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
( r < g & g in dom (f1 + f2) ) ) & ex r being Real st f2 | (right_open_halfline r) is bounded_below implies f1 + f2 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
( r < g & g in dom (f1 + f2) ) ; :: thesis: ( for r being Real holds not f2 | (right_open_halfline r) is bounded_below or f1 + f2 is divergent_in+infty_to+infty )
given r1 being Real such that A3: f2 | (right_open_halfline r1) is bounded_below ; :: 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 that
A4: seq is divergent_to+infty and
A5: rng seq c= dom (f1 + f2) ; :: thesis: (f1 + f2) /* seq is divergent_to+infty
consider k being Element of NAT such that
A6: for n being Element of NAT st k <= n holds
r1 < seq . n by A4, Def4;
A7: rng (seq ^\ k) c= rng seq by VALUED_0:21;
dom (f1 + f2) = (dom f1) /\ (dom f2) by A5, Lm2;
then rng (seq ^\ k) c= (dom f1) /\ (dom f2) by A5, A7, XBOOLE_1:1;
then A8: (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) = (f1 + f2) /* (seq ^\ k) by RFUNCT_2:23
.= ((f1 + f2) /* seq) ^\ k by A5, VALUED_0:27 ;
consider r2 being real number such that
A9: for g being set st g in (right_open_halfline r1) /\ (dom f2) holds
r2 <= f2 . g by A3, RFUNCT_1:88;
A10: rng seq c= dom f2 by A5, Lm2;
then A11: rng (seq ^\ k) c= dom f2 by A7, XBOOLE_1:1;

now

let n be Element of NAT ; :: thesis: (- (abs r2)) - 1 < (f2 /* (seq ^\ k)) . n
r1 < seq . (n + k) by A6, NAT_1:12;
then ( (seq ^\ k) . n < +infty & r1 < (seq ^\ k) . n ) by NAT_1:def 3, XXREAL_0:9;
then ( (seq ^\ k) . n in rng (seq ^\ k) & (seq ^\ k) . n in right_open_halfline r1 ) by VALUED_0:28, XXREAL_1:4;
then (seq ^\ k) . n in (right_open_halfline r1) /\ (dom f2) by A11, XBOOLE_0:def 4;
then r2 <= f2 . ((seq ^\ k) . n) by A9;
then A12: r2 <= (f2 /* (seq ^\ k)) . n by A10, A7, FUNCT_2:185, XBOOLE_1:1;
- (abs r2) <= r2 by ABSVALUE:11;
then (- (abs r2)) - 1 < r2 - 0 by XREAL_1:17;
hence (- (abs r2)) - 1 < (f2 /* (seq ^\ k)) . n by A12, XXREAL_0:2; :: thesis: verum

end;

then A13: f2 /* (seq ^\ k) is bounded_below by SEQ_2:def 4;
rng seq c= dom f1 by A5, Lm2;
then A14: rng (seq ^\ k) c= dom f1 by A7, XBOOLE_1:1;
seq ^\ k is divergent_to+infty by A4, Th53;
then f1 /* (seq ^\ k) is divergent_to+infty by A1, A14, Def7;
then (f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) is divergent_to+infty by A13, Th36;
hence (f1 + f2) /* seq is divergent_to+infty by A8, Th34; :: thesis: verum

end;

hence f1 + f2 is divergent_in+infty_to+infty by A2, Def7; :: thesis: verum