:: INTEGRA3 semantic presentation

REAL is non empty V12() V37() V58() V59() V60() V64() non bounded_below non bounded_above interval set
NAT is non empty V12() epsilon-transitive epsilon-connected ordinal V37() cardinal limit_cardinal V58() V59() V60() V61() V62() V63() V64() left_end bounded_below Element of K19(REAL)
K19(REAL) is V12() V37() set
COMPLEX is non empty V12() V37() V58() V64() set
omega is non empty V12() epsilon-transitive epsilon-connected ordinal V37() cardinal limit_cardinal V58() V59() V60() V61() V62() V63() V64() left_end bounded_below set
K19(omega) is V12() V37() set
K19(NAT) is V12() V37() set
INT is non empty V12() V37() V58() V59() V60() V61() V62() V64() set
K164(NAT) is V36() set
{} is Relation-like non-empty empty-yielding RAT -valued functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() real ext-real non positive non negative V37() V38() V41() cardinal {} -element FinSequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V58() V59() V60() V61() V62() V63() V64() bounded_below bounded_above real-bounded interval set
RAT is non empty V12() V37() V58() V59() V60() V61() V64() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
{{},1} is non empty V37() V41() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded set
K20(REAL,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(REAL,REAL)) is V12() V37() set
K20(COMPLEX,COMPLEX) is Relation-like V12() V37() complex-valued set
K19(K20(COMPLEX,COMPLEX)) is V12() V37() set
K20(K20(COMPLEX,COMPLEX),COMPLEX) is Relation-like V12() V37() complex-valued set
K19(K20(K20(COMPLEX,COMPLEX),COMPLEX)) is V12() V37() set
K20(K20(REAL,REAL),REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(K20(REAL,REAL),REAL)) is V12() V37() set
K20(RAT,RAT) is Relation-like RAT -valued V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(RAT,RAT)) is V12() V37() set
K20(K20(RAT,RAT),RAT) is Relation-like RAT -valued V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(K20(RAT,RAT),RAT)) is V12() V37() set
K20(INT,INT) is Relation-like RAT -valued INT -valued V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(INT,INT)) is V12() V37() set
K20(K20(INT,INT),INT) is Relation-like RAT -valued INT -valued V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(K20(INT,INT),INT)) is V12() V37() set
K20(NAT,NAT) is Relation-like RAT -valued INT -valued V12() V37() complex-valued ext-real-valued real-valued natural-valued set
K20(K20(NAT,NAT),NAT) is Relation-like RAT -valued INT -valued V12() V37() complex-valued ext-real-valued real-valued natural-valued set
K19(K20(K20(NAT,NAT),NAT)) is V12() V37() set
K333() is set
ExtREAL is non empty V59() interval set
K20(NAT,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(NAT,REAL)) is V12() V37() set
K20(COMPLEX,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(COMPLEX,REAL)) is V12() V37() set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
3 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
0 is Relation-like non-empty empty-yielding RAT -valued functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V22() real ext-real non positive non negative V37() V38() V41() cardinal {} -element FinSequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V58() V59() V60() V61() V62() V63() V64() V69() V70() bounded_below bounded_above real-bounded interval Element of NAT
Seg 1 is non empty V12() V37() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
{1} is non empty V12() V37() V41() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded set
Seg 2 is non empty V37() 2 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
{1,2} is non empty V37() V41() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded set
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
A -' A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(A -' A) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
A - A is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
chi (A,A) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
upper_volume ((chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),f))) is V22() real ext-real set
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
divs A is non empty set
K20(NAT,(divs A)) is Relation-like V12() V37() set
K19(K20(NAT,(divs A))) is V12() V37() set
f is Relation-like Function-like non empty total V30( NAT , divs A) Element of K19(K20(NAT,(divs A)))
T is Relation-like Function-like non empty total V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
e is Relation-like Function-like non empty total V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . e is V22() real ext-real Element of REAL
e . e is V22() real ext-real Element of REAL
f . e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,(f . e)) is V22() real ext-real Element of REAL
chi (A,A) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
upper_volume ((chi (A,A)),(f . e)) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),(f . e))) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),(f . e)))) is V22() real ext-real set
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,f) is V22() real ext-real Element of REAL
chi (A,A) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
