REAL is non empty V2() V32() V129() V130() V131() V135() V160() V161() V163() set
NAT is non empty V2() epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() V158() V160() Element of K6(REAL)
K6(REAL) is V2() V32() set
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(NAT,(REAL *)) is V2() Relation-like V32() set
K6(K7(NAT,(REAL *))) is V2() V32() set
omega is non empty V2() epsilon-transitive epsilon-connected ordinal V32() V37() V38() V129() V130() V131() V132() V133() V134() V135() V158() V160() set
K6(omega) is V2() V32() set
K6(NAT) is V2() V32() set
COMPLEX is non empty V2() V32() V129() V135() set
RAT is non empty V2() V32() V129() V130() V131() V132() V135() set
INT is non empty V2() V32() V129() V130() V131() V132() V133() V135() set
K7(REAL,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(REAL,REAL)) is V2() V32() set
K316() is non empty V75() L8()
the carrier of K316() is non empty set
K321() is non empty L8()
K322() is non empty V75() M13(K321())
K323() is non empty V75() V97() V157() M16(K321(),K322())
K325() is non empty V75() V97() V99() V101() L8()
K326() is non empty V75() V97() V157() M13(K325())
K7(COMPLEX,COMPLEX) is V2() Relation-like V32() complex-valued set
K6(K7(COMPLEX,COMPLEX)) is V2() V32() set
K7(COMPLEX,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(COMPLEX,REAL)) is V2() V32() set
K7(NAT,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(NAT,REAL)) is V2() V32() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
K7(1,1) is Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued natural-valued set
K6(K7(1,1)) is V32() V36() set
K7(K7(1,1),1) is Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued natural-valued set
K6(K7(K7(1,1),1)) is V32() V36() set
K7(K7(1,1),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7(K7(1,1),REAL)) is set
K7(K7(REAL,REAL),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(K7(REAL,REAL),REAL)) is V2() V32() set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
K7(2,2) is Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued natural-valued set
K7(K7(2,2),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7(K7(2,2),REAL)) is set
TOP-REAL 2 is non empty V73() V142() V168() V169() V170() V171() V172() V173() V174() TopSpace-like V219() L20()
the carrier of (TOP-REAL 2) is non empty set
REAL 1 is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = 1 } is set
K7((REAL 1),(REAL 1)) is Relation-like set
K6(K7((REAL 1),(REAL 1))) is set
REAL-NS 1 is non empty V52() V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS 1) is non empty V2() set
K436((REAL-NS 1),(REAL-NS 1)) is non empty NORMSTR
the carrier of K436((REAL-NS 1),(REAL-NS 1)) is non empty set
K7(K7(COMPLEX,COMPLEX),COMPLEX) is V2() Relation-like V32() complex-valued set
K6(K7(K7(COMPLEX,COMPLEX),COMPLEX)) is V2() V32() set
K7(RAT,RAT) is V2() Relation-like RAT -valued V32() complex-valued ext-real-valued real-valued set
K6(K7(RAT,RAT)) is V2() V32() set
K7(K7(RAT,RAT),RAT) is V2() Relation-like RAT -valued V32() complex-valued ext-real-valued real-valued set
K6(K7(K7(RAT,RAT),RAT)) is V2() V32() set
K7(INT,INT) is V2() Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued set
K6(K7(INT,INT)) is V2() V32() set
K7(K7(INT,INT),INT) is V2() Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued set
K6(K7(K7(INT,INT),INT)) is V2() V32() set
K7(NAT,NAT) is V2() Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued natural-valued set
K7(K7(NAT,NAT),NAT) is V2() Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued natural-valued set
K6(K7(K7(NAT,NAT),NAT)) is V2() V32() set
ExtREAL is non empty V130() V163() set
the_set_of_RealSequences is non empty set
K7(the_set_of_RealSequences,the_set_of_RealSequences) is Relation-like set
K7(K7(the_set_of_RealSequences,the_set_of_RealSequences),the_set_of_RealSequences) is Relation-like set
K6(K7(K7(the_set_of_RealSequences,the_set_of_RealSequences),the_set_of_RealSequences)) is set
K7(REAL,the_set_of_RealSequences) is V2() Relation-like V32() set
K7(K7(REAL,the_set_of_RealSequences),the_set_of_RealSequences) is V2() Relation-like V32() set
K6(K7(K7(REAL,the_set_of_RealSequences),the_set_of_RealSequences)) is V2() V32() set
Linear_Space_of_RealSequences is L13()
the carrier of Linear_Space_of_RealSequences is set
K6( the carrier of Linear_Space_of_RealSequences) is set
the_set_of_l2RealSequences is Element of K6( the carrier of Linear_Space_of_RealSequences)
K7(the_set_of_l2RealSequences,the_set_of_l2RealSequences) is Relation-like set
K7(K7(the_set_of_l2RealSequences,the_set_of_l2RealSequences),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7(K7(the_set_of_l2RealSequences,the_set_of_l2RealSequences),REAL)) is set
K682() is Element of K6( the carrier of Linear_Space_of_RealSequences)
K7(K682(),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7(K682(),REAL)) is set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V36() V37() V39( {} ) FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V129() V130() V131() V132() V133() V134() V135() V160() V161() V162() V163() V255() V256() V257() V258() V259() V260() V261() V262() V263() V264() V265() V266() bounded set
{{},1} is non empty V32() V36() V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() set
K200() is V96() L11()
the carrier of K200() is set
the carrier of K200() * is non empty functional FinSequence-membered FinSequenceSet of the carrier of K200()
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V36() V37() V39( {} ) FinSequence-like FinSubsequence-like FinSequence-membered V44() V45() ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V129() V130() V131() V132() V133() V134() V135() V160() V161() V162() V163() V255() V256() V257() V258() V259() V260() V261() V262() V263() V264() V265() V266() bounded Element of NAT
3 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
- 1 is V11() real ext-real non positive Element of REAL
Seg 1 is non empty V2() V32() V39(1) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= 1 ) } is set
{1} is non empty V2() V32() V36() V39(1) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() set
Seg 2 is non empty V32() V39(2) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= 2 ) } is set
{1,2} is non empty V32() V36() V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() set
<*> REAL is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional V32() V33() V36() V37() V39( {} ) FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V129() V130() V131() V132() V133() V134() V135() V160() V161() V162() V163() V255() V256() V257() V258() V259() V260() V261() V262() V263() V264() V265() V266() bounded FinSequence of REAL
Sum (<*> REAL) is V11() real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
b - x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K777(REAL,REAL,(REAL a),(REAL a),b,x0) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
- x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
b + (- x0) is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K771(REAL,REAL,(REAL a),(REAL a),b,(- x0)) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
['a,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
[.a,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( a <= b1 & b1 <= x0 ) } is set
[.b,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( b <= b1 & b1 <= x0 ) } is set
[.a,b.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( a <= b1 & b1 <= b ) } is set
a is set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
['b,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
['x0,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
min (x0,n) is V11() real ext-real set
max (x0,n) is V11() real ext-real set
['(min (x0,n)),(max (x0,n))'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
[.a,b.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( a <= b1 & b1 <= b ) } is set
['x0,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
['n,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
['n,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
a is set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
min (n,f) is V11() real ext-real set
max (n,f) is V11() real ext-real set
['(min (n,f)),(max (n,f))'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
['b,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
F + c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K771(REAL,REAL,(REAL a),(REAL a),F,c7) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
dom (F + c7) is V129() V130() V131() Element of K6(REAL)
(dom F) /\ (dom c7) is V129() V130() V131() Element of K6(REAL)
['b,f'] /\ ['b,f'] is V129() V130() V131() V163() Element of K6(REAL)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
['b,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
F - c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K777(REAL,REAL,(REAL a),(REAL a),F,c7) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
dom (F - c7) is V129() V130() V131() Element of K6(REAL)
(dom F) /\ (dom c7) is V129() V130() V131() Element of K6(REAL)
['b,f'] /\ ['b,f'] is V129() V130() V131() V163() Element of K6(REAL)
a is non empty set
b is non empty set
K7(a,b) is Relation-like set
K6(K7(a,b)) is set
x0 is non empty set
K7(x0,b) is Relation-like set
K6(K7(x0,b)) is set
n is Relation-like a -defined b -valued Function-like Element of K6(K7(a,b))
dom n is Element of K6(a)
K6(a) is set
n | x0 is Relation-like a -defined x0 -defined a -defined b -valued Function-like Element of K6(K7(a,b))
rng n is Element of K6(b)
K6(b) is set
K7((dom n),b) is Relation-like set
K6(K7((dom n),b)) is set
a is V11() real ext-real set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
['a,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['a,n'] is Relation-like REAL -defined ['a,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom F is V129() V130() V131() Element of K6(REAL)
f (#) F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(f (#) F) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
a is V11() real ext-real set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
['a,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
f | ['a,n'] is Relation-like REAL -defined ['a,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['a,n'] is Relation-like REAL -defined ['a,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom f is V129() V130() V131() Element of K6(REAL)
dom F is V129() V130() V131() Element of K6(REAL)
f - F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
- F is Relation-like REAL -defined Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) F is Relation-like REAL -defined Function-like complex-valued ext-real-valued real-valued set
f + (- F) is Relation-like REAL -defined Function-like complex-valued ext-real-valued real-valued set
(f - F) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
- 1 is V11() real V44() V45() ext-real non positive Element of INT
(- 1) (#) F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((- 1) (#) F) | ['a,n'] is Relation-like REAL -defined ['a,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom ((- 1) (#) F) is V129() V130() V131() Element of K6(REAL)
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
min (x0,n) is V11() real ext-real set
max (x0,n) is V11() real ext-real set
['(min (x0,n)),(max (x0,n))'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n - x0 is V11() real ext-real Element of REAL
abs (n - x0) is V11() real ext-real Element of REAL
f is V11() real ext-real set
f * (abs (n - x0)) is V11() real ext-real Element of REAL
F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom F is V129() V130() V131() Element of K6(REAL)
integral (F,x0,n) is V11() real ext-real Element of REAL
abs (integral (F,x0,n)) is V11() real ext-real Element of REAL
abs F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued bounded_below Element of K6(K7(REAL,REAL))
rng (abs F) is V129() V130() V131() Element of K6(REAL)
dom (abs F) is V129() V130() V131() Element of K6(REAL)
K7((dom (abs F)),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7((dom (abs F)),REAL)) is set
K7(['(min (x0,n)),(max (x0,n))'],REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(['(min (x0,n)),(max (x0,n))'],REAL)) is V2() V32() set
(abs F) | ['(min (x0,n)),(max (x0,n))'] is Relation-like REAL -defined ['(min (x0,n)),(max (x0,n))'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued bounded_below Element of K6(K7(REAL,REAL))
vol ['(min (x0,n)),(max (x0,n))'] is V11() real ext-real Element of REAL
x is non empty Relation-like ['(min (x0,n)),(max (x0,n))'] -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
{f} is non empty V2() V32() V39(1) V129() V130() V131() V158() V159() V160() V161() V162() set
R is non empty Relation-like ['(min (x0,n)),(max (x0,n))'] -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
rng R is non empty V129() V130() V131() Element of K6(REAL)
R | ['(min (x0,n)),(max (x0,n))'] is Relation-like ['(min (x0,n)),(max (x0,n))'] -defined ['(min (x0,n)),(max (x0,n))'] -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
R is V11() real ext-real Element of REAL
x . R is V11() real ext-real Element of REAL
(abs F) . R is V11() real ext-real Element of REAL
F . R is V11() real ext-real Element of REAL
abs (F . R) is V11() real ext-real Element of REAL
R . R is V11() real ext-real Element of REAL
integral ((abs F),(min (x0,n)),(max (x0,n))) is V11() real ext-real Element of REAL
integral ((abs F),['(min (x0,n)),(max (x0,n))']) is V11() real ext-real Element of REAL
K576((abs F),['(min (x0,n)),(max (x0,n))']) is Relation-like ['(min (x0,n)),(max (x0,n))'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued bounded_below Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
integral K576((abs F),['(min (x0,n)),(max (x0,n))']) is V11() real ext-real Element of REAL
x | ['(min (x0,n)),(max (x0,n))'] is Relation-like ['(min (x0,n)),(max (x0,n))'] -defined ['(min (x0,n)),(max (x0,n))'] -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
integral R is V11() real ext-real Element of REAL
f * (vol ['(min (x0,n)),(max (x0,n))']) is V11() real ext-real Element of REAL
a is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
b is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom b is V129() V130() V131() Element of K6(REAL)
b | a is Relation-like REAL -defined a -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K7(a,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(a,REAL)) is V2() V32() set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
['b,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
['n,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
f | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom f is V129() V130() V131() Element of K6(REAL)
integral (f,b,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (f,b,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (f,n,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(integral (f,b,n)) + (integral (f,n,x0)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
(proj (F,a)) * f is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (F,a)) * (f | ['b,x0']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (F,a)) * f) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (proj (F,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng f is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (F,a)) * f) is V129() V130() V131() Element of K6(REAL)
integral (((proj (F,a)) * f),b,x0) is V11() real ext-real Element of REAL
integral (((proj (F,a)) * f),b,n) is V11() real ext-real Element of REAL
integral (((proj (F,a)) * f),n,x0) is V11() real ext-real Element of REAL
(integral (((proj (F,a)) * f),b,n)) + (integral (((proj (F,a)) * f),n,x0)) is V11() real ext-real Element of REAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (F,a)) * f is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (F,a)) * f is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
dom (integral (f,b,x0)) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(integral (f,b,x0)) . F is V11() real ext-real Element of REAL
(proj (F,a)) * f is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (F,a)) * f),b,x0) is V11() real ext-real Element of REAL
integral (((proj (F,a)) * f),b,n) is V11() real ext-real Element of REAL
integral (((proj (F,a)) * f),n,x0) is V11() real ext-real Element of REAL
(integral (((proj (F,a)) * f),b,n)) + (integral (((proj (F,a)) * f),n,x0)) is V11() real ext-real Element of REAL
(integral (f,b,n)) . F is V11() real ext-real Element of REAL
((integral (f,b,n)) . F) + (integral (((proj (F,a)) * f),n,x0)) is V11() real ext-real Element of REAL
(integral (f,n,x0)) . F is V11() real ext-real Element of REAL
((integral (f,b,n)) . F) + ((integral (f,n,x0)) . F) is V11() real ext-real Element of REAL
((integral (f,b,n)) + (integral (f,n,x0))) . F is V11() real ext-real Element of REAL
len ((integral (f,b,n)) + (integral (f,n,x0))) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom ((integral (f,b,n)) + (integral (f,n,x0))) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
['b,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
F | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
F | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (c7,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
(proj (c7,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (c7,a)) * (F | ['b,f']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (c7,a)) * F) | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (proj (c7,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng F is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (c7,a)) * F) is V129() V130() V131() Element of K6(REAL)
((proj (c7,a)) * F) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (c7,a)) * (F | ['x0,n']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
c7 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (c7,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (c7,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
c7 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (c7,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (c7,a)) * (F | ['x0,n']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
