:: INTEGR15 semantic presentation

begin

definition
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;
let D be ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ;
mode middle_volume of f,D -> ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) means :: INTEGR15:def 1
( len it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) = len D : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) & ( for i being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) st i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) in dom D : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
ex r being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) st
( r : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) in rng (f : ( ( ) ( ) Element of K6(K6(A : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | (divset (D : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) ,i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) )) : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) TopStruct ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(A : ( ( ) ( ) TopStruct ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V129() V130() V131() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) & it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) . i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * (vol (divset (D : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) ,i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) )) : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) );
end;

definition
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;
let D be ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ;
let F be ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of f : ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) ;
func middle_sum (f,F) -> ( ( ) ( V11() real ext-real ) Real) equals :: INTEGR15:def 2
 Sum F : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;
end;

theorem :: INTEGR15:1
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for D being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )
for F being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) st f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded_below holds
lower_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) <= middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,F : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) ) : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: INTEGR15:2
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for D being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )
for F being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) st f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded_above holds
middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,F : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) ) : ( ( ) ( V11() real ext-real ) Real) <= upper_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;

theorem :: INTEGR15:3
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for D being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )
for e being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded_below & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) < e : ( ( ) ( V11() real ext-real ) Real) holds
ex F being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) st middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,F : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) ) : ( ( ) ( V11() real ext-real ) Real) <= (lower_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) + e : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;

theorem :: INTEGR15:4
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for D being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )
for e being ( ( ) ( V11() real ext-real ) Real) st f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded_above & 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) < e : ( ( ) ( V11() real ext-real ) Real) holds
ex F being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) st (upper_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) - e : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) <= middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,F : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) ) : ( ( ) ( V11() real ext-real ) Real) ;

definition
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;
let T be ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ;
mode middle_volume_Sequence of f,T -> ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) means :: INTEGR15:def 3
 for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( FinSequence-like ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) is ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of f : ( ( ) ( ) Element of K6(K6(A : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,T : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( ) ( ) TopStruct ) ) ) ;
end;

definition
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;
let T be ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ;
let S be ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) ;
let k be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
:: original: .
redefine func S . k -> ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of f : ( ( ) ( ) Element of K6(K6(A : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,T : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) . k : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) TopStruct ) ) V17(T : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(A : ( ( ) ( ) TopStruct ) ,T : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( ) ( ) TopStruct ) ) ) ;
end;

definition
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;
let T be ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ;
let S be ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) ;
func middle_sum (f,S) -> ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) means :: INTEGR15:def 4
 for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds it : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) TopStruct ) ) V17(T : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(A : ( ( ) ( ) TopStruct ) ,T : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real V13() V17( RAT : ( ( ) ( non empty V32() V129() V130() V131() V132() V135() ) set ) ) Function-like ext-real V119() V120() V121() ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = middle_sum (f : ( ( ) ( ) Element of K6(K6(A : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(S : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) middle_volume of f : ( ( ) ( ) Element of K6(K6(A : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,T : ( ( ) ( ) Element of A : ( ( ) ( ) TopStruct ) ) . b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( ) ( ) TopStruct ) ) ) ) : ( ( ) ( V11() real ext-real ) Real) ;
end;

theorem :: INTEGR15:5
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for T being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) )
for S being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded_below holds
(lower_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) <= (middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;

theorem :: INTEGR15:6
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for T being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) )
for S being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded_above holds
(middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) <= (upper_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;

theorem :: INTEGR15:7
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for T being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) < e : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) & f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded_below holds
ex S being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) st
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds (middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) <= ((lower_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) + e : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;

theorem :: INTEGR15:8
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for T being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) )
for e being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) st 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) < e : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) & f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded_above holds
ex S being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) st
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds ((upper_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) - e : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) <= (middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ;

theorem :: INTEGR15:9
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) )
for T being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) )
for S being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) st f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) is bounded & f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) is integrable & delta T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is convergent & lim (delta T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds
( middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) is convergent & lim (middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b3 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = integral f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;