upper_volume ((chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),f))) is V22() real ext-real set
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,T) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),T)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),T))) is V22() real ext-real set
dom (upper_volume ((chi (A,A)),T)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (A,A)),T)) . e is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (T,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (T,e)) is V22() real ext-real Element of REAL
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (f,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (f,e)) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),f)) . e is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol A is V22() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
T is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (f,T) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (f,T)) is V22() real ext-real Element of REAL
T is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (f,T) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (f,T)) is V22() real ext-real Element of REAL
lower_bound (divset (f,T)) is V22() real ext-real Element of REAL
vol (divset (f,T)) is V22() real ext-real Element of REAL
(upper_bound (divset (f,T))) - (lower_bound (divset (f,T))) is V22() real ext-real Element of REAL
(len f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len f) - 1 is V22() real ext-real Element of REAL
divset (f,(len f)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (f,(len f))) is V22() real ext-real Element of REAL
f . (len f) is V22() real ext-real Element of REAL
lower_bound (divset (f,(len f))) is V22() real ext-real Element of REAL
f . ((len f) - 1) is V22() real ext-real Element of REAL
divset (f,(len f)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (f,(len f))) is V22() real ext-real Element of REAL
f . (len f) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
lower_bound (divset (f,(len f))) is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
(lower_bound A) + 0 is V22() real ext-real Element of REAL
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
lower_bound f is V22() real ext-real Element of REAL
upper_bound f is V22() real ext-real Element of REAL
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . e is V22() real ext-real Element of REAL
divset (T,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (T,e)) is V22() real ext-real Element of REAL
upper_bound (divset (T,e)) is V22() real ext-real Element of REAL
e is V22() real ext-real Element of REAL
y is V22() real ext-real Element of REAL
lower_bound (divset (T,e)) is V22() real ext-real Element of REAL
upper_bound (divset (T,e)) is V22() real ext-real Element of REAL
e is V22() real ext-real Element of REAL
y is V22() real ext-real Element of REAL
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (T,(len T)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
T . (len T) is V22() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
divset (T,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (T,e)) is V22() real ext-real Element of REAL
divset (T,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (T,1)) is V22() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
divset (T,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (T,e)) is V22() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
divset (T,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (T,e)) is V22() real ext-real Element of REAL
T . e is V22() real ext-real Element of REAL
e - 1 is V22() real ext-real Element of REAL
T . (e - 1) is V22() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (T,e)) is V22() real ext-real Element of REAL
T . e is V22() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (T,(e + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (T,(e + 1))) is V22() real ext-real Element of REAL
(e + 1) - 1 is V22() real ext-real Element of REAL
T . ((e + 1) - 1) is V22() real ext-real Element of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng f is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng f) \/ (rng T) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
f ^ T is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (f ^ T) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
card (rng (f ^ T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of omega
e is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng e is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
len e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e is Relation-like NAT -defined REAL -valued Function-like one-to-one V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y . (len y) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
f . (len f) is V22() real ext-real Element of REAL
dom y is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . D is V22() real ext-real Element of REAL
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng D is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
card (rng f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of omega
len D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
card (rng T) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of omega
