let f1, f2 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 (f1 (#) f2) ) ) & ex r, r1 being Real st
( 0 < r & ( for g being Real st g in (dom f2) /\ (left_open_halfline r1) holds
r <= f2 . g ) ) implies f1 (#) f2 is divergent_in-infty_to+infty )
assume A1: ( f1 is divergent_in-infty_to+infty & ( for r being Real ex g being Real st
( g < r & g in dom (f1 (#) f2) ) ) ) ; :: thesis: ( for r, r1 being Real holds
( not 0 < r or ex g being Real st
( g in (dom f2) /\ (left_open_halfline r1) & not r <= f2 . g ) ) or f1 (#) f2 is divergent_in-infty_to+infty )
given r2, r1 being Real such that A2: ( 0 < r2 & ( for g being Real st g in (dom f2) /\ (left_open_halfline r1) holds
r2 <= f2 . g ) ) ; :: thesis: f1 (#) f2 is divergent_in-infty_to+infty

now

let seq be Real_Sequence; :: thesis: ( seq is divergent_to-infty & rng seq c= dom (f1 (#) f2) implies (f1 (#) f2) /* seq is divergent_to+infty )
assume A3: ( seq is divergent_to-infty & rng seq c= dom (f1 (#) f2) ) ; :: thesis: (f1 (#) f2) /* seq is divergent_to+infty
then A4: ( dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 ) by Lm3;
consider k being Element of NAT such that
A5: for n being Element of NAT st k <= n holds
seq . n < r1 by A3, Def5;
A6: seq ^\ k is divergent_to-infty by A3, Th54;
A7: rng (seq ^\ k) c= rng seq by VALUED_0:21;
then A8: ( rng (seq ^\ k) c= dom f1 & rng (seq ^\ k) c= dom f2 ) by A4, XBOOLE_1:1;
then A9: f1 /* (seq ^\ k) is divergent_to+infty by A1, A6, Def10;

now

thus 0 < r2 by A2; :: thesis: for n being Element of NAT holds r2 <= (f2 /* (seq ^\ k)) . n
let n be Element of NAT ; :: thesis: r2 <= (f2 /* (seq ^\ k)) . n
A10: (seq ^\ k) . n in rng (seq ^\ k) by VALUED_0:28;
seq . (n + k) < r1 by A5, NAT_1:12;
then (seq ^\ k) . n < r1 by NAT_1:def 3;
then (seq ^\ k) . n in { g2 where g2 is Real : g2 < r1 } ;
then (seq ^\ k) . n in left_open_halfline r1 by XXREAL_1:229;
then (seq ^\ k) . n in (dom f2) /\ (left_open_halfline r1) by A8, A10, XBOOLE_0:def 4;
then r2 <= f2 . ((seq ^\ k) . n) by A2;
hence r2 <= (f2 /* (seq ^\ k)) . n by A4, A7, FUNCT_2:185, XBOOLE_1:1; :: thesis: verum

end;

then A11: (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) is divergent_to+infty by A9, Th49;
rng (seq ^\ k) c= (dom f1) /\ (dom f2) by A3, A4, A7, XBOOLE_1:1;
then (f1 /* (seq ^\ k)) (#) (f2 /* (seq ^\ k)) = (f1 (#) f2) /* (seq ^\ k) by RFUNCT_2:23
.= ((f1 (#) f2) /* seq) ^\ k by A3, VALUED_0:27 ;
hence (f1 (#) f2) /* seq is divergent_to+infty by A11, Th34; :: thesis: verum

end;

hence f1 (#) f2 is divergent_in-infty_to+infty by A1, Def10; :: thesis: verum