theorem :: INTEGR15:10
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) st f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) is bounded holds
( f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) is integrable iff ex I being ( ( ) ( V11() real ext-real ) Real) st
for T being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) )
for S being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) st delta T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is convergent & lim (delta T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds
( middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b4 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) is convergent & lim (middle_sum (f : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) middle_volume_Sequence of b2 : ( ( Function-like V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27(b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Function of b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,b4 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b1 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = I : ( ( ) ( V11() real ext-real ) Real) ) ) ;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
let D be ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ;
mode middle_volume of f,D -> ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) means :: INTEGR15:def 5
( len it : ( ( Function-like ) ( V13() V16(n : ( ( ) ( ) TopStruct ) ) V17(f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(n : ( ( ) ( ) TopStruct ) ,f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) = len D : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) & ( for i being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) st i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) in dom D : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
ex r being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( ) TopStruct ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) st
( r : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) in rng (f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) | (divset (D : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ,i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) )) : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like ) Element of K6(K7(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) ) & it : ( ( Function-like ) ( V13() V16(n : ( ( ) ( ) TopStruct ) ) V17(f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(n : ( ( ) ( ) TopStruct ) ,f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( V13() V17( RAT : ( ( ) ( non empty V32() V129() V130() V131() V132() V135() ) set ) ) Function-like V119() V120() V121() ) set ) = (vol (divset (D : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ,i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) )) : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * r : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( ) TopStruct ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) );
end;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
let D be ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ;
let F be ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) middle_volume of f : ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,D : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) ) ;
func middle_sum (f,F) -> ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( ) TopStruct ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) means :: INTEGR15:def 6
 for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) in Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
ex Fi being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) st
( Fi : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = (proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( ) TopStruct ) )) : ( ( Function-like V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) * F : ( ( Function-like ) ( V13() V16(n : ( ( ) ( ) TopStruct ) ) V17(f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(n : ( ( ) ( ) TopStruct ) ,f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) & it : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17(D : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,D : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real V13() V17( RAT : ( ( ) ( non empty V32() V129() V130() V131() V132() V135() ) set ) ) Function-like ext-real V119() V120() V121() ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = Sum Fi : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) FinSequence of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) );
end;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
let T be ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ;
mode middle_volume_Sequence of f,T -> ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) means :: INTEGR15:def 7
 for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds it : ( ( Function-like ) ( V13() V16(n : ( ( ) ( ) TopStruct ) ) V17(f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(n : ( ( ) ( ) TopStruct ) ,f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( RAT : ( ( ) ( non empty V32() V129() V130() V131() V132() V135() ) set ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) is ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) middle_volume of f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ,T : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ;
end;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
let T be ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ;
let S be ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) ;
let k be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
:: original: .
redefine func S . k -> ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) middle_volume of f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ,T : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) . k : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17(T : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,T : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ;
end;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
let T be ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ;
let S be ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) ;
func middle_sum (f,S) -> ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS n : ( ( ) ( ) TopStruct ) ) : ( ( non empty strict ) ( non empty strict ) L19()) : ( ( ) ( ) set ) ) ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( the carrier of (REAL-NS n : ( ( ) ( ) TopStruct ) ) : ( ( non empty strict ) ( non empty strict ) L19()) : ( ( ) ( ) set ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS n : ( ( ) ( ) TopStruct ) ) : ( ( non empty strict ) ( non empty strict ) L19()) : ( ( ) ( ) set ) ) ) sequence of ( ( ) ( ) set ) ) means :: INTEGR15:def 8
 for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds it : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17(T : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,T : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V13() V17( RAT : ( ( ) ( non empty V32() V129() V130() V131() V132() V135() ) set ) ) Function-like V119() V120() V121() ) Element of the carrier of (REAL-NS n : ( ( ) ( ) TopStruct ) ) : ( ( non empty strict ) ( non empty strict ) L19()) : ( ( ) ( ) set ) ) = middle_sum (f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ,(S : ( ( Function-like ) ( V13() V16(n : ( ( ) ( ) TopStruct ) ) V17(f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(n : ( ( ) ( ) TopStruct ) ,f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( RAT : ( ( ) ( non empty V32() V129() V130() V131() V132() V135() ) set ) ) V17( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like V119() V120() V121() complex-functions-valued ext-real-functions-valued real-functions-valued ) middle_volume of f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ,T : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) . b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like one-to-one V32() FinSequence-like FinSubsequence-like V119() V120() V121() V123() V125() ) Division of A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( ) TopStruct ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
end;

definition
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ;
let Z be ( ( ) ( ) set ) ;
let f, g be ( ( Function-like ) ( V13() V16(Z : ( ( ) ( ) set ) ) V17( REAL n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
func f + g -> ( ( Function-like ) ( V13() V16(Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) equals :: INTEGR15:def 9
f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) <++> g : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) : ( ( Function-like ) ( V13() V16(Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /\ Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) V17( R_PFuncs ((DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) /\ (DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7((Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /\ Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ,(R_PFuncs ((DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) /\ (DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
func f - g -> ( ( Function-like ) ( V13() V16(Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) equals :: INTEGR15:def 10
f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) <--> g : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) : ( ( Function-like ) ( V13() V16(Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /\ Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) V17( R_PFuncs ((DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) /\ (DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7((Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /\ Z : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ,(R_PFuncs ((DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) /\ (DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ;
let r be ( ( real ) ( V11() real ext-real ) number ) ;
let Z be ( ( ) ( ) set ) ;
let f be ( ( Function-like ) ( V13() V16(Z : ( ( ) ( ) set ) ) V17( REAL n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
func r (#) f -> ( ( Function-like ) ( V13() V16(Z : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) V17( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) equals :: INTEGR15:def 11
f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) [#] r : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V13() V16(Z : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) V17( R_PFuncs (DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(Z : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ,(R_PFuncs (DOMS (REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional complex-functions-membered ext-real-functions-membered real-functions-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