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
chi (A,A) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,f) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),f))) is V22() real ext-real set
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
upper_volume ((chi (A,A)),T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),T)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
inf (rng (upper_volume ((chi (A,A)),T))) is V22() real ext-real set
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
e is V22() real ext-real Element of REAL
e is V22() real ext-real Element of REAL
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (f,y) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(rng T) /\ (divset (f,y)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . D is V22() real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . p is V22() real ext-real Element of REAL
(T . D) - (T . p) is V22() real ext-real Element of REAL
abs ((T . D) - (T . p)) is V22() real ext-real Element of REAL
D + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
T . (D + 1) is V22() real ext-real Element of REAL
(T . D) - (T . (D + 1)) is V22() real ext-real Element of REAL
- ((T . D) - (T . (D + 1))) is V22() real ext-real Element of REAL
- ((T . D) - (T . p)) is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),T))) is non empty V37() len (upper_volume ((chi (A,A)),T)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),T)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),T)) . (D + 1) is V22() real ext-real Element of REAL
divset (T,(D + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (T,(D + 1))) is V22() real ext-real Element of REAL
- (abs ((T . D) - (T . p))) is V22() real ext-real Element of REAL
lower_bound (divset (T,(D + 1))) is V22() real ext-real Element of REAL
(D + 1) - 1 is V22() real ext-real Element of REAL
T . ((D + 1) - 1) is V22() real ext-real Element of REAL
vol (divset (T,(D + 1))) is V22() real ext-real Element of REAL
(T . (D + 1)) - (T . D) is V22() real ext-real Element of REAL
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
T . (p + 1) is V22() real ext-real Element of REAL
(T . (p + 1)) - (T . p) is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),T))) is non empty V37() len (upper_volume ((chi (A,A)),T)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),T)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),T)) . (p + 1) is V22() real ext-real Element of REAL
divset (T,(p + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (T,(p + 1))) is V22() real ext-real Element of REAL
lower_bound (divset (T,(p + 1))) is V22() real ext-real Element of REAL
(p + 1) - 1 is V22() real ext-real Element of REAL
T . ((p + 1) - 1) is V22() real ext-real Element of REAL
vol (divset (T,(p + 1))) is V22() real ext-real Element of REAL
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume ((chi (A,A)),f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),f))) is non empty V37() len (upper_volume ((chi (A,A)),f)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),f)) . y is V22() real ext-real Element of REAL
upper_bound (divset (f,y)) is V22() real ext-real Element of REAL
lower_bound (divset (f,y)) is V22() real ext-real Element of REAL
(T . p) - (lower_bound (divset (f,y))) is V22() real ext-real Element of REAL
(upper_bound (divset (f,y))) - (lower_bound (divset (f,y))) is V22() real ext-real Element of REAL
(T . p) - (T . D) is V22() real ext-real Element of REAL
vol (divset (f,y)) is V22() real ext-real Element of REAL
- ((T . D) - (T . p)) is V22() real ext-real Element of REAL
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume ((chi (A,A)),f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),f))) is non empty V37() len (upper_volume ((chi (A,A)),f)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),f)) . y is V22() real ext-real Element of REAL
upper_bound (divset (f,y)) is V22() real ext-real Element of REAL
lower_bound (divset (f,y)) is V22() real ext-real Element of REAL
(T . D) - (lower_bound (divset (f,y))) is V22() real ext-real Element of REAL
(upper_bound (divset (f,y))) - (lower_bound (divset (f,y))) is V22() real ext-real Element of REAL
vol (divset (f,y)) is V22() real ext-real Element of REAL
A is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng A is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
f is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng f is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
len A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of T
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of T
indx (e,e,A) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (e,e,f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e . A is V22() real ext-real Element of REAL
e . (indx (e,e,A)) is V22() real ext-real Element of REAL
e . f is V22() real ext-real Element of REAL
e . (indx (e,e,f)) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of T
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of T
indx (e,e,f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (e,e,A) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . A is V22() real ext-real Element of REAL
