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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
- f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
<> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
/ (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= ((4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (((a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
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
(f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= ((4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((a : ( ( ) ( 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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
/ ((a : ( ( ) ( 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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
x : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
/ (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) &
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
x : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) * (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f2,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
- 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ^) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
+ (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (x : ( ( ) ( V22() V23() ext-real ) Real) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
exp_R : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ number_e : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
exp_R : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
- 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
(#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
/ (exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - (x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
((- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) #R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - (x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
f2 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
+ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) ((- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
/ (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> number_e : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> number_e : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
/ (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
/ ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
(exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
/ ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
(f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
- (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
+ f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
((((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
* (exp_R (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
+ f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
((((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
* (exp_R (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
(exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
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
(f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
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
(f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
(f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
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
(f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
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
(f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= exp_R : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
+ (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= exp_R : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
* ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
(- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= sin (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
sin : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
cos : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
< 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
cos : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> - 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real V68() )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) holds
( 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
(#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
sin : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
cos : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
< 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
cos : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> - 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real V68() )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) holds
(
(- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real V68() )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
+ (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
= f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
+ (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* (cos ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
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()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
- ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (sin (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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
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()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) &
f : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
f1 : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
PartFunc of ,)
+ ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (cos (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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) &
n : ( ( ) (
V15()
V16()
V17()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
> 0 : ( ( ) (
empty V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
(#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
(((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= - (((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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 (((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- cos : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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
((((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
|^ 3 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
sin : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) 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() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set ) )
|^ 3 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
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()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
sin : ( (
V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V30(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ln : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,x : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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 (- (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
(
- (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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()
V70() )
set ) )
= (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,x : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;