begin
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 (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 ) & ( 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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) 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 ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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) + 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 (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 ) & ( 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)
- a : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) 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 ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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) - 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
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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) 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 ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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) - 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,
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 ((id 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 ) ) - (a : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ,)
= 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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
+ x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(id 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 ) )
- (a : ( ( ) ( V22() V23() ext-real ) Real) (#) 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
(((id 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 ) ) - (a : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
/ (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 ) ) ) ) ;
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) ) - (id 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 ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
+ x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) holds
(
((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) )
- (id 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 ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) ) - (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - 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) + 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,
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 ((id 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 ) ) - ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ,)
= 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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
+ a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(id 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 ) )
- ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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
(((id 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 ) ) - ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) )
= (x : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) + a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
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 ((id 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 ) ) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ,)
= 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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
- a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(id 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 ) )
+ ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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
(((id 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 ) ) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) )
= (x : ( ( ) ( V22() V23() ext-real ) Real) + a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a,
b 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 ((id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
+ b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(id 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 ) )
+ ((a : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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
(((id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a,
b 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 ((id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
- b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(id 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 ) )
+ ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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
(((id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a,
b 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 ((id 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 ) ) - ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
+ b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(id 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 ) )
- ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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
(((id 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 ) ) - ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
b,
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 ((id 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 ) ) + ((b : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,)
= 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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
- b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(id 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 ) )
+ ((b : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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
(((id 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 ) ) + ((b : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
c,
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
+ (b : ( ( ) ( 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 ) ) ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ,)
+ (c : ( ( ) ( 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= b : ( ( ) (
V22()
V23()
ext-real )
Real)
+ ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( 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 ) ) ) ) ;
theorem
for
c,
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
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 ) ) * (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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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)
+ (b : ( ( ) ( 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 ) ) &
(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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 : ( ( ) (
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 ) ) ) ) ) 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 ) )
* (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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) * (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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= (b : ( ( ) ( V22() V23() ext-real ) Real) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( 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 ) )
/ ((a : ( ( ) ( V22() V23() ext-real ) Real) + (b : ( ( ) ( 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 ) ) + (c : ( ( ) ( V22() V23() ext-real ) Real) * (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) (
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) (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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(- 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
(#) (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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) (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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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) + x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( 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
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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
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 (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 ) ) : ( ( ) ( )
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 : ( ( ) (
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 ) ) ) ) ) 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 ) )
* (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
((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 ) ) `| 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ ((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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
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 (- (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 ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 : ( ( ) (
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 ) ) ) ) ) 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 ) ) * (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
((- (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 ) )
= (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ ((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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
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) ) &
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 3 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) )
= 3 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
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 (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 ) ) : ( ( ) ( )
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 3 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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) &
(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 : ( ( ) (
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 ) ) ) ) ) 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 ) )
* (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
((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 ) ) `| 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 ) )
= (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ (a : ( ( ) ( V22() V23() ext-real ) Real) + (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
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 (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 ) ) : ( ( ) ( )
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)
+ x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) &
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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) 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 ) )
* (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
((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 ) ) `| 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ ((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) : ( ( ) ( 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,
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 (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 ) ) : ( ( ) ( )
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)
- a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) &
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)
+ a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 ) )
> 0 : ( ( ) (
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 ) ) ) ) ) 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 ) )
* (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
((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 ) ) `| 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) ^2) : ( ( ) ( 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 ) ) ) : ( ( ) (
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,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
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 ) ) * (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 ) & ( 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)
- a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) &
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)
- b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 ) )
> 0 : ( ( ) (
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 ) ) ) ) ) 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 ) )
* (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
((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 ) ) `| 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) - b : ( ( ) ( 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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (x : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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,
b 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - b : ( ( ) ( 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 ) ) (#) 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 ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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)
- a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 : ( ( ) (
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 ) ) ) &
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)
- b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 ) )
> 0 : ( ( ) (
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 ) ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
- b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
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 ) ) ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - b : ( ( ) ( 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 ) )
(#) 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - b : ( ( ) ( 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 ) ) (#) 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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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) - 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) - b : ( ( ) ( 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,
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 (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 ) ) : ( ( ) ( )
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
- a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
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 : ( ( ) (
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 ) ) ) &
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 : ( ( ) (
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)
<> 0 : ( ( ) (
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 ) ) ) ) ) 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 ) )