e . (indx (e,e,A)) is V22() real ext-real Element of REAL
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e . f is V22() real ext-real Element of REAL
e . (indx (e,e,f)) is V22() real ext-real Element of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,f) is V22() real ext-real Element of REAL
chi (A,A) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
upper_volume ((chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),f))) is V22() real ext-real set
rng f is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
T is V22() real ext-real Element of REAL
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
f . e is V22() real ext-real Element of REAL
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume ((chi (A,A)),f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),f))) is non empty V37() len (upper_volume ((chi (A,A)),f)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),f)) . e is V22() real ext-real Element of REAL
divset (f,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (f,e)) is V22() real ext-real Element of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
f is Relation-like Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
f | A is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng f is non empty V58() V59() V60() Element of K19(REAL)
lower_bound (rng f) is V22() real ext-real Element of REAL
upper_bound (rng f) is V22() real ext-real Element of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T is Relation-like Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
T | A is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng T is non empty V58() V59() V60() Element of K19(REAL)
lower_bound (rng T) is V22() real ext-real Element of REAL
T | f is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (T | f) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (T | f)) is V22() real ext-real Element of REAL
upper_bound (rng (T | f)) is V22() real ext-real Element of REAL
upper_bound (rng T) is V22() real ext-real Element of REAL
dom T is non empty set
dom (T | f) is set
e is V22() real ext-real Element of REAL
(T | f) . e is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (T,A) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (T,A)) is V22() real ext-real Element of REAL
(f,T) is V22() real ext-real Element of REAL
chi (f,f) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
K20(f,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(f,REAL)) is V12() V37() set
upper_volume ((chi (f,f)),T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (f,f)),T)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (f,f)),T))) is V22() real ext-real set
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume ((chi (f,f)),T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (f,f)),T))) is non empty V37() len (upper_volume ((chi (f,f)),T)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (f,f)),T)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (f,f)),T)) . A is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
{A} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len T) - 1 is V22() real ext-real Element of REAL
divset (T,(len T)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng T) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (e,T,e) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e | (indx (e,T,e)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (e | (indx (e,T,e))) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
T | e is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (T | e) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
lower_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
T . e is V22() real ext-real Element of REAL
y is set
dom (e | (indx (e,T,e))) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(e | (indx (e,T,e))) . D is V22() real ext-real Element of REAL
len (e | (indx (e,T,e))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len (e | (indx (e,T,e)))) is V37() len (e | (indx (e,T,e))) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (e,T,e)) is V37() indx (e,T,e) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
len (T | e) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . D is V22() real ext-real Element of REAL
e . (indx (e,T,e)) is V22() real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . p is V22() real ext-real Element of REAL
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg e is V37() e -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(T | e) . p is V22() real ext-real Element of REAL
dom (T | e) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
dom (T | e) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(T | e) . e is V22() real ext-real Element of REAL
Seg e is V37() e -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
y is set
dom (T | e) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(T | e) . D is V22() real ext-real Element of REAL
len (T | e) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len (T | e)) is V37() len (T | e) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg e is V37() e -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
T . D is V22() real ext-real Element of REAL
indx (e,T,D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . (indx (e,T,D)) is V22() real ext-real Element of REAL