['b,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
F | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
F + c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K771(REAL,REAL,(REAL a),(REAL a),F,c7) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
(F + c7) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
(proj (x,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * (F | ['b,f']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * F) | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (proj (x,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng F is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (x,a)) * F) is V129() V130() V131() Element of K6(REAL)
(proj (x,a)) * c7 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * (c7 | ['b,f']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * c7) | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
rng c7 is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (x,a)) * c7) is V129() V130() V131() Element of K6(REAL)
((proj (x,a)) * F) + ((proj (x,a)) * c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(((proj (x,a)) * F) + ((proj (x,a)) * c7)) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * (F + c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * (F + c7)) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * ((F + c7) | ['x0,n']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (x,a)) * (F + c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (x,a)) * ((F + c7) | ['x0,n']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
['b,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
F is V11() real ext-real set
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
F (#) c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K740(REAL,(REAL a),c7,F) is Relation-like REAL -defined K703(K698((REAL a))) -valued Function-like V261() V262() V263() Element of K6(K7(REAL,K703(K698((REAL a)))))
K698((REAL a)) is set
K703(K698((REAL a))) is functional V255() V256() V257() set
K7(REAL,K703(K698((REAL a)))) is Relation-like set
K6(K7(REAL,K703(K698((REAL a))))) is set
(F (#) c7) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
(proj (x,a)) * c7 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * (c7 | ['b,f']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * c7) | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (proj (x,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng c7 is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (x,a)) * c7) is V129() V130() V131() Element of K6(REAL)
F (#) ((proj (x,a)) * c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(F (#) ((proj (x,a)) * c7)) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * (F (#) c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * (F (#) c7)) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * ((F (#) c7) | ['x0,n']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (x,a)) * (F (#) c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (x,a)) * ((F (#) c7) | ['x0,n']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
['b,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
F | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
- F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
(- F) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
- 1 is V11() real V44() V45() ext-real non positive Element of INT
(- 1) (#) F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K740(REAL,(REAL a),F,(- 1)) is Relation-like REAL -defined K703(K698((REAL a))) -valued Function-like V261() V262() V263() Element of K6(K7(REAL,K703(K698((REAL a)))))
K698((REAL a)) is set
K703(K698((REAL a))) is functional V255() V256() V257() set
K7(REAL,K703(K698((REAL a)))) is Relation-like set
K6(K7(REAL,K703(K698((REAL a))))) is set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
['b,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
F | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
F - c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K777(REAL,REAL,(REAL a),(REAL a),F,c7) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
(F - c7) | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
- c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom (- c7) is V129() V130() V131() Element of K6(REAL)
F + (- c7) is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K771(REAL,REAL,(REAL a),(REAL a),F,(- c7)) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
(- c7) | ['b,f'] is Relation-like REAL -defined ['b,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
a is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
b is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL b is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
b -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = b } is set
K7(a,(REAL b)) is Relation-like set
K6(K7(a,(REAL b))) is set
x0 is non empty Relation-like a -defined REAL b -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(a,(REAL b)))
|.x0.| is Relation-like a -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
K7(a,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(a,REAL)) is V2() V32() set
Seg b is non empty V32() V39(b) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= b ) } is set
n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
proj (n,b) is non empty Relation-like REAL b -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL b),REAL))
K7((REAL b),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL b),REAL)) is V2() V32() set
(proj (n,b)) * x0 is non empty Relation-like a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
dom ((proj (n,b)) * x0) is non empty V129() V130() V131() Element of K6(a)
K6(a) is set
F is V11() real ext-real set
c7 is set
abs F is V11() real ext-real Element of REAL
((proj (n,b)) * x0) . c7 is V11() real ext-real Element of REAL
abs (((proj (n,b)) * x0) . c7) is V11() real ext-real Element of REAL
n is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom n is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (f,b) is non empty Relation-like REAL b -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL b),REAL))
K7((REAL b),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL b),REAL)) is V2() V32() set
n . f is V11() real ext-real Element of REAL
(proj (f,b)) * x0 is non empty Relation-like a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
dom ((proj (f,b)) * x0) is non empty V129() V130() V131() Element of K6(a)
K6(a) is set
c7 is set
((proj (f,b)) * x0) . c7 is V11() real ext-real Element of REAL
abs (((proj (f,b)) * x0) . c7) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
len n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f . F is V11() real ext-real Element of REAL
Sum f is V11() real ext-real Element of REAL
dom |.x0.| is V129() V130() V131() Element of K6(a)
b * (Sum f) is V11() real ext-real Element of REAL
(b * (Sum f)) + 1 is V11() real ext-real Element of REAL
c7 is set
|.x0.| . c7 is V11() real ext-real Element of REAL
abs (|.x0.| . c7) is V11() real ext-real Element of REAL
|.x0.| /. c7 is V11() real ext-real Element of REAL
x0 /. c7 is Relation-like NAT -defined Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
|.(x0 /. c7).| is V11() real ext-real non negative Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,b) is non empty Relation-like REAL b -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL b),REAL))
dom x0 is non empty V129() V130() V131() Element of K6(a)
x0 . c7 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
rng x0 is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL b))
K6((REAL b)) is set
dom (proj (x,b)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL b))
(proj (x,b)) * x0 is non empty Relation-like a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
dom ((proj (x,b)) * x0) is non empty V129() V130() V131() Element of K6(a)
f . x is V11() real ext-real Element of REAL
((proj (x,b)) * x0) . c7 is V11() real ext-real Element of REAL
abs (((proj (x,b)) * x0) . c7) is V11() real ext-real Element of REAL
(proj (x,b)) . (x0 . c7) is V11() real ext-real Element of REAL
abs ((proj (x,b)) . (x0 . c7)) is V11() real ext-real Element of REAL
(proj (x,b)) . (x0 /. c7) is V11() real ext-real Element of REAL
|.((proj (x,b)) . (x0 /. c7)).| is V11() real ext-real Element of REAL
0 + (abs (|.x0.| . c7)) is V11() real ext-real Element of REAL
n is V11() real ext-real set
f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (f,b) is non empty Relation-like REAL b -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL b),REAL))
(proj (f,b)) * x0 is non empty Relation-like a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
dom ((proj (f,b)) * x0) is non empty V129() V130() V131() Element of K6(a)
c7 is set
((proj (f,b)) * x0) . c7 is V11() real ext-real Element of REAL
abs (((proj (f,b)) * x0) . c7) is V11() real ext-real Element of REAL
dom x0 is non empty V129() V130() V131() Element of K6(a)
x0 . c7 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (f,b)) . (x0 . c7) is V11() real ext-real Element of REAL
x0 /. c7 is Relation-like NAT -defined Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
(proj (f,b)) . (x0 /. c7) is V11() real ext-real Element of REAL
|.((proj (f,b)) . (x0 /. c7)).| is V11() real ext-real Element of REAL
|.(x0 /. c7).| is V11() real ext-real non negative Element of REAL
|.x0.| . c7 is V11() real ext-real Element of REAL
abs (|.x0.| . c7) is V11() real ext-real Element of REAL
|.x0.| /. c7 is V11() real ext-real Element of REAL
abs (|.x0.| /. c7) is V11() real ext-real Element of REAL
abs |.(x0 /. c7).| is V11() real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
b is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x0 is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
len x0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
Sum x0 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
n is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
len n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom n is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
(Seg a) --> 0 is Relation-like Seg a -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg a),NAT))
K7((Seg a),NAT) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg a),NAT)) is set
dom ((Seg a) --> 0) is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
f is set
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (F,a)) * x0 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n . F is V11() real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum x is V11() real ext-real Element of REAL
n . f is V11() real ext-real Element of REAL
((Seg a) --> 0) . f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative Element of REAL
0* a is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
a |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of a -tuples_on REAL
(Seg a) --> 0 is Relation-like Seg a -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V261() Element of K6(K7((Seg a),{0}))
{0} is non empty V2() functional V32() V36() V39(1) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() V255() set
K7((Seg a),{0}) is Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg a),{0})) is V32() V36() set
x0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
x0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
n is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
len n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
n | x0 is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
Seg x0 is V32() V39(x0) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= x0 ) } is set
n | (Seg x0) is Relation-like NAT -defined Seg x0 -defined NAT -defined REAL a -valued Function-like V32() FinSubsequence-like V261() set
Seg (x0 + 1) is non empty V32() V39(x0 + 1) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= x0 + 1 ) } is set
dom n is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
n . (x0 + 1) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
rng n is functional V32() FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (x,a)) * (n | x0) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((proj (x,a)) * (n | x0)) is V11() real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom x is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
len x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (R,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (R,a)) * (n | x0) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
R . R is V11() real ext-real Element of REAL
len (n | x0) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
len f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
R + c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
len (R + c7) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
proj (R,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (R,a)) * n is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f . R is V11() real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum R is V11() real ext-real Element of REAL
(proj (R,a)) * (n | x0) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
R . R is V11() real ext-real Element of REAL
N1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum N1 is V11() real ext-real Element of REAL
dom (proj (R,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom R is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
Seg (len n) is V32() V39( len n) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len n ) } is set
c7 . R is V11() real ext-real Element of REAL
<*(c7 . R)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant V32() V39(1) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing bounded FinSequence of REAL
[1,(c7 . R)] is set
{1,(c7 . R)} is non empty V32() V129() V130() V131() V158() V159() V160() V161() V162() set
{{1,(c7 . R)},{1}} is non empty V32() V36() set
{[1,(c7 . R)]} is non empty V2() Relation-like Function-like constant V32() V39(1) set
len <*(c7 . R)*> is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
(len (n | x0)) + (len <*(c7 . R)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
rng (n | x0) is functional V32() FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom N1 is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
dom (n | x0) is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
Seg (len (n | x0)) is V32() V39( len (n | x0)) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len (n | x0) ) } is set
len N1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
(len N1) + (len <*(c7 . R)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
Seg ((len N1) + (len <*(c7 . R)*>)) is non empty V32() V39((len N1) + (len <*(c7 . R)*>)) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= (len N1) + (len <*(c7 . R)*>) ) } is set
e0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
R . e0 is V11() real ext-real Element of REAL
N1 . e0 is V11() real ext-real Element of REAL
(n | x0) . e0 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (R,a)) . ((n | x0) . e0) is V11() real ext-real Element of REAL
n . e0 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (R,a)) . (n . e0) is V11() real ext-real Element of REAL
dom <*(c7 . R)*> is non empty V2() V32() V39(1) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
e0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
(len N1) + e0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
R . ((len N1) + e0) is V11() real ext-real Element of REAL
<*(c7 . R)*> . e0 is V11() real ext-real Element of REAL
Seg (len <*(c7 . R)*>) is non empty V32() V39( len <*(c7 . R)*>) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len <*(c7 . R)*> ) } is set
(proj (R,a)) . (n . (x0 + 1)) is V11() real ext-real Element of REAL
N1 ^ <*(c7 . R)*> is non empty Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(R . R) + (c7 . R) is V11() real ext-real Element of REAL
(R + c7) . R is V11() real ext-real Element of REAL
n . (len n) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Sum n is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
Sum (n | x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(Sum (n | x0)) + R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
n is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
len n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
Sum n is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x0 is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
Sum x0 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
len x0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
b is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.b.| is V11() real ext-real non negative Element of REAL
x0 is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
len x0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom x0 is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
n is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
Sum n is V11() real ext-real Element of REAL
Sum x0 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
Seg (len x0) is V32() V39( len x0) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len x0 ) } is set
f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
x0 . f is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x0 /. f is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.(x0 /. f).| is V11() real ext-real non negative Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom f is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
x0 . F is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f . F is V11() real ext-real Element of REAL
len f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
|.(Sum x0).| is V11() real ext-real non negative Element of REAL
Sum f is V11() real ext-real Element of REAL
(len x0) -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = len x0 } is set
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
x0 /. R is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.(x0 /. R).| is V11() real ext-real non negative Element of REAL
n /. R is V11() real ext-real Element of REAL
dom n is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
n . R is V11() real ext-real Element of REAL
x0 . R is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f . R is V11() real ext-real Element of REAL
