begin
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
+ x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
- x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
<> 0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
/ f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
/ ((a : ( ( ) ( V22() V23() ext-real ) Real) - x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
- a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
+ a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
<> 0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
/ f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
/ ((x : ( ( ) ( V22() V23() ext-real ) Real) + a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
- a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
- b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
<> 0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
/ f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
/ ((x : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
sin : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
cos : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
* (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
(id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
- ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
(id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
+ ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
(sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
* (((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) - ((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) holds
(
(sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
* (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) holds
(
(cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
* (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) holds
(
(cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= - ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) holds
(
(sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
* (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) sin : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
- ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) cos : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (- ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
- ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
sin : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
+ (#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
+ ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (x : ( ( ) ( V22() V23() ext-real ) Real) #R (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
g being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
g : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= #Z 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) holds
(
g : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
(#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
- (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
g being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st not
0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
g : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= #Z 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) holds
(
g : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
(#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
+ (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
(id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
+ (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
g being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
g : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= #Z 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
x : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
g : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
(#) ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
+ ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= #Z 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
> 0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
/ (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= ((4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
/ ((a : ( ( ) ( V22() V23() ext-real ) Real) - (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= #Z 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
> 0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
> 0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= ((4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
/ ((a : ( ( ) ( V22() V23() ext-real ) Real) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
x : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real non
negative V49()
V50()
V51()
V52()
V53()
V54()
V60()
V61() )
Element of
NAT : ( ( ) ( non
empty V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
= (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) )
* (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V62() )
set ) ) ) ) ;