e . (indx (e,T,e)) is V22() real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . p is V22() real ext-real Element of REAL
Seg (indx (e,T,e)) is V37() indx (e,T,e) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
len (e | (indx (e,T,e))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len (e | (indx (e,T,e)))) is V37() len (e | (indx (e,T,e))) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
dom (e | (indx (e,T,e))) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(e | (indx (e,T,e))) . p is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
{A} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(f,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(f,REAL)) is V12() V37() set
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (T,(len T)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng T) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(f,T) is V22() real ext-real Element of REAL
chi (f,f) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
upper_volume ((chi (f,f)),T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (f,f)),T)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (f,f)),T))) is V22() real ext-real set
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
e is Relation-like Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
e | f is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
lower_volume (e,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (lower_volume (e,e)) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(lower_volume (e,e)),K263()) is V22() real ext-real Element of REAL
lower_volume (e,T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (lower_volume (e,T)) is V22() real ext-real Element of REAL
K190(REAL,(lower_volume (e,T)),K263()) is V22() real ext-real Element of REAL
(Sum (lower_volume (e,e))) - (Sum (lower_volume (e,T))) is V22() real ext-real Element of REAL
rng e is non empty V58() V59() V60() Element of K19(REAL)
upper_bound (rng e) is V22() real ext-real Element of REAL
lower_bound (rng e) is V22() real ext-real Element of REAL
(upper_bound (rng e)) - (lower_bound (rng e)) is V22() real ext-real Element of REAL
((upper_bound (rng e)) - (lower_bound (rng e))) * (f,T) is V22() real ext-real Element of REAL
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
indx (e,T,(len T)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len T) - 1 is V22() real ext-real Element of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
mid (T,1,D) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
T | D is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
indx (e,T,D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
mid (e,1,(indx (e,T,D))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
e | (indx (e,T,D)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (e | (indx (e,T,D))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
rng (e | (indx (e,T,D))) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
rng (T | D) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (e,T,p) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg D is V37() D -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (e,T,D)) is V37() indx (e,T,D) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(e | (indx (e,T,D))) . (indx (e,T,p)) is V22() real ext-real Element of REAL
e . (indx (e,T,p)) is V22() real ext-real Element of REAL
T . p is V22() real ext-real Element of REAL
T . (indx (e,T,p)) is V22() real ext-real Element of REAL
Seg D is V37() D -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
T . p is V22() real ext-real Element of REAL
(T | D) . p is V22() real ext-real Element of REAL
e . (indx (e,T,p)) is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
Seg (indx (e,T,D)) is V37() indx (e,T,D) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(lower_volume (e,T)) | D is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((lower_volume (e,T)) | D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(lower_volume (e,e)) | (indx (e,T,D)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (lower_volume (e,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (lower_volume (e,e))) is non empty V37() len (lower_volume (e,e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (e,e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
((lower_volume (e,T)) | D) . p is V22() real ext-real Element of REAL
((lower_volume (e,e)) | (indx (e,T,D))) . p is V22() real ext-real Element of REAL
len (lower_volume (e,T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
indx (e,T,H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg D is V37() D -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (e,T,D)) is V37() indx (e,T,D) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
T . H is V22() real ext-real Element of REAL
e . (indx (e,T,H)) is V22() real ext-real Element of REAL
divset (T,H) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (T,H)) is V22() real ext-real Element of REAL
divset (e,(indx (e,T,H))) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,(indx (e,T,H)))) is V22() real ext-real Element of REAL
upper_bound (divset (T,H)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(indx (e,T,H)))) is V22() real ext-real Element of REAL