c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39( len x0) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len x0) -tuples_on REAL
c7 . R is V11() real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V32() V39( len x0) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len x0) -tuples_on REAL
x . R is V11() real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.R.| is V11() real ext-real non negative Element of REAL
Sum c7 is V11() real ext-real Element of REAL
Sum x is V11() real ext-real Element of REAL
a is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(a,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(a,REAL)) is V2() V32() set
b is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL b is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
b -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = b } is set
K7(a,(REAL b)) is Relation-like set
K6(K7(a,(REAL b))) is set
x0 is non empty Relation-like a -defined REAL b -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(a,(REAL b)))
|.x0.| is Relation-like a -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
integral x0 is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
|.(integral x0).| is V11() real ext-real non negative Element of REAL
n is non empty Relation-like a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
integral n is V11() real ext-real Element of REAL
divs a is non empty set
K7(NAT,(divs a)) is V2() Relation-like V32() set
K6(K7(NAT,(divs a))) is V2() V32() set
f is non empty Relation-like NAT -defined divs a -valued Function-like total quasi_total Element of K6(K7(NAT,(divs a)))
delta f is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim (delta f) is V11() real ext-real Element of REAL
(REAL b) * is non empty functional FinSequence-membered FinSequenceSet of REAL b
the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f is non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f
dom n is non empty V129() V130() V131() Element of K6(a)
K6(a) is set
dom x0 is non empty V129() V130() V131() Element of K6(a)
c7 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f . c7 is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of a
len (f . c7) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
dom (f . c7) is non empty V32() V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . c7 is Relation-like NAT -defined REAL b -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() middle_volume of x0,f . c7
Seg (len (f . c7)) is non empty V32() V39( len (f . c7)) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len (f . c7) ) } is set
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
divset ((f . c7),x) is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 | (divset ((f . c7),x)) is Relation-like a -defined divset ((f . c7),x) -defined a -defined REAL b -valued Function-like V261() V262() V263() Element of K6(K7(a,(REAL b)))
dom (x0 | (divset ((f . c7),x))) is V129() V130() V131() Element of K6(a)
( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . c7) . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
vol (divset ((f . c7),x)) is V11() real ext-real Element of REAL
rng (x0 | (divset ((f . c7),x))) is functional FinSequence-membered V255() V256() V257() Element of K6((REAL b))
K6((REAL b)) is set
R is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
(vol (divset ((f . c7),x))) * R is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
R is set
(x0 | (divset ((f . c7),x))) . R is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
N is V11() real ext-real Element of REAL
(x0 | (divset ((f . c7),x))) . N is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom x is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
len x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
c7 is non empty Relation-like NAT -defined REAL * -valued Function-like total quasi_total Element of K6(K7(NAT,(REAL *)))
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f . x is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of a
dom (f . x) is non empty V32() V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
c7 . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like Element of REAL *
len (f . x) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
Seg (len (f . x)) is non empty V32() V39( len (f . x)) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len (f . x) ) } is set
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
(c7 . x) . R is set
divset ((f . x),R) is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n | (divset ((f . x),R)) is Relation-like a -defined divset ((f . x),R) -defined a -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
dom (n | (divset ((f . x),R))) is V129() V130() V131() Element of K6(a)
(n | (divset ((f . x),R))) . ((c7 . x) . R) is V11() real ext-real Element of REAL
vol (divset ((f . x),R)) is V11() real ext-real Element of REAL
the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . x is Relation-like NAT -defined REAL b -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() middle_volume of x0,f . x
R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
R . R is V11() real ext-real Element of REAL
x0 | (divset ((f . x),R)) is Relation-like a -defined divset ((f . x),R) -defined a -defined REAL b -valued Function-like V261() V262() V263() Element of K6(K7(a,(REAL b)))
dom (x0 | (divset ((f . x),R))) is V129() V130() V131() Element of K6(a)
(n | (divset ((f . x),R))) . (R . R) is V11() real ext-real Element of REAL
(vol (divset ((f . x),R))) * ((n | (divset ((f . x),R))) . (R . R)) is V11() real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom R is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
len R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
(c7 . x) . R is set
divset ((f . x),R) is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n | (divset ((f . x),R)) is Relation-like a -defined divset ((f . x),R) -defined a -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
dom (n | (divset ((f . x),R))) is V129() V130() V131() Element of K6(a)
(n | (divset ((f . x),R))) . ((c7 . x) . R) is V11() real ext-real Element of REAL
R . R is V11() real ext-real Element of REAL
vol (divset ((f . x),R)) is V11() real ext-real Element of REAL
N is V11() real ext-real Element of REAL
(vol (divset ((f . x),R))) * N is V11() real ext-real Element of REAL
rng (n | (divset ((f . x),R))) is V129() V130() V131() Element of K6(REAL)
R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued middle_volume of n,f . x
x is non empty Relation-like NAT -defined REAL * -valued Function-like total quasi_total Element of K6(K7(NAT,(REAL *)))
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f . R is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of a
x . R is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like Element of REAL *
dom (f . R) is non empty V32() V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
c7 . R is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like Element of REAL *
R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued middle_volume of n,f . R
R is non empty Relation-like NAT -defined REAL * -valued Function-like total quasi_total middle_volume_Sequence of n,f
middle_sum (n,R) is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim (middle_sum (n,R)) is V11() real ext-real Element of REAL
REAL-NS b is non empty V52() V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f) is non empty Relation-like NAT -defined the carrier of (REAL-NS b) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS b)))
the carrier of (REAL-NS b) is non empty V2() set
K7(NAT, the carrier of (REAL-NS b)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS b))) is V2() V32() set
lim (middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f)) is Element of the carrier of (REAL-NS b)
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
(middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f)) . R is Element of the carrier of (REAL-NS b)
||.((middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f)) . R).|| is V11() real ext-real Element of REAL
(middle_sum (n,R)) . R is V11() real ext-real Element of REAL
f . R is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of a
R . R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued middle_volume of n,f . R
middle_sum (n,(R . R)) is V11() real ext-real Element of REAL
Sum (R . R) is V11() real ext-real Element of REAL
the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R is Relation-like NAT -defined REAL b -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() middle_volume of x0,f . R
middle_sum (x0,( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
Seg b is non empty V32() V39(b) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= b ) } is set
N is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (N,b) is non empty Relation-like REAL b -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL b),REAL))
K7((REAL b),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL b),REAL)) is V2() V32() set
(proj (N,b)) * ( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(middle_sum (x0,( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R))) . N is V11() real ext-real Element of REAL
c7 . R is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like Element of REAL *
len (f . R) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
dom (f . R) is non empty V32() V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
len (R . R) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
len ( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
N is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
dom ( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R) is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
Seg (len (f . R)) is non empty V32() V39( len (f . R)) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len (f . R) ) } is set
dom (R . R) is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
(c7 . R) . N is set
divset ((f . R),N) is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 | (divset ((f . R),N)) is Relation-like a -defined divset ((f . R),N) -defined a -defined REAL b -valued Function-like V261() V262() V263() Element of K6(K7(a,(REAL b)))
dom (x0 | (divset ((f . R),N))) is V129() V130() V131() Element of K6(a)
R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
(x0 | (divset ((f . R),N))) . ((c7 . R) . N) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R) . N is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
vol (divset ((f . R),N)) is V11() real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
R is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
(vol (divset ((f . R),N))) * R is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
n | (divset ((f . R),N)) is Relation-like a -defined divset ((f . R),N) -defined a -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
dom (n | (divset ((f . R),N))) is V129() V130() V131() Element of K6(a)
(n | (divset ((f . R),N))) . ((c7 . R) . N) is V11() real ext-real Element of REAL
(R . R) . N is V11() real ext-real Element of REAL
N1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued middle_volume of n,f . R
N1 is V11() real ext-real Element of REAL
(vol (divset ((f . R),N))) * N1 is V11() real ext-real Element of REAL
( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R) /. N is Relation-like NAT -defined Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
|.(( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R) /. N).| is V11() real ext-real non negative Element of REAL
abs (vol (divset ((f . R),N))) is V11() real ext-real Element of REAL
|.R.| is V11() real ext-real non negative Element of REAL
(abs (vol (divset ((f . R),N)))) * |.R.| is V11() real ext-real Element of REAL
(vol (divset ((f . R),N))) * |.R.| is V11() real ext-real Element of REAL
x0 . ((c7 . R) . N) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x0 /. ((c7 . R) . N) is Relation-like NAT -defined Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
n . ((c7 . R) . N) is V11() real ext-real Element of REAL
|.x0.| /. ((c7 . R) . N) is V11() real ext-real Element of REAL
|.(x0 /. ((c7 . R) . N)).| is V11() real ext-real non negative Element of REAL
(R . R) /. N is V11() real ext-real Element of REAL
|.(middle_sum (x0,( the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f . R))).| is V11() real ext-real non negative Element of REAL
||.(middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f)).|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
||.(middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f)).|| . R is V11() real ext-real Element of REAL
(middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f)) . R is Element of the carrier of (REAL-NS b)
||.((middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f)) . R).|| is V11() real ext-real Element of REAL
(middle_sum (n,R)) . R is V11() real ext-real Element of REAL
lim ||.(middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f)).|| is V11() real ext-real Element of REAL
||.(lim (middle_sum (x0, the non empty Relation-like NAT -defined (REAL b) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f))).|| is V11() real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom x0 is V129() V130() V131() Element of K6(REAL)
x0 | b is Relation-like REAL -defined b -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom x0 is V129() V130() V131() Element of K6(REAL)
x0 | b is Relation-like REAL -defined b -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
proj (n,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (n,a)) * (x0 | b) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (n,a)) * x0 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom ((proj (n,a)) * x0) is V129() V130() V131() Element of K6(REAL)
F is V11() real ext-real set
c7 is set
b /\ (dom ((proj (n,a)) * x0)) is V129() V130() V131() Element of K6(REAL)
((proj (n,a)) * x0) . c7 is V11() real ext-real Element of REAL
abs (((proj (n,a)) * x0) . c7) is V11() real ext-real Element of REAL
((proj (n,a)) * x0) | b is Relation-like REAL -defined b -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
n is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
proj (f,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (f,a)) * n is non empty Relation-like b -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(b,REAL))
K7(b,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(b,REAL)) is V2() V32() set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
|.x0.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
n is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
|.n.| is Relation-like b -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(b,REAL))
K7(b,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(b,REAL)) is V2() V32() set
dom |.x0.| is V129() V130() V131() Element of K6(REAL)
dom x0 is V129() V130() V131() Element of K6(REAL)
dom |.n.| is V129() V130() V131() Element of K6(b)
K6(b) is set
f is set
|.x0.| . f is V11() real ext-real Element of REAL
|.x0.| /. f is V11() real ext-real Element of REAL
x0 /. f is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.(x0 /. f).| is V11() real ext-real non negative Element of REAL
|.n.| /. f is V11() real ext-real Element of REAL
|.n.| . f is V11() real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom x0 is V129() V130() V131() Element of K6(REAL)
x0 | b is Relation-like REAL -defined b -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
|.(x0 | b).| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
|.x0.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
|.x0.| | b is Relation-like REAL -defined b -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (x0 | b) is V129() V130() V131() Element of K6(b)
K6(b) is set
dom (|.x0.| | b) is V129() V130() V131() Element of K6(b)
dom |.x0.| is V129() V130() V131() Element of K6(REAL)
(dom |.x0.|) /\ b is V129() V130() V131() Element of K6(REAL)
(dom x0) /\ b is V129() V130() V131() Element of K6(REAL)
dom |.(x0 | b).| is V129() V130() V131() Element of K6(REAL)
n is set
x0 /. n is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x0 . n is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x0 | b) . n is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(x0 | b) /. n is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(|.x0.| | b) . n is V11() real ext-real Element of REAL
|.x0.| . n is V11() real ext-real Element of REAL
|.x0.| /. n is V11() real ext-real Element of REAL
|.(x0 /. n).| is V11() real ext-real non negative Element of REAL
|.(x0 | b).| /. n is V11() real ext-real Element of REAL
|.(x0 | b).| . n is V11() real ext-real Element of REAL
a is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
b is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL b is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
b -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = b } is set
K7(REAL,(REAL b)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL b))) is V2() V32() set
x0 is Relation-like REAL -defined REAL b -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL b)))
dom x0 is V129() V130() V131() Element of K6(REAL)
x0 | a is Relation-like REAL -defined a -defined REAL -defined REAL b -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL b)))
|.x0.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
|.x0.| | a is Relation-like REAL -defined a -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
|.(x0 | a).| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K7(a,(REAL b)) is Relation-like set
K6(K7(a,(REAL b))) is set
n is non empty Relation-like a -defined REAL b -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(a,(REAL b)))
|.n.| is Relation-like a -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
K7(a,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(a,REAL)) is V2() V32() set
a is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
b is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL b is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
b -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = b } is set
