begin
theorem
for
f2,
f1 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
for
s being ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence)
for
X being ( ( ) ( )
set ) st
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= (dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ,REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ X : ( ( ) ( )
set ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
(
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= X : ( ( ) ( )
set ) &
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= dom f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ X : ( ( ) ( )
set ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
rng (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) /* s : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() V15() V16() V17() V78() V79() V80() V81() V82() V83() V84() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V11() total quasi_total V33() V34() V35() ) Real_Sequence) ) : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
f2,
f1 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
for
s being ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence)
for
X being ( ( ) ( )
set ) st
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= (dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ,REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
\ X : ( ( ) ( )
set ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
(
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= dom f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
rng s : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Real_Sequence) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
\ X : ( ( ) ( )
set ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
rng (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) /* s : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() V15() V16() V17() V78() V79() V80() V81() V82() V83() V84() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V11() total quasi_total V33() V34() V35() ) Real_Sequence) ) : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V11()
total quasi_total V33()
V34()
V35() )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
c= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to-infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to-infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to-infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to-infty ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to+infty_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to-infty_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to+infty_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to-infty_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to+infty_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to-infty_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to+infty_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to-infty_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to+infty_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to-infty_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to-infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to+infty_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to-infty_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to-infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to+infty_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to-infty_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to-infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to+infty_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to-infty_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to-infty ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to+infty & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to-infty & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to+infty & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to-infty & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to+infty_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to-infty_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to+infty_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
> lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to-infty_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
> lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to+infty_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to-infty_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to+infty_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to-infty_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in+infty_to-infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to+infty_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to+infty ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to-infty_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
divergent_in-infty_to-infty ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to+infty_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to-infty_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to+infty_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to-infty_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in+infty &
lim_in+infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in+infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in+infty_to-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in+infty &
lim_in+infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in-infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in-infty &
lim_in-infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in+infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
divergent_in-infty_to-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in-infty &
lim_in-infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in-infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_left (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in+infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_left (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in-infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_right (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in+infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_right (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in-infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_left (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_left (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_right (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_right (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_left (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_left (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_right (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_right (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in+infty &
lim_in+infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_in+infty f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in+infty &
lim_in+infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_in+infty f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in-infty &
lim_in-infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_in-infty f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in-infty &
lim_in-infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_in-infty f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to+infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in+infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_divergent_to-infty_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_in-infty f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in+infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_in+infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in+infty &
lim_in+infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_in+infty f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) is
convergent_in-infty &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
for
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline r : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. g : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_in-infty f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) is
convergent_in-infty &
lim_in-infty (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_in-infty f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_left (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_left (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
< f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
g : ( ( ) (
V22()
V23()
ext-real )
Real)
< r : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g : ( ( ) (
V22()
V23()
ext-real )
Real) &
g : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim_right (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim_right (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
is_convergent_in lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) & ( for
r1,
r2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) holds
ex
g1,
g2 being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
r1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g1 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
< x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g1 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
< r2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
x0 : ( ( ) (
V22()
V23()
ext-real )
Real)
< g2 : ( ( ) (
V22()
V23()
ext-real )
Real) &
g2 : ( ( ) (
V22()
V23()
ext-real )
Real)
in dom (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
g being ( ( ) (
V22()
V23()
ext-real )
Real) st
(
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V78()
V79()
V80()
V81()
V82()
V83()
V84() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) )
< g : ( ( ) (
V22()
V23()
ext-real )
Real) & ( for
r being ( ( ) (
V22()
V23()
ext-real )
Real) st
r : ( ( ) (
V22()
V23()
ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) )
/\ (].(x0 : ( ( ) ( V22() V23() ext-real ) Real) - g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) \/ ].x0 : ( ( ) ( V22() V23() ext-real ) Real) ,(x0 : ( ( ) ( V22() V23() ext-real ) Real) + g : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) .[ : ( ( ) ( V78() V79() V80() ) Element of K19(REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V78()
V79()
V80() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
. r : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
<> lim (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ) holds
(
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,)
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) )
is_convergent_in x0 : ( ( ) (
V22()
V23()
ext-real )
Real) &
lim (
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) : ( ( ) (
V33()
V34()
V35() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
= lim (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) )
V6()
V33()
V34()
V35() )
PartFunc of ,) ,
(lim (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V78() V79() V80() V84() ) set ) ) V6() V33() V34() V35() ) PartFunc of ,) ,x0 : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V78()
V79()
V80()
V84() )
set ) ) ) ;