lower_bound f is V22() real ext-real Element of REAL
H - 1 is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,H)) - 1 is V22() real ext-real Element of REAL
e . ((indx (e,T,H)) - 1) is V22() real ext-real Element of REAL
T . (H - 1) is V22() real ext-real Element of REAL
e . (H - 1) is V22() real ext-real Element of REAL
indx (e,T,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . (indx (e,T,h)) is V22() real ext-real Element of REAL
[.(lower_bound (divset (T,H))),(upper_bound (divset (T,H))).] is V58() V59() V60() interval Element of K19(REAL)
Seg (len (lower_volume (e,T))) is non empty V37() len (lower_volume (e,T)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (e,T)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
((lower_volume (e,T)) | D) . H is V22() real ext-real Element of REAL
(lower_volume (e,T)) . H is V22() real ext-real Element of REAL
e | (divset (e,(indx (e,T,H)))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (e,(indx (e,T,H))))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (e | (divset (e,(indx (e,T,H)))))) is V22() real ext-real Element of REAL
vol (divset (e,(indx (e,T,H)))) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (e,(indx (e,T,H))))))) * (vol (divset (e,(indx (e,T,H))))) is V22() real ext-real Element of REAL
((lower_volume (e,e)) | (indx (e,T,D))) . H is V22() real ext-real Element of REAL
((lower_volume (e,e)) | (indx (e,T,D))) . (indx (e,T,H)) is V22() real ext-real Element of REAL
(lower_volume (e,e)) . (indx (e,T,H)) is V22() real ext-real Element of REAL
len (lower_volume (e,T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len H₁(T) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(T) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len T) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
PartSums (lower_volume (e,T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(PartSums (lower_volume (e,T))) . D is V22() real ext-real Element of REAL
H₁(T) | D is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(T) | D) is V22() real ext-real Element of REAL
K190(REAL,(H₁(T) | D),K263()) is V22() real ext-real Element of REAL
mid (H₁(T),(len T),(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(T),(len T),(len T))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(T),(len T),(len T))),K263()) is V22() real ext-real Element of REAL
H₂(T,D) + (Sum (mid (H₁(T),(len T),(len T)))) is V22() real ext-real Element of REAL
(H₁(T) | D) ^ (mid (H₁(T),(len T),(len T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((H₁(T) | D) ^ (mid (H₁(T),(len T),(len T)))) is V22() real ext-real Element of REAL
K190(REAL,((H₁(T) | D) ^ (mid (H₁(T),(len T),(len T)))),K263()) is V22() real ext-real Element of REAL
mid (H₁(T),1,D) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
D + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (H₁(T),(D + 1),(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (H₁(T),1,D)) ^ (mid (H₁(T),(D + 1),(len T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(T),1,D)) ^ (mid (H₁(T),(D + 1),(len T)))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(T),1,D)) ^ (mid (H₁(T),(D + 1),(len T)))),K263()) is V22() real ext-real Element of REAL
mid (H₁(T),1,(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(T),1,(len T))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(T),1,(len T))),K263()) is V22() real ext-real Element of REAL
H₁(T) | (len T) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(T) | (len T)) is V22() real ext-real Element of REAL
K190(REAL,(H₁(T) | (len T)),K263()) is V22() real ext-real Element of REAL
mid ((lower_volume (e,T)),(len T),(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid ((lower_volume (e,T)),(len T),(len T))) is V22() real ext-real Element of REAL
K190(REAL,(mid ((lower_volume (e,T)),(len T),(len T))),K263()) is V22() real ext-real Element of REAL
H₂(T,D) + (Sum (mid ((lower_volume (e,T)),(len T),(len T)))) is V22() real ext-real Element of REAL
(PartSums (lower_volume (e,T))) . (len T) is V22() real ext-real Element of REAL
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(indx (e,T,D)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
K190(REAL,(mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))),K263()) is V22() real ext-real Element of REAL
(Sum (mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) - (Sum (mid ((lower_volume (e,T)),(len T),(len T)))) is V22() real ext-real Element of REAL
(indx (e,T,(len T))) - (indx (e,T,D)) is V22() real ext-real Element of REAL
((indx (e,T,D)) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,D)) + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (indx (e,T,(len T))) is V22() real ext-real Element of REAL
e . (((indx (e,T,D)) + 1) + 1) is V22() real ext-real Element of REAL
T . D is V22() real ext-real Element of REAL
e . (indx (e,T,D)) is V22() real ext-real Element of REAL
T . (len T) is V22() real ext-real Element of REAL
e . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . h is V22() real ext-real Element of REAL
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . h is V22() real ext-real Element of REAL
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid ((lower_volume (e,T)),(len T),(len T))) . 1 is V22() real ext-real Element of REAL
