let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_left_convergent_in x0 & f2 is_left_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st
( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds
f1 . r < lim_left (f1,x0) ) ) holds
f2 * f1 is_left_divergent_to-infty_in x0
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( f1 is_left_convergent_in x0 & f2 is_left_divergent_to-infty_in lim_left (f1,x0) & ( for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ) & ex g being Real st
( 0 < g & ( for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds
f1 . r < lim_left (f1,x0) ) ) implies f2 * f1 is_left_divergent_to-infty_in x0 )
assume that
A1: f1 is_left_convergent_in x0 and
A2: f2 is_left_divergent_to-infty_in lim_left (f1,x0) and
A3: for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom (f2 * f1) ) ; :: thesis: ( for g being Real holds
( not 0 < g or ex r being Real st
( r in (dom f1) /\ ].(x0 - g),x0.[ & not f1 . r < lim_left (f1,x0) ) ) or f2 * f1 is_left_divergent_to-infty_in x0 )
given g being Real such that A4: 0 < g and
A5: for r being Real st r in (dom f1) /\ ].(x0 - g),x0.[ holds
f1 . r < lim_left (f1,x0) ; :: thesis: f2 * f1 is_left_divergent_to-infty_in x0
now :: thesis: for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) holds
(f2 * f1) /* s is divergent_to-infty
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) implies (f2 * f1) /* s is divergent_to-infty )
assume that
A6: ( s is convergent & lim s = x0 ) and
A7: rng s c= (dom (f2 * f1)) /\ (left_open_halfline x0) ; :: thesis: (f2 * f1) /* s is divergent_to-infty
consider k being Nat such that
A8: for n being Nat st k <= n holds
x0 - g < s . n by A4, A6, Lm1, LIMFUNC2:1;
set q = (f1 /* s) ^\ k;
A9: rng s c= dom (f2 * f1) by A7, Th1;
rng (f1 /* s) c= dom f2 by A7, Th1;
then A10: f2 /* ((f1 /* s) ^\ k) = (f2 /* (f1 /* s)) ^\ k by VALUED_0:27
.= ((f2 * f1) /* s) ^\ k by A9, VALUED_0:31 ;
A11: rng s c= (dom f1) /\ (left_open_halfline x0) by A7, Th1;
then A12: f1 /* s is convergent by A1, A2, A6, LIMFUNC2:def 7;
A13: rng s c= dom f1 by A7, Th1;
A14: rng s c= left_open_halfline x0 by A7, Th1;
now :: thesis: for x being object st x in rng ((f1 /* s) ^\ k) holds
x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0)))
let x be object ; :: thesis: ( x in rng ((f1 /* s) ^\ k) implies x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) )
assume x in rng ((f1 /* s) ^\ k) ; :: thesis: x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0)))
then consider n being Element of NAT such that
A15: ((f1 /* s) ^\ k) . n = x by FUNCT_2:113;
A16: n + k in NAT by ORDINAL1:def 12;
A17: f1 . (s . (n + k)) = (f1 /* s) . (n + k) by A13, FUNCT_2:108, A16
.= x by A15, NAT_1:def 3 ;
A18: x0 - g < s . (n + k) by A8, NAT_1:12;
A19: s . (n + k) in rng s by VALUED_0:28;
then s . (n + k) in left_open_halfline x0 by A14;
then s . (n + k) in { g1 where g1 is Real : g1 < x0 } by XXREAL_1:229;
then ex g1 being Real st
( g1 = s . (n + k) & g1 < x0 ) ;
then s . (n + k) in { g2 where g2 is Real : ( x0 - g < g2 & g2 < x0 ) } by A18;
then s . (n + k) in ].(x0 - g),x0.[ by RCOMP_1:def 2;
then s . (n + k) in (dom f1) /\ ].(x0 - g),x0.[ by A13, A19, XBOOLE_0:def 4;
then f1 . (s . (n + k)) < lim_left (f1,x0) by A5;
then f1 . (s . (n + k)) in { r1 where r1 is Real : r1 < lim_left (f1,x0) } ;
then A20: f1 . (s . (n + k)) in left_open_halfline (lim_left (f1,x0)) by XXREAL_1:229;
f1 . (s . (n + k)) in dom f2 by A9, A19, FUNCT_1:11;
hence x in (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) by A20, A17, XBOOLE_0:def 4; :: thesis: verum
end;
then A21: rng ((f1 /* s) ^\ k) c= (dom f2) /\ (left_open_halfline (lim_left (f1,x0))) ;
lim (f1 /* s) = lim_left (f1,x0) by A1, A6, A11, LIMFUNC2:def 7;
then lim ((f1 /* s) ^\ k) = lim_left (f1,x0) by A12, SEQ_4:20;
then f2 /* ((f1 /* s) ^\ k) is divergent_to-infty by A2, A12, A21, LIMFUNC2:def 3;
hence (f2 * f1) /* s is divergent_to-infty by A10, LIMFUNC1:7; :: thesis: verum
end;
hence f2 * f1 is_left_divergent_to-infty_in x0 by A3, LIMFUNC2:def 3; :: thesis: verum