begin
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= - ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= 2 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
(#) (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= 2 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
(#) (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
(cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
(cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
(cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
(cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
(#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(((- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
/ (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= sin : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ exp_R : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
(#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
/ (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= cos : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ exp_R : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= exp_R : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
(#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= exp_R : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
(#) (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (- ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
+ (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (- ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
+ (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
+ (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
+ (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= - ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
/ ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= - (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ (x : ( ( ) ( V28() V29() ext-real ) Real) * ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= exp_R : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= - (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
/ ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= exp_R : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= - (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) &
arctan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
> 0 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
n being ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= n : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
(#) (((#Z (n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
n being ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= - (n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (((#Z (n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
n being ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- ((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- ((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) #Z (n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
/ (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= n : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
(#) (((#Z (n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- ((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- ((#Z n : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= arctan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= - (arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
/ (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= arccot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) ((#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= arctan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
+ ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= arccot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
- ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= - ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
/ (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f,
f1 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b5 : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
b5 : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2,
f,
h being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b7 : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
b7 : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
/ (a : ( ( ) ( V28() V29() ext-real ) Real) (#) (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
h : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
/ a : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
a : ( ( ) (
V28()
V29()
ext-real )
Real)
<> 0 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* h : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= (((a : ( ( ) ( V28() V29() ext-real ) Real) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (((a : ( ( ) ( V28() V29() ext-real ) Real) / 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) * (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
+ (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) * (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= 1 : ( ( ) (
V27()
V28()
V29()
V30()
ext-real V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) (
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= (arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
+ (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + (#Z 2 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ,1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
V51()
V52()
V53()
open )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b1 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) )
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V1()
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V76() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
= ((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) )
- ((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b4 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V62() )
set ) ) ;