K7(REAL,(REAL b)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL b))) is V2() V32() set
x0 is Relation-like REAL -defined REAL b -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL b)))
dom x0 is V129() V130() V131() Element of K6(REAL)
x0 | a is Relation-like REAL -defined a -defined REAL -defined REAL b -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL b)))
|.x0.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (x0,a) is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
|.(integral (x0,a)).| is V11() real ext-real non negative Element of REAL
integral (|.x0.|,a) is V11() real ext-real Element of REAL
K576(|.x0.|,a) is Relation-like a -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
K7(a,REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(a,REAL)) is V2() V32() set
integral K576(|.x0.|,a) is V11() real ext-real Element of REAL
K7(a,(REAL b)) is Relation-like set
K6(K7(a,(REAL b))) is set
f is non empty Relation-like a -defined REAL b -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(a,(REAL b)))
integral f is Relation-like NAT -defined REAL -valued Function-like V32() V39(b) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL b
dom |.x0.| is V129() V130() V131() Element of K6(REAL)
|.x0.| | a is Relation-like REAL -defined a -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F is non empty Relation-like a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
|.f.| is Relation-like a -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(a,REAL))
|.(x0 | a).| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL x0 is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
x0 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = x0 } is set
K7(REAL,(REAL x0)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL x0))) is V2() V32() set
n is Relation-like REAL -defined REAL x0 -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL x0)))
dom n is V129() V130() V131() Element of K6(REAL)
|.n.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
n | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL x0 -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL x0)))
integral (n,a,b) is Relation-like NAT -defined REAL -valued Function-like V32() V39(x0) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL x0
|.(integral (n,a,b)).| is V11() real ext-real non negative Element of REAL
integral (|.n.|,a,b) is V11() real ext-real Element of REAL
[.a,b.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( a <= b1 & b1 <= b ) } is set
integral (n,['a,b']) is Relation-like NAT -defined REAL -valued Function-like V32() V39(x0) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL x0
integral (|.n.|,['a,b']) is V11() real ext-real Element of REAL
K576(|.n.|,['a,b']) is Relation-like ['a,b'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(['a,b'],REAL))
K7(['a,b'],REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(['a,b'],REAL)) is V2() V32() set
integral K576(|.n.|,['a,b']) is V11() real ext-real Element of REAL
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL f is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
K7(REAL,(REAL f)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL f))) is V2() V32() set
F is Relation-like REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
|.F.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
dom F is V129() V130() V131() Element of K6(REAL)
|.F.| | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (F,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
|.(integral (F,x0,n)).| is V11() real ext-real non negative Element of REAL
integral (|.F.|,x0,n) is V11() real ext-real Element of REAL
[.a,b.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( a <= b1 & b1 <= b ) } is set
F | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
dom |.F.| is V129() V130() V131() Element of K6(REAL)
|.F.| | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
min (x0,n) is V11() real ext-real set
max (x0,n) is V11() real ext-real set
['(min (x0,n)),(max (x0,n))'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL f is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
K7(REAL,(REAL f)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL f))) is V2() V32() set
F is Relation-like REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
|.F.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
dom F is V129() V130() V131() Element of K6(REAL)
|.F.| | ['(min (x0,n)),(max (x0,n))'] is Relation-like REAL -defined ['(min (x0,n)),(max (x0,n))'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (F,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
|.(integral (F,x0,n)).| is V11() real ext-real non negative Element of REAL
integral (|.F.|,(min (x0,n)),(max (x0,n))) is V11() real ext-real Element of REAL
['n,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
[.n,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( n <= b1 & b1 <= x0 ) } is set
integral (F,['n,x0']) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (integral (F,['n,x0'])) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (F,['n,x0'])) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
integral (F,n,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (integral (F,n,x0)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
K38(1) (#) (integral (F,n,x0)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
|.(integral (F,n,x0)).| is V11() real ext-real non negative Element of REAL
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL f is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
K7(REAL,(REAL f)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL f))) is V2() V32() set
F is Relation-like REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
|.F.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
dom F is V129() V130() V131() Element of K6(REAL)
|.F.| | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (F,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
|.(integral (F,x0,n)).| is V11() real ext-real non negative Element of REAL
integral (|.F.|,x0,n) is V11() real ext-real Element of REAL
min (x0,n) is V11() real ext-real set
max (x0,n) is V11() real ext-real set
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
f is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL f is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
K7(REAL,(REAL f)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL f))) is V2() V32() set
F is Relation-like REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
|.F.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
dom F is V129() V130() V131() Element of K6(REAL)
integral (F,n,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
|.(integral (F,n,x0)).| is V11() real ext-real non negative Element of REAL
integral (|.F.|,x0,n) is V11() real ext-real Element of REAL
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
[.x0,n.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( x0 <= b1 & b1 <= n ) } is set
integral (F,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
|.(integral (F,x0,n)).| is V11() real ext-real non negative Element of REAL
integral (F,['x0,n']) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
- (integral (F,['x0,n'])) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (F,['x0,n'])) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
f is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL f is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
f -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = f } is set
K7(REAL,(REAL f)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL f))) is V2() V32() set
a is V11() real ext-real set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
F is Relation-like REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
|.F.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL f -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL f)))
dom F is V129() V130() V131() Element of K6(REAL)
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
|.F.| | ['x0,n'] is Relation-like REAL -defined ['x0,n'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (F,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
|.(integral (F,x0,n)).| is V11() real ext-real non negative Element of REAL
integral (|.F.|,x0,n) is V11() real ext-real Element of REAL
integral (F,n,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(f) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL f
|.(integral (F,n,x0)).| is V11() real ext-real non negative Element of REAL
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
min (x0,n) is V11() real ext-real set
max (x0,n) is V11() real ext-real set
['(min (x0,n)),(max (x0,n))'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n - x0 is V11() real ext-real Element of REAL
abs (n - x0) is V11() real ext-real Element of REAL
f is V11() real ext-real set
f * (abs (n - x0)) is V11() real ext-real Element of REAL
F is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL F is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
F -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = F } is set
K7(REAL,(REAL F)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL F))) is V2() V32() set
c7 is Relation-like REAL -defined REAL F -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL F)))
|.c7.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
c7 | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL F -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL F)))
dom c7 is V129() V130() V131() Element of K6(REAL)
integral (c7,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
|.(integral (c7,x0,n)).| is V11() real ext-real non negative Element of REAL
rng |.c7.| is V129() V130() V131() Element of K6(REAL)
dom |.c7.| is V129() V130() V131() Element of K6(REAL)
K7((dom |.c7.|),REAL) is Relation-like complex-valued ext-real-valued real-valued set
K6(K7((dom |.c7.|),REAL)) is set
K7(['(min (x0,n)),(max (x0,n))'],REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7(['(min (x0,n)),(max (x0,n))'],REAL)) is V2() V32() set
|.c7.| | ['(min (x0,n)),(max (x0,n))'] is Relation-like REAL -defined ['(min (x0,n)),(max (x0,n))'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
vol ['(min (x0,n)),(max (x0,n))'] is V11() real ext-real Element of REAL
R is non empty Relation-like ['(min (x0,n)),(max (x0,n))'] -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
{f} is non empty V2() V32() V39(1) V129() V130() V131() V158() V159() V160() V161() V162() set
R is non empty Relation-like ['(min (x0,n)),(max (x0,n))'] -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
rng R is non empty V129() V130() V131() Element of K6(REAL)
R | ['(min (x0,n)),(max (x0,n))'] is Relation-like ['(min (x0,n)),(max (x0,n))'] -defined ['(min (x0,n)),(max (x0,n))'] -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
N is V11() real ext-real Element of REAL
R . N is V11() real ext-real Element of REAL
|.c7.| . N is V11() real ext-real Element of REAL
|.c7.| /. N is V11() real ext-real Element of REAL
c7 /. N is Relation-like NAT -defined Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
|.(c7 /. N).| is V11() real ext-real non negative Element of REAL
R . N is V11() real ext-real Element of REAL
integral (|.c7.|,(min (x0,n)),(max (x0,n))) is V11() real ext-real Element of REAL
integral (|.c7.|,['(min (x0,n)),(max (x0,n))']) is V11() real ext-real Element of REAL
K576(|.c7.|,['(min (x0,n)),(max (x0,n))']) is Relation-like ['(min (x0,n)),(max (x0,n))'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
integral K576(|.c7.|,['(min (x0,n)),(max (x0,n))']) is V11() real ext-real Element of REAL
R | ['(min (x0,n)),(max (x0,n))'] is Relation-like ['(min (x0,n)),(max (x0,n))'] -defined ['(min (x0,n)),(max (x0,n))'] -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(['(min (x0,n)),(max (x0,n))'],REAL))
integral R is V11() real ext-real Element of REAL
f * (vol ['(min (x0,n)),(max (x0,n))']) is V11() real ext-real Element of REAL
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n - x0 is V11() real ext-real Element of REAL
f is V11() real ext-real set
f * (n - x0) is V11() real ext-real Element of REAL
F is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL F is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
F -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = F } is set
K7(REAL,(REAL F)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL F))) is V2() V32() set
c7 is Relation-like REAL -defined REAL F -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL F)))
|.c7.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
c7 | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL F -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL F)))
dom c7 is V129() V130() V131() Element of K6(REAL)
integral (c7,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
|.(integral (c7,x0,n)).| is V11() real ext-real non negative Element of REAL
abs (n - x0) is V11() real ext-real Element of REAL
min (x0,n) is V11() real ext-real set
max (x0,n) is V11() real ext-real set
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is V11() real ext-real set
n is V11() real ext-real set
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n - x0 is V11() real ext-real Element of REAL
f is V11() real ext-real set
f * (n - x0) is V11() real ext-real Element of REAL
F is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL F is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
F -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = F } is set
K7(REAL,(REAL F)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL F))) is V2() V32() set
c7 is Relation-like REAL -defined REAL F -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL F)))
|.c7.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
c7 | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL F -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL F)))
dom c7 is V129() V130() V131() Element of K6(REAL)
integral (c7,n,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
|.(integral (c7,n,x0)).| is V11() real ext-real non negative Element of REAL
[.x0,n.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( x0 <= b1 & b1 <= n ) } is set
integral (c7,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
|.(integral (c7,x0,n)).| is V11() real ext-real non negative Element of REAL
integral (c7,['x0,n']) is Relation-like NAT -defined REAL -valued Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
- (integral (c7,['x0,n'])) is Relation-like NAT -defined REAL -valued Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (c7,['x0,n'])) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
F is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL F is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
F -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = F } is set
K7(REAL,(REAL F)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL F))) is V2() V32() set
a is V11() real ext-real set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
c7 is Relation-like REAL -defined REAL F -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL F)))
|.c7.| is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
c7 | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined REAL F -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL F)))
dom c7 is V129() V130() V131() Element of K6(REAL)
['x0,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f is V11() real ext-real set
integral (c7,x0,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
|.(integral (c7,x0,n)).| is V11() real ext-real non negative Element of REAL
n - x0 is V11() real ext-real Element of REAL
f * (n - x0) is V11() real ext-real Element of REAL
integral (c7,n,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(F) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL F
|.(integral (c7,n,x0)).| is V11() real ext-real non negative Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is V11() real ext-real set
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
F (#) c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K740(REAL,(REAL a),c7,F) is Relation-like REAL -defined K703(K698((REAL a))) -valued Function-like V261() V262() V263() Element of K6(K7(REAL,K703(K698((REAL a)))))
K698((REAL a)) is set
K703(K698((REAL a))) is functional V255() V256() V257() set
K7(REAL,K703(K698((REAL a)))) is Relation-like set
K6(K7(REAL,K703(K698((REAL a))))) is set
integral ((F (#) c7),n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (c7,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
F * (integral (c7,n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
(proj (x,a)) * c7 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * (c7 | ['b,x0']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * c7) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (proj (x,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng c7 is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (x,a)) * c7) is V129() V130() V131() Element of K6(REAL)
(proj (x,a)) * (F (#) c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F (#) ((proj (x,a)) * c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * (F (#) c7)),n,f) is V11() real ext-real Element of REAL
integral (((proj (x,a)) * c7),n,f) is V11() real ext-real Element of REAL
F * (integral (((proj (x,a)) * c7),n,f)) is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
dom (integral ((F (#) c7),n,f)) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(integral ((F (#) c7),n,f)) . x is V11() real ext-real Element of REAL
(proj (x,a)) * (F (#) c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * (F (#) c7)),n,f) is V11() real ext-real Element of REAL
(proj (x,a)) * c7 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * c7),n,f) is V11() real ext-real Element of REAL
F * (integral (((proj (x,a)) * c7),n,f)) is V11() real ext-real Element of REAL
(integral (c7,n,f)) . x is V11() real ext-real Element of REAL
F * ((integral (c7,n,f)) . x) is V11() real ext-real Element of REAL
(F * (integral (c7,n,f))) . x is V11() real ext-real Element of REAL
len (F * (integral (c7,n,f))) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom (F * (integral (c7,n,f))) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