begin

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
attr f is bounded means :: INTEGR15:def 12
 for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) in Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
(proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( ) TopStruct ) )) : ( ( Function-like V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) is bounded ;
end;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
attr f is integrable means :: INTEGR15:def 13
 for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) in Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
(proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( ) TopStruct ) )) : ( ( Function-like V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) is integrable ;
end;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;
func integral f -> ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( ) TopStruct ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) means :: INTEGR15:def 14
( dom it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) = Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) & ( for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) in Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = integral ((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( ) TopStruct ) )) : ( ( Function-like V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) : ( ( Function-like ) ( V13() V16(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) );
end;

theorem :: INTEGR15:11
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) )
for T being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) )
for S being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) st f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) is bounded & f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) is integrable & delta T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is convergent & lim (delta T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds
( middle_sum (f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) middle_volume_Sequence of b3 : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,b4 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) ) sequence of ( ( ) ( ) set ) ) is convergent & lim (middle_sum (f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) middle_volume_Sequence of b3 : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,b4 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) ) sequence of ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) = integral f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ;

theorem :: INTEGR15:12
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) st f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) is bounded holds
( f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) is integrable iff ex I being ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) st
for T being ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) )
for S being ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) middle_volume_Sequence of f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) st delta T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is convergent & lim (delta T : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) holds
( middle_sum (f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) middle_volume_Sequence of b3 : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,b5 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) ) sequence of ( ( ) ( ) set ) ) is convergent & lim (middle_sum (f : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,S : ( ( ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17((REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) FinSequenceSet of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) middle_volume_Sequence of b3 : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ,b5 : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ( non empty V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) Function-like total V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , divs b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( ) ( non empty ) set ) ) ) DivSequence of ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) Function-like V27( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) , the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) ) sequence of ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( non empty strict ) ( non empty V73() V194() V195() V196() V197() V198() V199() V200() V216() V217() strict V219() V225() ) L19()) : ( ( ) ( ) set ) ) = I : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ) ;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
attr f is bounded means :: INTEGR15:def 15
 for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) in Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
(proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( ) TopStruct ) )) : ( ( Function-like V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded ;
end;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
pred f is_integrable_on A means :: INTEGR15:def 16
 for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) in Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
(proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( ) TopStruct ) )) : ( ( Function-like V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) is_integrable_on A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let A be ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
func integral (f,A) -> ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( ) TopStruct ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) means :: INTEGR15:def 17
( dom it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) = Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) & ( for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) in Seg n : ( ( ) ( ) TopStruct ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = integral (((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( ) TopStruct ) )) : ( ( Function-like V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL n : ( ( ) ( ) TopStruct ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( ) ( ) Element of n : ( ( ) ( ) TopStruct ) ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) ,A : ( ( ) ( ) Element of K6(K6(n : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) );
end;

theorem :: INTEGR15:13
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for g being ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V32() ) set ) ) = g : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) holds
( f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) iff g : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) is integrable ) ;

theorem :: INTEGR15:14
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for g being ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) st f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V32() ) set ) ) = g : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) holds
integral (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = integral g : ( ( Function-like V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ( non empty V13() V16(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like total V27(b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) complex-functions-valued ext-real-functions-valued real-functions-valued ) Function of b2 : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) , REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;

definition
let a, b be ( ( real ) ( V11() real ext-real ) number ) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;
let f be ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
func integral (f,a,b) -> ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) means :: INTEGR15:def 18
( dom it : ( ( Function-like ) ( V13() V16(a : ( ( ) ( ) TopStruct ) ) V17(n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(a : ( ( ) ( ) TopStruct ) ,n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) = Seg n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) & ( for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) in Seg n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V129() V130() V131() V132() V133() V134() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V32() ) set ) ) holds
it : ( ( Function-like ) ( V13() V16(a : ( ( ) ( ) TopStruct ) ) V17(n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued rational-functions-valued ) Element of K6(K7(a : ( ( ) ( ) TopStruct ) ,n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V11() real V13() V17( RAT : ( ( ) ( non empty V32() V129() V130() V131() V132() V135() ) set ) ) Function-like ext-real V119() V120() V121() ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) = integral (((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) )) : ( ( Function-like V27( REAL n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL n : ( ( ) ( ) Element of a : ( ( ) ( ) TopStruct ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) ,a : ( ( ) ( ) TopStruct ) ,b : ( ( ) ( ) Element of K6(K6(a : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) );
end;

