begin
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st ( for
h being ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) st
rng c : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
= {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) (
V51()
V71()
V72()
V73()
V99()
V100()
V101() )
set ) &
rng (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
h : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
. n : ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
< 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) &
{x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) (
V51()
V71()
V72()
V73()
V99()
V100()
V101() )
set )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) holds
for
h1,
h2 being ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) st
rng c : ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
= {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) (
V51()
V71()
V72()
V73()
V99()
V100()
V101() )
set ) &
rng (h1 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
h1 : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
< 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
rng (h2 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
h2 : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
< 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
lim ((h1 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h1 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
= lim ((h2 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h2 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st ( for
h being ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) st
rng c : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
= {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) (
V51()
V71()
V72()
V73()
V99()
V100()
V101() )
set ) &
rng (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
h : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
. n : ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
convergent ) &
{x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) (
V51()
V71()
V72()
V73()
V99()
V100()
V101() )
set )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) holds
for
h1,
h2 being ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) st
rng c : ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
= {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) (
V51()
V71()
V72()
V73()
V99()
V100()
V101() )
set ) &
rng (h1 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) &
rng (h2 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
h1 : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
h2 : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
lim ((h1 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h1 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
= lim ((h2 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h2 : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0,
g2 being ( ( ) (
V30()
real ext-real )
Real) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_Lcontinuous_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
<> g2 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) &
[.(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .] : ( ( ) (
V71()
V72()
V73()
V102() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
ex
r1 being ( ( ) (
V30()
real ext-real )
Real) st
(
r1 : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) &
[.(x0 : ( ( ) ( V30() real ext-real ) Real) - r1 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .] : ( ( ) (
V71()
V72()
V73()
V102() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in [.(x0 : ( ( ) ( V30() real ext-real ) Real) - r1 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .] : ( ( ) (
V71()
V72()
V73()
V102() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
<> g2 : ( ( ) (
V30()
real ext-real )
Real) ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0,
g2 being ( ( ) (
V30()
real ext-real )
Real) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_Rcontinuous_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
<> g2 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) &
[.x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) .] : ( ( ) (
V71()
V72()
V73()
V102() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
ex
r1 being ( ( ) (
V30()
real ext-real )
Real) st
(
r1 : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) &
[.x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r1 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) .] : ( ( ) (
V71()
V72()
V73()
V102() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in [.x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r1 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) .] : ( ( ) (
V71()
V72()
V73()
V102() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
<> g2 : ( ( ) (
V30()
real ext-real )
Real) ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0,
g being ( ( ) (
V30()
real ext-real )
Real) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Ldiff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= g : ( ( ) (
V30()
real ext-real )
Real) iff ( ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence) &
[.(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .] : ( ( ) (
V71()
V72()
V73()
V102() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
h being ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) st
rng c : ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
= {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) (
V51()
V71()
V72()
V73()
V99()
V100()
V101() )
set ) &
rng (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
h : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
< 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
convergent &
lim ((h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
= g : ( ( ) (
V30()
real ext-real )
Real) ) ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Ldiff (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (Ldiff (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
+ (Ldiff (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
- f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Ldiff (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (Ldiff (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
- (Ldiff (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Ldiff (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= ((Ldiff (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) * (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
+ ((Ldiff (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
<> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
/ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Ldiff (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (((Ldiff (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) * (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) - ((Ldiff (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
/ ((f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ^2) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
<> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Ldiff (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= - ((Ldiff (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) / ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ^2) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0,
g1 being ( ( ) (
V30()
real ext-real )
Real) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Rdiff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= g1 : ( ( ) (
V30()
real ext-real )
Real) iff ( ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) &
[.x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) .] : ( ( ) (
V71()
V72()
V73()
V102() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
h being ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) st
rng c : ( (
V6()
constant quasi_total ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
constant V11()
total quasi_total complex-valued ext-real-valued real-valued convergent )
Real_Sequence) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
= {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) (
V51()
V71()
V72()
V73()
V99()
V100()
V101() )
set ) &
rng (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) & ( for
n being ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
h : ( (
non-zero V6()
quasi_total 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V1()
non-zero V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
. n : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
convergent &
lim ((h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* (h : ( ( non-zero V6() quasi_total 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V1() non-zero V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V30() real ext-real V71() V72() V73() V74() V75() V76() V91() V92() V99() ) Element of NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) /* c : ( ( V6() constant quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() constant V11() total quasi_total complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ) : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() V11() total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( V11() epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V97() V99() ) Element of K19(REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
V11()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
= g1 : ( ( ) (
V30()
real ext-real )
Real) ) ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Rdiff (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (Rdiff (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
+ (Rdiff (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
- f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Rdiff (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (Rdiff (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
- (Rdiff (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Rdiff (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= ((Rdiff (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) * (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
+ ((Rdiff (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
<> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
/ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Rdiff (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (((Rdiff (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) * (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) - ((Rdiff (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
/ ((f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ^2) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
<> 0 : ( ( ) (
epsilon-transitive epsilon-connected ordinal natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V91()
V92()
V99() )
Element of
NAT : ( ( ) (
V11()
epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V97()
V99() )
Element of
K19(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Rdiff (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ,
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= - ((Rdiff (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) ( V30() real ext-real ) Real) / ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V5( REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ^2) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( V11() V12() V49() V51() V71() V72() V73() V77() V99() V100() V102() ) set ) ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
Rdiff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= Ldiff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
diff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
= Rdiff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) &
diff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
= Ldiff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_differentiable_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
diff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
= Rdiff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) &
diff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
= Ldiff (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V5(
REAL : ( ( ) (
V11()
V12()
V49()
V51()
V71()
V72()
V73()
V77()
V99()
V100()
V102() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) ) ;