- F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral ((- F),n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (F,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
- (integral (F,n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (F,n,f)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- 1 is V11() real V44() V45() ext-real non positive Element of INT
(- 1) (#) F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K740(REAL,(REAL a),F,(- 1)) is Relation-like REAL -defined K703(K698((REAL a))) -valued Function-like V261() V262() V263() Element of K6(K7(REAL,K703(K698((REAL a)))))
K698((REAL a)) is set
K703(K698((REAL a))) is functional V255() V256() V257() set
K7(REAL,K703(K698((REAL a)))) is Relation-like set
K6(K7(REAL,K703(K698((REAL a))))) is set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
integral (F,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
F + c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K771(REAL,REAL,(REAL a),(REAL a),F,c7) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
integral ((F + c7),n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (c7,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(integral (F,n,f)) + (integral (c7,n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
(proj (x,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * (F | ['b,x0']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * F) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * c7 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (x,a)) * (c7 | ['b,x0']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * c7) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (proj (x,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng F is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (x,a)) * F) is V129() V130() V131() Element of K6(REAL)
rng c7 is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (x,a)) * c7) is V129() V130() V131() Element of K6(REAL)
(proj (x,a)) * (F + c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * F) + ((proj (x,a)) * c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * (F + c7)),n,f) is V11() real ext-real Element of REAL
integral (((proj (x,a)) * F),n,f) is V11() real ext-real Element of REAL
integral (((proj (x,a)) * c7),n,f) is V11() real ext-real Element of REAL
(integral (((proj (x,a)) * F),n,f)) + (integral (((proj (x,a)) * c7),n,f)) is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
dom (integral ((F + c7),n,f)) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(integral ((F + c7),n,f)) . x is V11() real ext-real Element of REAL
(proj (x,a)) * (F + c7) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * (F + c7)),n,f) is V11() real ext-real Element of REAL
(proj (x,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * F),n,f) is V11() real ext-real Element of REAL
(proj (x,a)) * c7 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * c7),n,f) is V11() real ext-real Element of REAL
(integral (((proj (x,a)) * F),n,f)) + (integral (((proj (x,a)) * c7),n,f)) is V11() real ext-real Element of REAL
(integral (F,n,f)) . x is V11() real ext-real Element of REAL
((integral (F,n,f)) . x) + (integral (((proj (x,a)) * c7),n,f)) is V11() real ext-real Element of REAL
(integral (c7,n,f)) . x is V11() real ext-real Element of REAL
((integral (F,n,f)) . x) + ((integral (c7,n,f)) . x) is V11() real ext-real Element of REAL
((integral (F,n,f)) + (integral (c7,n,f))) . x is V11() real ext-real Element of REAL
len ((integral (F,n,f)) + (integral (c7,n,f))) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom ((integral (F,n,f)) + (integral (c7,n,f))) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
integral (F,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
F - c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K777(REAL,REAL,(REAL a),(REAL a),F,c7) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
integral ((F - c7),n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (c7,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(integral (F,n,f)) - (integral (c7,n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
- (integral (c7,n,f)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (c7,n,f)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(integral (F,n,f)) + (- (integral (c7,n,f))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
(- c7) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom (- c7) is V129() V130() V131() Element of K6(REAL)
F + (- c7) is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K771(REAL,REAL,(REAL a),(REAL a),F,(- c7)) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
integral ((- c7),n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(integral (F,n,f)) + (integral ((- c7),n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 - b is V11() real ext-real Element of REAL
n is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom n is V129() V130() V131() Element of K6(REAL)
n | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral (n,b,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
f is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(x0 - b) * f is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
dom (proj (F,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng n is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
(proj (F,a)) * n is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom ((proj (F,a)) * n) is V129() V130() V131() Element of K6(REAL)
(proj (F,a)) . f is V11() real ext-real Element of REAL
x is V11() real ext-real set
((proj (F,a)) * n) . x is V11() real ext-real Element of REAL
n . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
n /. x is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(proj (F,a)) . (n /. x) is V11() real ext-real Element of REAL
((proj (F,a)) * n) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (F,a)) * n),b,x0) is V11() real ext-real Element of REAL
((proj (F,a)) . f) * (x0 - b) is V11() real ext-real Element of REAL
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (F,a)) * n is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (integral (n,b,x0)) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (F,a)) * n is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (F,a)) * n) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (F,a)) * (n | ['b,x0']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
(integral (n,b,x0)) . F is V11() real ext-real Element of REAL
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (F,a)) * n is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (F,a)) * n),b,x0) is V11() real ext-real Element of REAL
(proj (F,a)) . f is V11() real ext-real Element of REAL
((proj (F,a)) . f) * (x0 - b) is V11() real ext-real Element of REAL
f . F is V11() real ext-real Element of REAL
(x0 - b) * (f . F) is V11() real ext-real Element of REAL
((x0 - b) * f) . F is V11() real ext-real Element of REAL
len ((x0 - b) * f) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom ((x0 - b) * f) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
f - n is V11() real ext-real Element of REAL
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
integral (F,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(f - n) * c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
dom (integral (F,n,f)) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
dom (proj (x,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng F is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
(proj (x,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom ((proj (x,a)) * F) is V129() V130() V131() Element of K6(REAL)
(proj (x,a)) . c7 is V11() real ext-real Element of REAL
R is V11() real ext-real set
((proj (x,a)) * F) . R is V11() real ext-real Element of REAL
F . R is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
F /. R is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(proj (x,a)) . (F /. R) is V11() real ext-real Element of REAL
integral (((proj (x,a)) * F),n,f) is V11() real ext-real Element of REAL
((proj (x,a)) . c7) * (f - n) is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(integral (F,n,f)) . x is V11() real ext-real Element of REAL
(proj (x,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * F),n,f) is V11() real ext-real Element of REAL
(proj (x,a)) . c7 is V11() real ext-real Element of REAL
((proj (x,a)) . c7) * (f - n) is V11() real ext-real Element of REAL
c7 . x is V11() real ext-real Element of REAL
(f - n) * (c7 . x) is V11() real ext-real Element of REAL
((f - n) * c7) . x is V11() real ext-real Element of REAL
len ((f - n) * c7) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom ((f - n) * c7) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom F is V129() V130() V131() Element of K6(REAL)
integral (F,b,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (F,b,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (F,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(integral (F,b,n)) + (integral (F,n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
c7 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (c7,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
(proj (c7,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(proj (c7,a)) * (F | ['b,x0']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (c7,a)) * F) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom (proj (c7,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng F is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom ((proj (c7,a)) * F) is V129() V130() V131() Element of K6(REAL)
integral (((proj (c7,a)) * F),b,f) is V11() real ext-real Element of REAL
integral (((proj (c7,a)) * F),b,n) is V11() real ext-real Element of REAL
integral (((proj (c7,a)) * F),n,f) is V11() real ext-real Element of REAL
(integral (((proj (c7,a)) * F),b,n)) + (integral (((proj (c7,a)) * F),n,f)) is V11() real ext-real Element of REAL
dom (integral (F,b,f)) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
c7 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
proj (c7,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(integral (F,b,f)) . c7 is V11() real ext-real Element of REAL
(proj (c7,a)) * F is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (c7,a)) * F),b,f) is V11() real ext-real Element of REAL
integral (((proj (c7,a)) * F),b,n) is V11() real ext-real Element of REAL
integral (((proj (c7,a)) * F),n,f) is V11() real ext-real Element of REAL
(integral (((proj (c7,a)) * F),b,n)) + (integral (((proj (c7,a)) * F),n,f)) is V11() real ext-real Element of REAL
(integral (F,b,n)) . c7 is V11() real ext-real Element of REAL
((integral (F,b,n)) . c7) + (integral (((proj (c7,a)) * F),n,f)) is V11() real ext-real Element of REAL
(integral (F,n,f)) . c7 is V11() real ext-real Element of REAL
((integral (F,b,n)) . c7) + ((integral (F,n,f)) . c7) is V11() real ext-real Element of REAL
((integral (F,b,n)) + (integral (F,n,f))) . c7 is V11() real ext-real Element of REAL
len ((integral (F,b,n)) + (integral (F,n,f))) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom ((integral (F,b,n)) + (integral (F,n,f))) is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
min (n,f) is V11() real ext-real set
max (n,f) is V11() real ext-real set
['(min (n,f)),(max (n,f))'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f - n is V11() real ext-real Element of REAL
abs (f - n) is V11() real ext-real Element of REAL
F is V11() real ext-real set
a * F is V11() real ext-real Element of REAL
(a * F) * (abs (f - n)) is V11() real ext-real Element of REAL
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
integral (c7,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.(integral (c7,n,f)).| is V11() real ext-real non negative Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
dom (proj (x,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng c7 is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
(proj (x,a)) * c7 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom ((proj (x,a)) * c7) is V129() V130() V131() Element of K6(REAL)
N is V11() real ext-real set
((proj (x,a)) * c7) . N is V11() real ext-real Element of REAL
|.(((proj (x,a)) * c7) . N).| is V11() real ext-real Element of REAL
c7 /. N is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.(c7 /. N).| is V11() real ext-real non negative Element of REAL
c7 . N is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (x,a)) . (c7 . N) is V11() real ext-real Element of REAL
(proj (x,a)) . (c7 /. N) is V11() real ext-real Element of REAL
|.((proj (x,a)) . (c7 /. N)).| is V11() real ext-real Element of REAL
(proj (x,a)) * (c7 | ['b,x0']) is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
((proj (x,a)) * c7) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * c7),n,f) is V11() real ext-real Element of REAL
|.(integral (((proj (x,a)) * c7),n,f)).| is V11() real ext-real Element of REAL
F * (abs (f - n)) is V11() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
(proj (x,a)) * c7 is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
integral (((proj (x,a)) * c7),n,f) is V11() real ext-real Element of REAL
|.(integral (((proj (x,a)) * c7),n,f)).| is V11() real ext-real Element of REAL
(integral (c7,n,f)) . x is V11() real ext-real Element of REAL
|.((integral (c7,n,f)) . x).| is V11() real ext-real Element of REAL
a * (F * (abs (f - n))) is V11() real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
n is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral (n,b,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (n,x0,b) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
- (integral (n,x0,b)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (n,x0,b)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
['x0,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
[.x0,b.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( x0 <= b1 & b1 <= b ) } is set
integral (n,['x0,b']) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
[.b,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( b <= b1 & b1 <= x0 ) } is set
integral (n,['b,x0']) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
- (integral (n,['b,x0'])) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
K38(1) (#) (integral (n,['b,x0'])) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
b is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
len b is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
x0 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
n is Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSequence-like FinSubsequence-like FinSequence of the carrier of (REAL-NS a)
Sum n is Element of the carrier of (REAL-NS a)
<*> the carrier of (REAL-NS a) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding NAT -defined the carrier of (REAL-NS a) -valued Function-like one-to-one constant functional V32() V33() V36() V37() V39( {} ) FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V129() V130() V131() V132() V133() V134() V135() V160() V161() V162() V163() V255() V256() V257() V258() V259() V260() V261() V262() V263() V264() V265() V266() bounded FinSequence of the carrier of (REAL-NS a)
len x0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
dom x0 is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
(Seg a) --> 0 is Relation-like Seg a -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K6(K7((Seg a),NAT))
K7((Seg a),NAT) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg a),NAT)) is set
dom ((Seg a) --> 0) is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
f is set
F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (F,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (F,a)) * b is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x0 . F is V11() real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum x is V11() real ext-real Element of REAL
x0 . f is V11() real ext-real Element of REAL
((Seg a) --> 0) . f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative Element of REAL
0* a is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
a |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of a -tuples_on REAL
(Seg a) --> 0 is Relation-like Seg a -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V261() Element of K6(K7((Seg a),{0}))
{0} is non empty V2() functional V32() V36() V39(1) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() V255() set
K7((Seg a),{0}) is Relation-like RAT -valued INT -valued V32() complex-valued ext-real-valued real-valued natural-valued set
K6(K7((Seg a),{0})) is V32() V36() set
0. (REAL-NS a) is V54( REAL-NS a) Element of the carrier of (REAL-NS a)
b is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
x0 is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
len x0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f is Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSequence-like FinSubsequence-like FinSequence of the carrier of (REAL-NS a)
n is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x0 | b is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
Seg b is V32() V39(b) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= b ) } is set
x0 | (Seg b) is Relation-like NAT -defined Seg b -defined NAT -defined REAL a -valued Function-like V32() FinSubsequence-like V261() set
Seg (b + 1) is non empty V32() V39(b + 1) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= b + 1 ) } is set
dom x0 is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
x0 . (b + 1) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
rng x0 is functional V32() FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
f | b is Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSequence-like FinSubsequence-like FinSequence of the carrier of (REAL-NS a)
f | (Seg b) is Relation-like NAT -defined Seg b -defined NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSubsequence-like set
dom f is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
f . (b + 1) is set
rng f is V32() Element of K6( the carrier of (REAL-NS a))
K6( the carrier of (REAL-NS a)) is set
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
proj (R,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (R,a)) * (x0 | b) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((proj (R,a)) * (x0 | b)) is V11() real ext-real Element of REAL