begin

theorem :: INTEGR15:15
for Z being ( ( ) ( ) set )
for n, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for f1, f2 being ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) holds
( (proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) )) : ( ( Function-like V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) * (f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) + f2 : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) = ((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) )) : ( ( Function-like V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) * f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) + ((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) )) : ( ( Function-like V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) * f2 : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) & (proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) )) : ( ( Function-like V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) * (f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) - f2 : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) = ((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) )) : ( ( Function-like V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) * f1 : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) - ((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) )) : ( ( Function-like V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) * f2 : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ;

theorem :: INTEGR15:16
for Z being ( ( ) ( ) set )
for n, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) holds (proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) )) : ( ( Function-like V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) * (r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) (#) ((proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) )) : ( ( Function-like V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ( non empty V13() V16( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like total V27( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) , REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V119() V120() V121() ) Element of K6(K7((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) * f : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16(b1 : ( ( ) ( ) set ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V119() V120() V121() ) Element of K6(K7(b1 : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V119() V120() V121() ) set ) ) : ( ( ) ( V32() ) set ) ) ;

theorem :: INTEGR15:17
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f1, f2 being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) st f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) & f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) & A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) c= dom f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( V129() V130() V131() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) & A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) c= dom f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( V129() V130() V131() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) & f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded & f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded holds
( f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) + f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) & f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) - f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) & integral ((f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) + f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = (integral (f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) + (integral (f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) & integral ((f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) - f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = (integral (f1 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) - (integral (f2 : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ;

theorem :: INTEGR15:18
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) st A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) c= dom f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( V129() V130() V131() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) & f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) & f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ,(REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V32() ) set ) ) : ( ( ) ( V32() ) set ) ) is bounded holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) & integral ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (integral (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ) ;

theorem :: INTEGR15:19
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for a, b being ( ( ) ( V11() real ext-real ) Real) st A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) = [.a : ( ( ) ( V11() real ext-real ) Real) ,b : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) ( V129() V130() V131() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) holds
integral (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = integral (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,a : ( ( ) ( V11() real ext-real ) Real) ,b : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;

theorem :: INTEGR15:20
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) )
for f being ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) )
for a, b being ( ( ) ( V11() real ext-real ) Real) st A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) = [.b : ( ( ) ( V11() real ext-real ) Real) ,a : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) ( V129() V130() V131() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) holds
- (integral (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V129() V130() V131() real-bounded interval V226() closed_interval ) Subset of ( ( ) ( V32() ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = integral (f : ( ( Function-like ) ( V13() V16( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) V17( REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,a : ( ( ) ( V11() real ext-real ) Real) ,b : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;

theorem :: INTEGR15:21
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat)
for Z, x being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6(b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
(f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) + g : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( ) set ) : ( ( ) ( V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = (f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( ) set ) ) : ( ( ) ( V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) + (g : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( ) set ) ) : ( ( ) ( V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;

theorem :: INTEGR15:22
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat)
for Z, x being ( ( ) ( ) set )
for f, g being ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) st x : ( ( ) ( ) set ) in dom (f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) - g : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6(b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
(f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) - g : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( ) set ) : ( ( ) ( V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = (f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( ) set ) ) : ( ( ) ( V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) - (g : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( ) set ) ) : ( ( ) ( V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;

theorem :: INTEGR15:23
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat)
for Z, x being ( ( ) ( ) set )
for f being ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for r being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( ) ( ) set ) in dom (r : ( ( real ) ( V11() real ext-real ) number ) (#) f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( ) Element of K6(b2 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
(r : ( ( real ) ( V11() real ext-real ) number ) (#) f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ) : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( ) set ) : ( ( ) ( V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) = r : ( ( real ) ( V11() real ext-real ) number ) * (f : ( ( Function-like ) ( V13() V16(b2 : ( ( ) ( ) set ) ) V17( REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( ) set ) ) : ( ( ) ( V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() ) Element of K6(REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) : ( ( ) ( V32() ) set ) ) ) V17( REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) Function-like V32() V39(b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) ) FinSequence-like FinSubsequence-like V119() V120() V121() ) Element of REAL b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative ) Nat) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V32() V129() V130() V131() V135() ) set ) ) ) ;