* (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
((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 ) ) `| 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) * (x : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() 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 ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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
(((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
* ((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 ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
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 ((- (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(- (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
(#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
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 (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
( 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ((- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= 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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
b,
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 ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
+ (b : ( ( ) ( 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 ) ) &
b : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
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 ) ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + (b : ( ( ) ( 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 ) )
#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
b,
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 ((- (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
- (b : ( ( ) ( 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 ) ) &
b : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
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 ) ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(- (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - (b : ( ( ) ( 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 ) )
#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
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 ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ,)
+ 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 ) ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
(#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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
(((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ^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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) #R (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) (
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 (- ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,)
- 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 ) ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) ) &
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 : ( ( ) (
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 ) ) ) ) ) holds
(
- ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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
((- ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ^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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) #R (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) (
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 (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,)
+ 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 ) ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real) &
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 : ( ( ) (
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 ) ) ) ) ) holds
( 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) ) : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a,
b 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 (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 ) ) : ( ( ) ( )
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 ) )
= (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 ) )
+ b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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 ) )
* 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
((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 ) ) `| 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)
* (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 ) ) . ((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 ) ) + b : ( ( ) ( 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,
b 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 (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 ) ) : ( ( ) ( )
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 ) )
= (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 ) )
+ b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
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 ) )
* 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
((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 ) ) `| 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) * (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 ) ) . ((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 ) ) + b : ( ( ) ( 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 ) ) st ( 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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
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 ) ) ) ) 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 ) )
^ : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
/ ((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 ) ) ^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 ) ) st ( 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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
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 ) ) ) ) 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 ) )
^ : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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 ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) / ((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 ) ) ^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 ) ) ) ) ;
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 ) ) (#) 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
(
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 ) )
(#) 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
((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 ) ) (#) 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 ) )
= cos (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) 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 ) ) * 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 ) & ( 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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
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 ) ) ) ) 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 ) )
* 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
((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 ) ) * 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 ) )
= - (tan 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 ) ) 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 ) ) * 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 ) ) : ( ( ) ( )
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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
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 ) ) ) ) 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 ) )
* 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 ) )
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 ) ) * 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 ) ) `| 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 ) )
= cot 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 ((- (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
(
(- (id 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 ) ) ) : ( (
V6() ) (
Relation-like REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-defined REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set )
-valued V6()
V39()
V40()
V41() )
Element of
K19(
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
(((- (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
= (- (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 ) )
+ (x : ( ( ) ( V22() V23() ext-real ) Real) * (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 ((id 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 ) ) (#) 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 ) ) : ( ( ) ( )
set ) holds
(
(id 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 ) )
(#) 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 ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) `| 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 ) )
+ (x : ( ( ) ( V22() V23() ext-real ) Real) * (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 ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) + 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 ) ) : ( ( ) ( )
set ) holds
(
((- (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) )
+ 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 ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((((- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) + 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 ) ) `| 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)
* (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 ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) + 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
(
((id 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 ) ) (#) 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 ) )
+ 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
((((id 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 ) ) (#) 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 ) ) + 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 ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (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 ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) : ( ( ) ( )
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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
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 ) ) ) ) holds
( 2 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) )
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) `| 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 ) )
* ((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 ) ) #R (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) (
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) ( )
set ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) `| 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 ) )
* (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 ) ) ) ) ;
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) & ( 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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
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 ) ) ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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
(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) - 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 ) & ( 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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
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 ) ) ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
- 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) - 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) & ( 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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
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 ) ) ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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
(
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;
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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) & ( 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 ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) ( non
empty V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
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 ) ) ) &
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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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
(
(- 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 ) )
- ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
(((- 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 ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
= ((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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
/ (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) : ( ( ) ( )
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 : ( ( ) (
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 ) ) ) holds
(
(1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) )
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) `| 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 ) ) #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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) )
* (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 ) ) ) ) ;
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)
- 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) ) ) 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 ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
* (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 (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 ) ) / (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 ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) ) / (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
((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 ) ) / (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 ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / 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 V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 : ( ( ) (
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 ) ) ) ) ) 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 ) ) / 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
((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 ) ) / 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 ) )
= 1 : ( ( ) ( non
empty V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) ( non
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 ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;