R is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom R is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
len R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
N is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (R,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (R,a)) * (x0 | b) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
N . R is V11() real ext-real Element of REAL
len (x0 | b) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
Sum (f | b) is Element of the carrier of (REAL-NS a)
len n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
N + c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
len (N + c7) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
proj (R,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (R,a)) * x0 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n . R is V11() real ext-real Element of REAL
e0 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum e0 is V11() real ext-real Element of REAL
(proj (R,a)) * (x0 | b) is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
N . R is V11() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum e is V11() real ext-real Element of REAL
dom (proj (R,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom e0 is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
Seg (len x0) is V32() V39( len x0) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len x0 ) } is set
(len (x0 | b)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
c7 . R is V11() real ext-real Element of REAL
<*(c7 . R)*> is non empty V2() Relation-like NAT -defined REAL -valued Function-like one-to-one constant V32() V39(1) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing bounded FinSequence of REAL
[1,(c7 . R)] is set
{1,(c7 . R)} is non empty V32() V129() V130() V131() V158() V159() V160() V161() V162() set
{{1,(c7 . R)},{1}} is non empty V32() V36() set
{[1,(c7 . R)]} is non empty V2() Relation-like Function-like constant V32() V39(1) set
len <*(c7 . R)*> is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
(len (x0 | b)) + (len <*(c7 . R)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
rng (x0 | b) is functional V32() FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom e is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
dom (x0 | b) is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
Seg (len (x0 | b)) is V32() V39( len (x0 | b)) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len (x0 | b) ) } is set
len e is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
(len e) + (len <*(c7 . R)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
Seg ((len e) + (len <*(c7 . R)*>)) is non empty V32() V39((len e) + (len <*(c7 . R)*>)) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= (len e) + (len <*(c7 . R)*>) ) } is set
p is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
e0 . p is V11() real ext-real Element of REAL
e . p is V11() real ext-real Element of REAL
(x0 | b) . p is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (R,a)) . ((x0 | b) . p) is V11() real ext-real Element of REAL
x0 . p is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (R,a)) . (x0 . p) is V11() real ext-real Element of REAL
dom <*(c7 . R)*> is non empty V2() V32() V39(1) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
p is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
(len e) + p is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
e0 . ((len e) + p) is V11() real ext-real Element of REAL
<*(c7 . R)*> . p is V11() real ext-real Element of REAL
Seg (len <*(c7 . R)*>) is non empty V32() V39( len <*(c7 . R)*>) V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= len <*(c7 . R)*> ) } is set
e0 . (b + 1) is V11() real ext-real Element of REAL
(proj (R,a)) . (x0 . (b + 1)) is V11() real ext-real Element of REAL
e ^ <*(c7 . R)*> is non empty Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(N . R) + (c7 . R) is V11() real ext-real Element of REAL
(N + c7) . R is V11() real ext-real Element of REAL
len f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
len (f | b) is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
(len (f | b)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
dom (f | b) is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
f | (dom (f | b)) is Relation-like NAT -defined dom (f | b) -defined NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSubsequence-like Element of K6(K7(NAT, the carrier of (REAL-NS a)))
K7(NAT, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS a))) is V2() V32() set
R is Element of the carrier of (REAL-NS a)
(Sum (f | b)) + R is Element of the carrier of (REAL-NS a)
Sum f is Element of the carrier of (REAL-NS a)
x0 is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
len x0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f is Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSequence-like FinSubsequence-like FinSequence of the carrier of (REAL-NS a)
n is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
Sum f is Element of the carrier of (REAL-NS a)
b is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
x0 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
n is Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSequence-like FinSubsequence-like FinSequence of the carrier of (REAL-NS a)
Sum n is Element of the carrier of (REAL-NS a)
len b is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
b is non empty set
a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() RealNormSpace-like NORMSTR
the carrier of a is non empty set
K7(b, the carrier of a) is Relation-like set
K6(K7(b, the carrier of a)) is set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x0 is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (n,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (n,a)) * b is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom ((proj (n,a)) * b) is V129() V130() V131() Element of K6(REAL)
F is V11() real ext-real set
c7 is V11() real ext-real Element of REAL
x is set
((proj (n,a)) * b) . x is V11() real ext-real Element of REAL
abs (((proj (n,a)) * b) . x) is V11() real ext-real Element of REAL
n is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom n is V32() V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
max n is V11() real ext-real Element of REAL
dom x0 is V129() V130() V131() Element of K6(REAL)
a * (max n) is V11() real ext-real Element of REAL
(a * (max n)) + 1 is V11() real ext-real Element of REAL
F is set
x0 /. F is Element of the carrier of (REAL-NS a)
||.(x0 /. F).|| is V11() real ext-real Element of REAL
b /. F is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
dom (proj (x,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng b is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
(proj (x,a)) * b is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom ((proj (x,a)) * b) is V129() V130() V131() Element of K6(REAL)
n /. x is V11() real ext-real Element of REAL
((proj (x,a)) * b) . F is V11() real ext-real Element of REAL
abs (((proj (x,a)) * b) . F) is V11() real ext-real Element of REAL
b . F is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (x,a)) . (b /. F) is V11() real ext-real Element of REAL
(b /. F) . x is V11() real ext-real Element of REAL
abs ((b /. F) . x) is V11() real ext-real Element of REAL
len n is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
n . x is V11() real ext-real Element of REAL
|.(b /. F).| is V11() real ext-real non negative Element of REAL
x0 . F is set
dom x0 is V129() V130() V131() Element of K6(REAL)
n is V11() real ext-real set
f is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
proj (f,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (f,a)) * b is Relation-like REAL -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
dom ((proj (f,a)) * b) is V129() V130() V131() Element of K6(REAL)
c7 is set
((proj (f,a)) * b) . c7 is V11() real ext-real Element of REAL
abs (((proj (f,a)) * b) . c7) is V11() real ext-real Element of REAL
dom (proj (f,a)) is non empty functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
rng b is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
dom b is V129() V130() V131() Element of K6(REAL)
x0 /. c7 is Element of the carrier of (REAL-NS a)
||.(x0 /. c7).|| is V11() real ext-real Element of REAL
b /. c7 is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x0 . c7 is set
|.(b /. c7).| is V11() real ext-real non negative Element of REAL
(b /. c7) . f is V11() real ext-real Element of REAL
abs ((b /. c7) . f) is V11() real ext-real Element of REAL
b . c7 is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(proj (f,a)) . (b /. c7) is V11() real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is set
x0 is set
b /\ x0 is set
n is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
n | b is Relation-like REAL -defined b -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
f is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
f | x0 is Relation-like REAL -defined x0 -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
n + f is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
(n + f) | (b /\ x0) is Relation-like REAL -defined b /\ x0 -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
n - f is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
(n - f) | (b /\ x0) is Relation-like REAL -defined b /\ x0 -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom (n | b) is Element of K6(b)
K6(b) is set
F is V11() real ext-real set
dom (f | x0) is Element of K6(x0)
K6(x0) is set
c7 is V11() real ext-real set
x is set
dom ((n + f) | (b /\ x0)) is Element of K6((b /\ x0))
K6((b /\ x0)) is set
dom (n + f) is V129() V130() V131() Element of K6(REAL)
dom n is V129() V130() V131() Element of K6(REAL)
dom f is V129() V130() V131() Element of K6(REAL)
(dom n) /\ (dom f) is V129() V130() V131() Element of K6(REAL)
((n + f) | (b /\ x0)) /. x is Element of the carrier of (REAL-NS a)
(n + f) /. x is Element of the carrier of (REAL-NS a)
n /. x is Element of the carrier of (REAL-NS a)
f /. x is Element of the carrier of (REAL-NS a)
(n /. x) + (f /. x) is Element of the carrier of (REAL-NS a)
(n | b) /. x is Element of the carrier of (REAL-NS a)
((n | b) /. x) + (f /. x) is Element of the carrier of (REAL-NS a)
(f | x0) /. x is Element of the carrier of (REAL-NS a)
((n | b) /. x) + ((f | x0) /. x) is Element of the carrier of (REAL-NS a)
||.(((n + f) | (b /\ x0)) /. x).|| is V11() real ext-real Element of REAL
||.((n | b) /. x).|| is V11() real ext-real Element of REAL
||.((f | x0) /. x).|| is V11() real ext-real Element of REAL
||.((n | b) /. x).|| + ||.((f | x0) /. x).|| is V11() real ext-real Element of REAL
F + c7 is V11() real ext-real Element of REAL
dom ((n - f) | (b /\ x0)) is V129() V130() V131() Element of K6(REAL)
x is set
((n - f) | (b /\ x0)) /. x is Element of the carrier of (REAL-NS a)
||.(((n - f) | (b /\ x0)) /. x).|| is V11() real ext-real Element of REAL
dom ((n - f) | (b /\ x0)) is Element of K6((b /\ x0))
dom (n - f) is V129() V130() V131() Element of K6(REAL)
(n - f) /. x is Element of the carrier of (REAL-NS a)
n /. x is Element of the carrier of (REAL-NS a)
f /. x is Element of the carrier of (REAL-NS a)
(n /. x) - (f /. x) is Element of the carrier of (REAL-NS a)
(n | b) /. x is Element of the carrier of (REAL-NS a)
((n | b) /. x) - (f /. x) is Element of the carrier of (REAL-NS a)
(f | x0) /. x is Element of the carrier of (REAL-NS a)
((n | b) /. x) - ((f | x0) /. x) is Element of the carrier of (REAL-NS a)
||.((n | b) /. x).|| is V11() real ext-real Element of REAL
||.((f | x0) /. x).|| is V11() real ext-real Element of REAL
||.((n | b) /. x).|| + ||.((f | x0) /. x).|| is V11() real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
x0 is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
n is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
f is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of b
F is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() FinSequence of REAL a
c7 is Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSequence-like FinSubsequence-like FinSequence of the carrier of (REAL-NS a)
len c7 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
len f is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
dom f is non empty V32() V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
divset (f,x) is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n | (divset (f,x)) is Relation-like b -defined divset (f,x) -defined b -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(b, the carrier of (REAL-NS a)))
rng (n | (divset (f,x))) is Element of K6( the carrier of (REAL-NS a))
K6( the carrier of (REAL-NS a)) is set
c7 . x is set
vol (divset (f,x)) is V11() real ext-real Element of REAL
x0 | (divset (f,x)) is Relation-like b -defined divset (f,x) -defined b -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(b,(REAL a)))
rng (x0 | (divset (f,x))) is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
F . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(vol (divset (f,x))) * R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
R is Element of the carrier of (REAL-NS a)
(vol (divset (f,x))) * R is Element of the carrier of (REAL-NS a)
len F is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
len f is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
dom f is non empty V32() V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of K6(NAT)
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() ext-real non negative set
divset (f,x) is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 | (divset (f,x)) is Relation-like b -defined divset (f,x) -defined b -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(b,(REAL a)))
rng (x0 | (divset (f,x))) is functional FinSequence-membered V255() V256() V257() Element of K6((REAL a))
K6((REAL a)) is set
F . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
vol (divset (f,x)) is V11() real ext-real Element of REAL
n | (divset (f,x)) is Relation-like b -defined divset (f,x) -defined b -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(b, the carrier of (REAL-NS a)))
rng (n | (divset (f,x))) is Element of K6( the carrier of (REAL-NS a))
K6( the carrier of (REAL-NS a)) is set
c7 . x is set
R is Element of the carrier of (REAL-NS a)
(vol (divset (f,x))) * R is Element of the carrier of (REAL-NS a)
R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(vol (divset (f,x))) * R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
x0 is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
n is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
f is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of b
F is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() middle_volume of x0,f
middle_sum (x0,F) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
c7 is Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSequence-like FinSubsequence-like middle_volume of n,f
middle_sum (n,c7) is Element of the carrier of (REAL-NS a)
Sum c7 is Element of the carrier of (REAL-NS a)
Seg a is V32() V39(a) V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of K6(NAT)
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT : ( 1 <= b1 & b1 <= a ) } is set
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
proj (x,a) is non empty Relation-like REAL a -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL a),REAL))
K7((REAL a),REAL) is V2() Relation-like V32() complex-valued ext-real-valued real-valued set
K6(K7((REAL a),REAL)) is V2() V32() set
(proj (x,a)) * F is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(middle_sum (x0,F)) . x is V11() real ext-real Element of REAL
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
(REAL a) * is non empty functional FinSequence-membered FinSequenceSet of REAL a
K7(NAT,((REAL a) *)) is V2() Relation-like V32() set
K6(K7(NAT,((REAL a) *))) is V2() V32() set
the carrier of (REAL-NS a) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (REAL-NS a)
K7(NAT,( the carrier of (REAL-NS a) *)) is V2() Relation-like V32() set
K6(K7(NAT,( the carrier of (REAL-NS a) *))) is V2() V32() set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
divs b is non empty set
K7(NAT,(divs b)) is V2() Relation-like V32() set
K6(K7(NAT,(divs b))) is V2() V32() set
x0 is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
n is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
f is non empty Relation-like NAT -defined divs b -valued Function-like total quasi_total Element of K6(K7(NAT,(divs b)))
F is non empty Relation-like NAT -defined (REAL a) * -valued Function-like total quasi_total Element of K6(K7(NAT,((REAL a) *)))
c7 is non empty Relation-like NAT -defined the carrier of (REAL-NS a) * -valued Function-like total quasi_total Element of K6(K7(NAT,( the carrier of (REAL-NS a) *)))
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
c7 . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like Element of the carrier of (REAL-NS a) *
f . x is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of b
F . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like Element of (REAL a) *
x is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
F . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like Element of (REAL a) *
f . x is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of b
c7 . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like Element of the carrier of (REAL-NS a) *
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
(REAL a) * is non empty functional FinSequence-membered FinSequenceSet of REAL a
the carrier of (REAL-NS a) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (REAL-NS a)
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
divs b is non empty set
K7(NAT,(divs b)) is V2() Relation-like V32() set
K6(K7(NAT,(divs b))) is V2() V32() set
x0 is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
n is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
f is non empty Relation-like NAT -defined divs b -valued Function-like total quasi_total Element of K6(K7(NAT,(divs b)))
F is non empty Relation-like NAT -defined (REAL a) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f
