:: LIMFUNC4 semantic presentation

begin

theorem :: LIMFUNC4:1
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 :: LIMFUNC4:2
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 :: LIMFUNC4:3
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 :: LIMFUNC4:4
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 :: LIMFUNC4:5
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 :: LIMFUNC4:6
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 :: LIMFUNC4:7
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 :: LIMFUNC4:8
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 :: LIMFUNC4:9
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 :: LIMFUNC4:10
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 :: LIMFUNC4:11
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 :: LIMFUNC4:12
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 :: LIMFUNC4:13
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 :: LIMFUNC4:14
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 :: LIMFUNC4:15
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 :: LIMFUNC4:16
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 :: LIMFUNC4:17
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 :: LIMFUNC4:18
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 :: LIMFUNC4:19
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 :: LIMFUNC4:20
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 :: LIMFUNC4:21
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 :: LIMFUNC4:22
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 :: LIMFUNC4:23
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 :: LIMFUNC4:24
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 :: LIMFUNC4:25
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 :: LIMFUNC4:26
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 :: LIMFUNC4:27
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 :: LIMFUNC4:28
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 :: LIMFUNC4:29
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 :: LIMFUNC4:30
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 :: LIMFUNC4:31
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 :: LIMFUNC4:32
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 :: LIMFUNC4:33
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 :: LIMFUNC4:34
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 :: LIMFUNC4:35
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 :: LIMFUNC4:36
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 :: LIMFUNC4:37
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 :: LIMFUNC4:38
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 :: LIMFUNC4:39
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 :: LIMFUNC4:40
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 :: LIMFUNC4:41
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 :: LIMFUNC4:42
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 :: LIMFUNC4:43
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 :: LIMFUNC4:44
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 :: LIMFUNC4:45
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 :: LIMFUNC4:46
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 :: LIMFUNC4:47
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 :: LIMFUNC4:48
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 :: LIMFUNC4:49
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 :: LIMFUNC4:50
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 :: LIMFUNC4:51
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 :: LIMFUNC4:52
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 :: LIMFUNC4:53
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 :: LIMFUNC4:54
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 :: LIMFUNC4:55
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 :: LIMFUNC4:56
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 :: LIMFUNC4:57
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 :: LIMFUNC4:58
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 :: LIMFUNC4:59
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 :: LIMFUNC4:60
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 :: LIMFUNC4:61
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 :: LIMFUNC4:62
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 :: LIMFUNC4:63
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 :: LIMFUNC4:64
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 :: LIMFUNC4:65
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 :: LIMFUNC4:66
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 :: LIMFUNC4:67
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 :: LIMFUNC4:68
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 :: LIMFUNC4:69
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 :: LIMFUNC4:70
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 :: LIMFUNC4:71
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 :: LIMFUNC4:72
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 :: LIMFUNC4:73
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 :: LIMFUNC4:74
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 :: LIMFUNC4:75
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 :: LIMFUNC4:76
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 :: LIMFUNC4:77
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 ) ) ) ;