:: INTEGR13 semantic presentation

begin

theorem :: INTEGR13:1
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 :: INTEGR13:2
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 :: INTEGR13:3
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 :: INTEGR13:4
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 :: INTEGR13:5
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 :: INTEGR13:6
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 :: INTEGR13:7
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 :: INTEGR13:8
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 :: INTEGR13:9
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 :: INTEGR13:10
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 :: INTEGR13:11
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 :: INTEGR13:12
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 :: INTEGR13:13
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 :: INTEGR13:14
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 :: INTEGR13:15
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 :: INTEGR13:16
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 :: INTEGR13:17
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 :: INTEGR13:18
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 :: INTEGR13:19
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 :: INTEGR13:20
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 :: INTEGR13:21
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 :: INTEGR13:22
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 :: INTEGR13:23
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 :: INTEGR13:24
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 :: INTEGR13:25
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 :: INTEGR13:26
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 :: INTEGR13:27
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 :: INTEGR13:28
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 :: INTEGR13:29
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 :: INTEGR13:30
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 :: INTEGR13:31
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 :: INTEGR13:32
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 :: INTEGR13:33
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 :: INTEGR13:34
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 :: INTEGR13:35
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 :: INTEGR13:36
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 :: INTEGR13:37
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 :: INTEGR13:38
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 :: INTEGR13:39
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 :: INTEGR13:40
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 :: INTEGR13:41
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 :: INTEGR13:42
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 :: INTEGR13:43
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 :: INTEGR13:44
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 :: INTEGR13:45
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 :: INTEGR13:46
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 :: INTEGR13:47
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 :: INTEGR13:48
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 :: INTEGR13:49
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 :: INTEGR13:50
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 :: INTEGR13:51
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 :: INTEGR13:52
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 :: INTEGR13:53
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 :: INTEGR13:54
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 :: INTEGR13:55
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 :: INTEGR13:56
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 :: INTEGR13:57
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 ) ) ;