middle_sum (x0,F) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
K7(NAT, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS a))) is V2() V32() set
c7 is non empty Relation-like NAT -defined the carrier of (REAL-NS a) * -valued Function-like total quasi_total middle_volume_Sequence of n,f
middle_sum (n,c7) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
dom (middle_sum (x0,F)) is non empty V129() V130() V131() V132() V133() V134() V158() V160() Element of K6(NAT)
dom (middle_sum (n,c7)) is non empty V129() V130() V131() V132() V133() V134() V158() V160() Element of K6(NAT)
x is set
R is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
f . R is non empty Relation-like NAT -defined REAL -valued Function-like one-to-one V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of b
F . R is Relation-like NAT -defined REAL a -valued Function-like V32() FinSequence-like FinSubsequence-like V261() V262() V263() middle_volume of x0,f . R
middle_sum (x0,(F . R)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
c7 . R is Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like V32() FinSequence-like FinSubsequence-like middle_volume of n,f . R
middle_sum (n,(c7 . R)) is Element of the carrier of (REAL-NS a)
Sum (c7 . R) is Element of the carrier of (REAL-NS a)
(middle_sum (x0,F)) . x is set
(middle_sum (n,c7)) . x is set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
(REAL a) * is non empty functional FinSequence-membered FinSequenceSet of REAL a
the carrier of (REAL-NS a) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (REAL-NS a)
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
divs b is non empty set
K7(NAT,(divs b)) is V2() Relation-like V32() set
K6(K7(NAT,(divs b))) is V2() V32() set
x0 is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
n is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
f is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
F is Element of the carrier of (REAL-NS a)
c7 is non empty Relation-like NAT -defined divs b -valued Function-like total quasi_total Element of K6(K7(NAT,(divs b)))
delta c7 is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim (delta c7) is V11() real ext-real Element of REAL
x is non empty Relation-like NAT -defined the carrier of (REAL-NS a) * -valued Function-like total quasi_total middle_volume_Sequence of n,c7
middle_sum (n,x) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
K7(NAT, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS a))) is V2() V32() set
lim (middle_sum (n,x)) is Element of the carrier of (REAL-NS a)
R is non empty Relation-like NAT -defined (REAL a) * -valued Function-like total quasi_total middle_volume_Sequence of x0,c7
middle_sum (x0,R) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
lim (middle_sum (x0,R)) is Element of the carrier of (REAL-NS a)
c7 is non empty Relation-like NAT -defined divs b -valued Function-like total quasi_total Element of K6(K7(NAT,(divs b)))
delta c7 is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim (delta c7) is V11() real ext-real Element of REAL
x is non empty Relation-like NAT -defined (REAL a) * -valued Function-like total quasi_total middle_volume_Sequence of x0,c7
middle_sum (x0,x) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
K7(NAT, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS a))) is V2() V32() set
lim (middle_sum (x0,x)) is Element of the carrier of (REAL-NS a)
R is non empty Relation-like NAT -defined the carrier of (REAL-NS a) * -valued Function-like total quasi_total middle_volume_Sequence of n,c7
middle_sum (n,R) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
lim (middle_sum (n,R)) is Element of the carrier of (REAL-NS a)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
x0 is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
n is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
divs b is non empty set
K7(NAT,(divs b)) is V2() Relation-like V32() set
K6(K7(NAT,(divs b))) is V2() V32() set
(REAL a) * is non empty functional FinSequence-membered FinSequenceSet of REAL a
f is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
F is Element of the carrier of (REAL-NS a)
the carrier of (REAL-NS a) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (REAL-NS a)
c7 is non empty Relation-like NAT -defined divs b -valued Function-like total quasi_total Element of K6(K7(NAT,(divs b)))
delta c7 is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim (delta c7) is V11() real ext-real Element of REAL
x is non empty Relation-like NAT -defined the carrier of (REAL-NS a) * -valued Function-like total quasi_total middle_volume_Sequence of n,c7
middle_sum (n,x) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
K7(NAT, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS a))) is V2() V32() set
lim (middle_sum (n,x)) is Element of the carrier of (REAL-NS a)
f is Element of the carrier of (REAL-NS a)
F is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
c7 is non empty Relation-like NAT -defined divs b -valued Function-like total quasi_total Element of K6(K7(NAT,(divs b)))
delta c7 is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim (delta c7) is V11() real ext-real Element of REAL
x is non empty Relation-like NAT -defined (REAL a) * -valued Function-like total quasi_total middle_volume_Sequence of x0,c7
middle_sum (x0,x) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
K7(NAT, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS a))) is V2() V32() set
lim (middle_sum (x0,x)) is Element of the carrier of (REAL-NS a)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
x0 is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
integral x0 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
n is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
integral n is Element of the carrier of (REAL-NS a)
divs b is non empty set
K7(NAT,(divs b)) is V2() Relation-like V32() set
K6(K7(NAT,(divs b))) is V2() V32() set
(REAL a) * is non empty functional FinSequence-membered FinSequenceSet of REAL a
f is non empty Relation-like NAT -defined divs b -valued Function-like total quasi_total Element of K6(K7(NAT,(divs b)))
delta f is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim (delta f) is V11() real ext-real Element of REAL
F is non empty Relation-like NAT -defined (REAL a) * -valued Function-like total quasi_total middle_volume_Sequence of x0,f
middle_sum (x0,F) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
K7(NAT, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS a))) is V2() V32() set
lim (middle_sum (x0,F)) is Element of the carrier of (REAL-NS a)
f is Element of the carrier of (REAL-NS a)
the carrier of (REAL-NS a) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (REAL-NS a)
F is non empty Relation-like NAT -defined divs b -valued Function-like total quasi_total Element of K6(K7(NAT,(divs b)))
delta F is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
lim (delta F) is V11() real ext-real Element of REAL
c7 is non empty Relation-like NAT -defined the carrier of (REAL-NS a) * -valued Function-like total quasi_total middle_volume_Sequence of n,F
middle_sum (n,c7) is non empty Relation-like NAT -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS a)))
K7(NAT, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS a))) is V2() V32() set
lim (middle_sum (n,c7)) is Element of the carrier of (REAL-NS a)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x0 | b is Relation-like REAL -defined b -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom x0 is V129() V130() V131() Element of K6(REAL)
n is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
f is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
F is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x0 | b is Relation-like REAL -defined b -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom x0 is V129() V130() V131() Element of K6(REAL)
integral (x0,b) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
n is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral (n,b) is Element of the carrier of (REAL-NS a)
K7(b,(REAL a)) is Relation-like set
K6(K7(b,(REAL a))) is set
K7(b, the carrier of (REAL-NS a)) is Relation-like set
K6(K7(b, the carrier of (REAL-NS a))) is set
f is non empty Relation-like b -defined REAL a -valued Function-like total quasi_total V261() V262() V263() Element of K6(K7(b,(REAL a)))
integral f is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
F is non empty Relation-like b -defined the carrier of (REAL-NS a) -valued Function-like total quasi_total Element of K6(K7(b, the carrier of (REAL-NS a)))
integral F is Element of the carrier of (REAL-NS a)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
n | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
dom n is V129() V130() V131() Element of K6(REAL)
integral (n,b,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
f is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral (f,b,x0) is Element of the carrier of (REAL-NS a)
[.b,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( b <= b1 & b1 <= x0 ) } is set
integral (f,['b,x0']) is Element of the carrier of (REAL-NS a)
integral (n,['b,x0']) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
f is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom n is V129() V130() V131() Element of K6(REAL)
dom f is V129() V130() V131() Element of K6(REAL)
n + f is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral ((n + f),b,x0) is Element of the carrier of (REAL-NS a)
integral (n,b,x0) is Element of the carrier of (REAL-NS a)
integral (f,b,x0) is Element of the carrier of (REAL-NS a)
(integral (n,b,x0)) + (integral (f,b,x0)) is Element of the carrier of (REAL-NS a)
n - f is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral ((n - f),b,x0) is Element of the carrier of (REAL-NS a)
(integral (n,b,x0)) - (integral (f,b,x0)) is Element of the carrier of (REAL-NS a)
integral ((n + f),['b,x0']) is Element of the carrier of (REAL-NS a)
integral (n,['b,x0']) is Element of the carrier of (REAL-NS a)
integral (f,['b,x0']) is Element of the carrier of (REAL-NS a)
(integral (n,['b,x0'])) + (integral (f,['b,x0'])) is Element of the carrier of (REAL-NS a)
integral ((n - f),['b,x0']) is Element of the carrier of (REAL-NS a)
(integral (n,['b,x0'])) - (integral (f,['b,x0'])) is Element of the carrier of (REAL-NS a)
[.b,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( b <= b1 & b1 <= x0 ) } is set
(integral (n,b,x0)) + (integral (f,['b,x0'])) is Element of the carrier of (REAL-NS a)
(integral (n,b,x0)) - (integral (f,['b,x0'])) is Element of the carrier of (REAL-NS a)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom n is V129() V130() V131() Element of K6(REAL)
integral (n,x0,b) is Element of the carrier of (REAL-NS a)
integral (n,b,x0) is Element of the carrier of (REAL-NS a)
- (integral (n,b,x0)) is Element of the carrier of (REAL-NS a)
[.b,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( b <= b1 & b1 <= x0 ) } is set
integral (n,['b,x0']) is Element of the carrier of (REAL-NS a)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom F is V129() V130() V131() Element of K6(REAL)
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral (F,n,f) is Element of the carrier of (REAL-NS a)
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral (c7,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
[.b,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( b <= b1 & b1 <= x0 ) } is set
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
['n,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
dom c7 is V129() V130() V131() Element of K6(REAL)
c7 | ['n,f'] is Relation-like REAL -defined ['n,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
['f,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
dom c7 is V129() V130() V131() Element of K6(REAL)
c7 | ['f,n'] is Relation-like REAL -defined ['f,n'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral (F,f,n) is Element of the carrier of (REAL-NS a)
integral (c7,f,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
- (integral (c7,f,n)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (c7,f,n)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- (integral (F,f,n)) is Element of the carrier of (REAL-NS a)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
c7 is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom F is V129() V130() V131() Element of K6(REAL)
dom c7 is V129() V130() V131() Element of K6(REAL)
F + c7 is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral ((F + c7),n,f) is Element of the carrier of (REAL-NS a)
integral (F,n,f) is Element of the carrier of (REAL-NS a)
integral (c7,n,f) is Element of the carrier of (REAL-NS a)
(integral (F,n,f)) + (integral (c7,n,f)) is Element of the carrier of (REAL-NS a)
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
x is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
R is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
R | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x + R is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K771(REAL,REAL,(REAL a),(REAL a),x,R) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
integral (x,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (R,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
dom (F + c7) is V129() V130() V131() Element of K6(REAL)
(x + R) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
(F + c7) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral ((x + R),n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(integral (x,n,f)) + (integral (R,n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
c7 is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom F is V129() V130() V131() Element of K6(REAL)
dom c7 is V129() V130() V131() Element of K6(REAL)
F - c7 is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral ((F - c7),n,f) is Element of the carrier of (REAL-NS a)
integral (F,n,f) is Element of the carrier of (REAL-NS a)
integral (c7,n,f) is Element of the carrier of (REAL-NS a)
(integral (F,n,f)) - (integral (c7,n,f)) is Element of the carrier of (REAL-NS a)
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
x is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
R is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
R | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x - R is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
K777(REAL,REAL,(REAL a),(REAL a),x,R) is Relation-like REAL /\ REAL -defined K703((K698((REAL a)) /\ K698((REAL a)))) -valued Function-like V261() V262() V263() Element of K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))))
REAL /\ REAL is V129() V130() V131() V163() set
K698((REAL a)) is set
K698((REAL a)) /\ K698((REAL a)) is set
K703((K698((REAL a)) /\ K698((REAL a)))) is functional V255() V256() V257() set
K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a))))) is Relation-like set
K6(K7((REAL /\ REAL),K703((K698((REAL a)) /\ K698((REAL a)))))) is set
integral (x,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (R,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
dom (F - c7) is V129() V130() V131() Element of K6(REAL)
(x - R) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
(F - c7) | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral ((x - R),n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(integral (x,n,f)) - (integral (R,n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
- (integral (R,n,f)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (R,n,f)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(integral (x,n,f)) + (- (integral (R,n,f))) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x0 - b is V11() real ext-real Element of REAL
n is Element of the carrier of (REAL-NS a)
(x0 - b) * n is Element of the carrier of (REAL-NS a)
f is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom f is V129() V130() V131() Element of K6(REAL)
f | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
integral (f,b,x0) is Element of the carrier of (REAL-NS a)
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
F is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x is V11() real ext-real set
F . x is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral (F,b,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(x0 - b) * c7 is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
f - n is V11() real ext-real Element of REAL
F is Element of the carrier of (REAL-NS a)
(f - n) * F is Element of the carrier of (REAL-NS a)
c7 is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
integral (c7,n,f) is Element of the carrier of (REAL-NS a)
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
x is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
R is V11() real ext-real set
x . R is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
x | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral (x,b,x0) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
x0 - b is V11() real ext-real Element of REAL
(x0 - b) * R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (x,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(f - n) * R is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
[.b,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( b <= b1 & b1 <= x0 ) } is set
['n,f'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
dom x is V129() V130() V131() Element of K6(REAL)
x | ['n,f'] is Relation-like REAL -defined ['n,f'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
['f,n'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
dom x is V129() V130() V131() Element of K6(REAL)
x | ['f,n'] is Relation-like REAL -defined ['f,n'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral (x,f,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (c7,f,n) is Element of the carrier of (REAL-NS a)
- (integral (x,f,n)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
K38(1) is V11() real ext-real non positive set
K38(1) (#) (integral (x,f,n)) is Relation-like NAT -defined Function-like V32() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
- (integral (c7,f,n)) is Element of the carrier of (REAL-NS a)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
