infty_to-infty by A2, Def11; :: thesis: verum