F is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
F | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom F is V129() V130() V131() Element of K6(REAL)
integral (F,b,f) is Element of the carrier of (REAL-NS a)
integral (F,b,n) is Element of the carrier of (REAL-NS a)
integral (F,n,f) is Element of the carrier of (REAL-NS a)
(integral (F,b,n)) + (integral (F,n,f)) is Element of the carrier of (REAL-NS a)
[.b,x0.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( b <= b1 & b1 <= x0 ) } is set
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
c7 is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
integral (c7,b,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (c7,b,n) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
integral (c7,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
(integral (c7,b,n)) + (integral (c7,n,f)) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
['b,x0'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
n is V11() real ext-real set
f is V11() real ext-real set
min (n,f) is V11() real ext-real set
max (n,f) is V11() real ext-real set
['(min (n,f)),(max (n,f))'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
f - n is V11() real ext-real Element of REAL
abs (f - n) is V11() real ext-real Element of REAL
F is V11() real ext-real set
a * F is V11() real ext-real Element of REAL
(a * F) * (abs (f - n)) is V11() real ext-real Element of REAL
c7 is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
c7 | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom c7 is V129() V130() V131() Element of K6(REAL)
integral (c7,n,f) is Element of the carrier of (REAL-NS a)
||.(integral (c7,n,f)).|| is V11() real ext-real Element of REAL
REAL a is non empty functional FinSequence-membered V255() V256() V257() FinSequenceSet of REAL
a -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like V32() FinSequence-like FinSubsequence-like Element of REAL * : len b1 = a } is set
K7(REAL,(REAL a)) is V2() Relation-like V32() set
K6(K7(REAL,(REAL a))) is V2() V32() set
x is Relation-like REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
x | ['b,x0'] is Relation-like REAL -defined ['b,x0'] -defined REAL -defined REAL a -valued Function-like V261() V262() V263() Element of K6(K7(REAL,(REAL a)))
R is V11() real ext-real set
x /. R is Relation-like NAT -defined Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.(x /. R).| is V11() real ext-real non negative Element of REAL
c7 /. R is Element of the carrier of (REAL-NS a)
||.(c7 /. R).|| is V11() real ext-real Element of REAL
c7 . R is set
integral (x,n,f) is Relation-like NAT -defined REAL -valued Function-like V32() V39(a) FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL a
|.(integral (x,n,f)).| is V11() real ext-real non negative Element of REAL
a is V11() real ext-real set
a " is V11() real ext-real Element of REAL
abs a is V11() real ext-real Element of REAL
b is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() RealNormSpace-like NORMSTR
the carrier of b is non empty set
x0 is Element of the carrier of b
(a ") * x0 is Element of the carrier of b
||.((a ") * x0).|| is V11() real ext-real Element of REAL
||.x0.|| is V11() real ext-real Element of REAL
||.x0.|| / (abs a) is V11() real ext-real Element of REAL
abs (a ") is V11() real ext-real Element of REAL
(abs (a ")) * ||.x0.|| is V11() real ext-real Element of REAL
(abs a) " is V11() real ext-real Element of REAL
((abs a) ") * ||.x0.|| is V11() real ext-real Element of REAL
1 / (abs a) is V11() real ext-real Element of REAL
(1 / (abs a)) * ||.x0.|| is V11() real ext-real Element of REAL
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
].a,b.[ is V129() V130() V131() V158() V159() V163() open Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( not b1 <= a & not b <= b1 ) } is set
x0 is V11() real ext-real set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL-NS n is non empty V52() V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS n) is non empty V2() set
K7(REAL, the carrier of (REAL-NS n)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS n))) is V2() V32() set
F is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
F | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom F is V129() V130() V131() Element of K6(REAL)
f is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom f is V129() V130() V131() Element of K6(REAL)
diff (f,x0) is Element of the carrier of (REAL-NS n)
F /. x0 is Element of the carrier of (REAL-NS n)
0. (REAL-NS n) is V54( REAL-NS n) Element of the carrier of (REAL-NS n)
c7 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS n)))
f /. x0 is Element of the carrier of (REAL-NS n)
x is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
R is Relation-like Function-like set
dom R is set
rng R is set
R is set
N is set
R . N is set
R_id N is V11() real ext-real Element of REAL
x0 + (R_id N) is V11() real ext-real Element of REAL
f /. (x0 + (R_id N)) is Element of the carrier of (REAL-NS n)
(f /. (x0 + (R_id N))) - (f /. x0) is Element of the carrier of (REAL-NS n)
x . (R_id N) is Element of the carrier of (REAL-NS n)
((f /. (x0 + (R_id N))) - (f /. x0)) - (x . (R_id N)) is Element of the carrier of (REAL-NS n)
R is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
[.a,b.] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( a <= b1 & b1 <= b ) } is set
N is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
R /* N is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
K7(NAT, the carrier of (REAL-NS n)) is V2() Relation-like V32() set
K6(K7(NAT, the carrier of (REAL-NS n))) is V2() V32() set
N " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(N ") (#) (R /* N) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
lim ((N ") (#) (R /* N)) is Element of the carrier of (REAL-NS n)
REAL --> (F /. x0) is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS n)))
e0 is V11() real ext-real Element of REAL
e0 / 2 is V11() real ext-real Element of REAL
(e0 / 2) / n is V11() real ext-real Element of REAL
lim N is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V32() V33() V36() V37() V39( {} ) FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V129() V130() V131() V132() V133() V134() V135() V160() V161() V162() V163() V255() V256() V257() V258() V259() V260() V261() V262() V263() V264() V265() V266() bounded Element of REAL
p is V11() real ext-real set
p / 2 is V11() real ext-real Element of REAL
N is V129() V130() V131() open Neighbourhood of x0
q is V11() real ext-real set
x0 - q is V11() real ext-real Element of REAL
x0 + q is V11() real ext-real Element of REAL
].(x0 - q),(x0 + q).[ is V129() V130() V131() V158() V159() V163() open Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( not b1 <= x0 - q & not x0 + q <= b1 ) } is set
q / 2 is V11() real ext-real Element of REAL
min ((p / 2),(q / 2)) is V11() real ext-real set
n0 is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
((N ") (#) (R /* N)) . m is Element of the carrier of (REAL-NS n)
(((N ") (#) (R /* N)) . m) - (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
||.((((N ") (#) (R /* N)) . m) - (0. (REAL-NS n))).|| is V11() real ext-real Element of REAL
x0 - (min ((p / 2),(q / 2))) is V11() real ext-real Element of REAL
x0 + (min ((p / 2),(q / 2))) is V11() real ext-real Element of REAL
].(x0 - (min ((p / 2),(q / 2)))),(x0 + (min ((p / 2),(q / 2)))).[ is V129() V130() V131() V158() V159() V163() open Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( not b1 <= x0 - (min ((p / 2),(q / 2))) & not x0 + (min ((p / 2),(q / 2))) <= b1 ) } is set
N . m is V11() real ext-real Element of REAL
(N . m) - 0 is V11() real ext-real Element of REAL
abs ((N . m) - 0) is V11() real ext-real Element of REAL
x0 + (N . m) is V11() real ext-real Element of REAL
(x0 + (N . m)) - x0 is V11() real ext-real Element of REAL
abs ((x0 + (N . m)) - x0) is V11() real ext-real Element of REAL
R_id (N . m) is V11() real ext-real Element of REAL
x0 + (R_id (N . m)) is V11() real ext-real Element of REAL
R . (N . m) is set
f /. (x0 + (R_id (N . m))) is Element of the carrier of (REAL-NS n)
(f /. (x0 + (R_id (N . m)))) - (f /. x0) is Element of the carrier of (REAL-NS n)
x . (R_id (N . m)) is Element of the carrier of (REAL-NS n)
((f /. (x0 + (R_id (N . m)))) - (f /. x0)) - (x . (R_id (N . m))) is Element of the carrier of (REAL-NS n)
f /. (x0 + (N . m)) is Element of the carrier of (REAL-NS n)
(f /. (x0 + (N . m))) - (f /. x0) is Element of the carrier of (REAL-NS n)
((f /. (x0 + (N . m))) - (f /. x0)) - (x . (R_id (N . m))) is Element of the carrier of (REAL-NS n)
x . (N . m) is Element of the carrier of (REAL-NS n)
((f /. (x0 + (N . m))) - (f /. x0)) - (x . (N . m)) is Element of the carrier of (REAL-NS n)
f . x0 is set
integral (F,a,x0) is Element of the carrier of (REAL-NS n)
f . (x0 + (N . m)) is set
integral (F,a,(x0 + (N . m))) is Element of the carrier of (REAL-NS n)
(integral (F,a,(x0 + (N . m)))) - (integral (F,a,x0)) is Element of the carrier of (REAL-NS n)
(N . m) * (F /. x0) is Element of the carrier of (REAL-NS n)
((integral (F,a,(x0 + (N . m)))) - (integral (F,a,x0))) - ((N . m) * (F /. x0)) is Element of the carrier of (REAL-NS n)
dom (REAL --> (F /. x0)) is non empty V129() V130() V131() Element of K6(REAL)
['a,b'] /\ ['a,b'] is V129() V130() V131() V163() Element of K6(REAL)
(dom F) /\ (dom (REAL --> (F /. x0))) is V129() V130() V131() Element of K6(REAL)
F - (REAL --> (F /. x0)) is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom (F - (REAL --> (F /. x0))) is V129() V130() V131() Element of K6(REAL)
integral (F,x0,(x0 + (N . m))) is Element of the carrier of (REAL-NS n)
(integral (F,a,x0)) + (integral (F,x0,(x0 + (N . m)))) is Element of the carrier of (REAL-NS n)
min (x0,(x0 + (N . m))) is V11() real ext-real set
max (x0,(x0 + (N . m))) is V11() real ext-real set
['(min (x0,(x0 + (N . m)))),(max (x0,(x0 + (N . m))))'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
x is V11() real ext-real set
(F - (REAL --> (F /. x0))) /. x is Element of the carrier of (REAL-NS n)
||.((F - (REAL --> (F /. x0))) /. x).|| is V11() real ext-real Element of REAL
[.(min (x0,(x0 + (N . m)))),(max (x0,(x0 + (N . m)))).] is V129() V130() V131() V163() closed Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( min (x0,(x0 + (N . m))) <= b1 & b1 <= max (x0,(x0 + (N . m))) ) } is set
x - x0 is V11() real ext-real Element of REAL
abs (x - x0) is V11() real ext-real Element of REAL
abs (N . m) is V11() real ext-real Element of REAL
xx is V11() real ext-real Element of REAL
xx is V11() real ext-real Element of REAL
xx is V11() real ext-real Element of REAL
xx is V11() real ext-real Element of REAL
x0 - x0 is V11() real ext-real Element of REAL
- (N . m) is V11() real ext-real Element of REAL
xx is V11() real ext-real Element of REAL
- (x - x0) is V11() real ext-real Element of REAL
F /. x is Element of the carrier of (REAL-NS n)
(F /. x) - (F /. x0) is Element of the carrier of (REAL-NS n)
||.((F /. x) - (F /. x0)).|| is V11() real ext-real Element of REAL
xx is V11() real ext-real Element of REAL
(REAL --> (F /. x0)) /. xx is Element of the carrier of (REAL-NS n)
(REAL --> (F /. x0)) . xx is Element of the carrier of (REAL-NS n)
(F /. x) - ((REAL --> (F /. x0)) /. xx) is Element of the carrier of (REAL-NS n)
||.((F /. x) - ((REAL --> (F /. x0)) /. xx)).|| is V11() real ext-real Element of REAL
x is V11() real ext-real set
(REAL --> (F /. x0)) . x is set
(REAL --> (F /. x0)) | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
(F - (REAL --> (F /. x0))) | (['a,b'] /\ ['a,b']) is Relation-like REAL -defined ['a,b'] /\ ['a,b'] -defined REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
rng N is non empty V129() V130() V131() Element of K6(REAL)
dom R is V129() V130() V131() Element of K6(REAL)
R /. (N . m) is Element of the carrier of (REAL-NS n)
(N . m) " is V11() real ext-real Element of REAL
((N . m) ") * (R /. (N . m)) is Element of the carrier of (REAL-NS n)
(R /* N) . m is Element of the carrier of (REAL-NS n)
((N . m) ") * ((R /* N) . m) is Element of the carrier of (REAL-NS n)
(N ") . m is V11() real ext-real Element of REAL
((N ") . m) * ((R /* N) . m) is Element of the carrier of (REAL-NS n)
dom N is non empty V129() V130() V131() V132() V133() V134() V158() V160() Element of K6(NAT)
abs (N . m) is V11() real ext-real Element of REAL
integral ((F - (REAL --> (F /. x0))),x0,(x0 + (N . m))) is Element of the carrier of (REAL-NS n)
||.(integral ((F - (REAL --> (F /. x0))),x0,(x0 + (N . m)))).|| is V11() real ext-real Element of REAL
n * ((e0 / 2) / n) is V11() real ext-real Element of REAL
(n * ((e0 / 2) / n)) * (abs ((x0 + (N . m)) - x0)) is V11() real ext-real Element of REAL
integral ((REAL --> (F /. x0)),x0,(x0 + (N . m))) is Element of the carrier of (REAL-NS n)
((x0 + (N . m)) - x0) * (F /. x0) is Element of the carrier of (REAL-NS n)
(integral (F,x0,(x0 + (N . m)))) - (integral ((REAL --> (F /. x0)),x0,(x0 + (N . m)))) is Element of the carrier of (REAL-NS n)
(integral (F,a,x0)) - (integral (F,a,x0)) is Element of the carrier of (REAL-NS n)
(integral (F,x0,(x0 + (N . m)))) + ((integral (F,a,x0)) - (integral (F,a,x0))) is Element of the carrier of (REAL-NS n)
((integral (F,x0,(x0 + (N . m)))) + ((integral (F,a,x0)) - (integral (F,a,x0)))) - ((N . m) * (F /. x0)) is Element of the carrier of (REAL-NS n)
(integral (F,x0,(x0 + (N . m)))) + (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
((integral (F,x0,(x0 + (N . m)))) + (0. (REAL-NS n))) - ((N . m) * (F /. x0)) is Element of the carrier of (REAL-NS n)
||.(((N . m) ") * (R /. (N . m))).|| is V11() real ext-real Element of REAL
||.(R /. (N . m)).|| is V11() real ext-real Element of REAL
||.(R /. (N . m)).|| / (abs (N . m)) is V11() real ext-real Element of REAL
(n * ((e0 / 2) / n)) * (abs (N . m)) is V11() real ext-real Element of REAL
((n * ((e0 / 2) / n)) * (abs (N . m))) / (abs (N . m)) is V11() real ext-real Element of REAL
||.(((N ") (#) (R /* N)) . m).|| is V11() real ext-real Element of REAL
N is V129() V130() V131() open Neighbourhood of x0
R is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
N1 is V11() real ext-real Element of REAL
f /. N1 is Element of the carrier of (REAL-NS n)
(f /. N1) - (f /. x0) is Element of the carrier of (REAL-NS n)
N1 - x0 is V11() real ext-real Element of REAL
x . (N1 - x0) is Element of the carrier of (REAL-NS n)
R /. (N1 - x0) is Element of the carrier of (REAL-NS n)
(x . (N1 - x0)) + (R /. (N1 - x0)) is Element of the carrier of (REAL-NS n)
x0 + (N1 - x0) is V11() real ext-real Element of REAL
R_id (N1 - x0) is V11() real ext-real Element of REAL
x0 + (R_id (N1 - x0)) is V11() real ext-real Element of REAL
R . (N1 - x0) is set
f /. (x0 + (R_id (N1 - x0))) is Element of the carrier of (REAL-NS n)
(f /. (x0 + (R_id (N1 - x0)))) - (f /. x0) is Element of the carrier of (REAL-NS n)
x . (R_id (N1 - x0)) is Element of the carrier of (REAL-NS n)
((f /. (x0 + (R_id (N1 - x0)))) - (f /. x0)) - (x . (R_id (N1 - x0))) is Element of the carrier of (REAL-NS n)
((f /. N1) - (f /. x0)) - (x . (N1 - x0)) is Element of the carrier of (REAL-NS n)
(R /. (N1 - x0)) + (x . (N1 - x0)) is Element of the carrier of (REAL-NS n)
(x . (N1 - x0)) - (x . (N1 - x0)) is Element of the carrier of (REAL-NS n)
((f /. N1) - (f /. x0)) - ((x . (N1 - x0)) - (x . (N1 - x0))) is Element of the carrier of (REAL-NS n)
((f /. N1) - (f /. x0)) - (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
x . 1 is Element of the carrier of (REAL-NS n)
N1 is V11() real ext-real Element of REAL
N1 * (F /. x0) is Element of the carrier of (REAL-NS n)
a is epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real non negative V129() V130() V131() V132() V133() V134() V160() V161() V162() Element of NAT
REAL-NS a is non empty V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS a) is non empty set
K7(REAL, the carrier of (REAL-NS a)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS a))) is V2() V32() set
b is V11() real ext-real set
x0 is V11() real ext-real set
].b,x0.[ is V129() V130() V131() V158() V159() V163() open Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( not b1 <= b & not x0 <= b1 ) } is set
0. (REAL-NS a) is V54( REAL-NS a) Element of the carrier of (REAL-NS a)
n is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
f is Relation-like Function-like set
dom f is set
F is set
rng f is set
c7 is set
f . c7 is set
x is V11() real ext-real Element of REAL
f . x is set
integral (n,b,x) is Element of the carrier of (REAL-NS a)
F is Relation-like REAL -defined the carrier of (REAL-NS a) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS a)))
dom F is V129() V130() V131() Element of K6(REAL)
c7 is V11() real ext-real set
F . c7 is set
integral (n,b,c7) is Element of the carrier of (REAL-NS a)
a is V11() real ext-real set
b is V11() real ext-real set
['a,b'] is non empty V129() V130() V131() V160() V161() V162() V163() closed_interval compact closed Element of K6(REAL)
].a,b.[ is V129() V130() V131() V158() V159() V163() open Element of K6(REAL)
{ b1 where b1 is V11() real ext-real Element of REAL : ( not b1 <= a & not b <= b1 ) } is set
x0 is V11() real ext-real set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real V32() V37() V44() V45() ext-real positive non negative V129() V130() V131() V132() V133() V134() V158() V159() V160() V161() V162() Element of NAT
REAL-NS n is non empty V52() V73() V168() V169() V170() V171() V172() V173() V174() V178() V179() strict RealNormSpace-like V187() NORMSTR
the carrier of (REAL-NS n) is non empty V2() set
K7(REAL, the carrier of (REAL-NS n)) is V2() Relation-like V32() set
K6(K7(REAL, the carrier of (REAL-NS n))) is V2() V32() set
f is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
f | ['a,b'] is Relation-like REAL -defined ['a,b'] -defined REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom f is V129() V130() V131() Element of K6(REAL)
f /. x0 is Element of the carrier of (REAL-NS n)
F is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom F is V129() V130() V131() Element of K6(REAL)
diff (F,x0) is Element of the carrier of (REAL-NS n)