(lower_volume (e,T)) . (len T) is V22() real ext-real Element of REAL
(len T) -' (len T) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((len T) -' (len T)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid ((lower_volume (e,T)),(len T),(len T))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
<*((lower_volume (e,T)) . (len T))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
len (lower_volume (e,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (lower_volume (e,e))) is non empty V37() len (lower_volume (e,e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(indx (e,T,(len T))) -' ((indx (e,T,D)) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,(len T))) - ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
((indx (e,T,(len T))) -' ((indx (e,T,D)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
T . (len T) is V22() real ext-real Element of REAL
e . (indx (e,T,(len T))) is V22() real ext-real Element of REAL
lower_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
T . D is V22() real ext-real Element of REAL
e . (indx (e,T,D)) is V22() real ext-real Element of REAL
[.(e . (indx (e,T,D))),(e . (indx (e,T,(len T)))).] is V58() V59() V60() interval Element of K19(REAL)
divset (e,(indx (e,T,(len T)))) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (e,(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
lower_bound (divset (e,(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
(indx (e,T,(len T))) - 1 is V22() real ext-real Element of REAL
e . ((indx (e,T,(len T))) - 1) is V22() real ext-real Element of REAL
(mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) . 1 is V22() real ext-real Element of REAL
(lower_volume (e,e)) . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
<*((lower_volume (e,e)) . ((indx (e,T,D)) + 1))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
divset (e,((indx (e,T,D)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e | (divset (e,((indx (e,T,D)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (e,((indx (e,T,D)) + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
vol (divset (e,((indx (e,T,D)) + 1))) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
(mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) . 1 is V22() real ext-real Element of REAL
(lower_volume (e,e)) . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
2 + ((indx (e,T,D)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) . 2 is V22() real ext-real Element of REAL
(2 + ((indx (e,T,D)) + 1)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
H₁(e) . ((2 + ((indx (e,T,D)) + 1)) -' 1) is V22() real ext-real Element of REAL
(2 + ((indx (e,T,D)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(e) . ((2 + ((indx (e,T,D)) + 1)) - 1) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,D)) + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H₁(e) . ((indx (e,T,D)) + (1 + 1)) is V22() real ext-real Element of REAL
(indx (e,T,D)) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(lower_volume (e,e)) . ((indx (e,T,D)) + 2) is V22() real ext-real Element of REAL
<*((lower_volume (e,e)) . ((indx (e,T,D)) + 1)),((lower_volume (e,e)) . ((indx (e,T,D)) + 2))*> is Relation-like NAT -defined Function-like non empty V37() 2 -element FinSequence-like FinSubsequence-like set
((lower_volume (e,e)) . ((indx (e,T,D)) + 1)) + ((lower_volume (e,e)) . ((indx (e,T,D)) + 2)) is V22() real ext-real Element of REAL
divset (e,((indx (e,T,D)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (e,((indx (e,T,D)) + 1))) is V22() real ext-real Element of REAL
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
T . (len T) is V22() real ext-real Element of REAL
e . (indx (e,T,(len T))) is V22() real ext-real Element of REAL
divset (e,((indx (e,T,D)) + 2)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (e,((indx (e,T,D)) + 2))) is V22() real ext-real Element of REAL
((indx (e,T,(len T))) - ((indx (e,T,D)) + 1)) + 1 is V22() real ext-real Element of REAL
lower_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
T . D is V22() real ext-real Element of REAL
e . (indx (e,T,D)) is V22() real ext-real Element of REAL
vol (divset (T,(len T))) is V22() real ext-real Element of REAL
e . ((indx (e,T,D)) + 2) is V22() real ext-real Element of REAL
e . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
(e . ((indx (e,T,D)) + 2)) - (e . ((indx (e,T,D)) + 1)) is V22() real ext-real Element of REAL
((e . ((indx (e,T,D)) + 2)) - (e . ((indx (e,T,D)) + 1))) + (e . ((indx (e,T,D)) + 1)) is V22() real ext-real Element of REAL
(((e . ((indx (e,T,D)) + 2)) - (e . ((indx (e,T,D)) + 1))) + (e . ((indx (e,T,D)) + 1))) - (e . (indx (e,T,D))) is V22() real ext-real Element of REAL
upper_bound (divset (e,((indx (e,T,D)) + 1))) is V22() real ext-real Element of REAL
((indx (e,T,D)) + 1) - 1 is V22() real ext-real Element of REAL
(indx (e,T,D)) + 0 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
lower_bound (divset (e,((indx (e,T,D)) + 1))) is V22() real ext-real Element of REAL
((indx (e,T,D)) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (e,T,D)) + 2) - 1 is V22() real ext-real Element of REAL
lower_bound (divset (e,((indx (e,T,D)) + 2))) is V22() real ext-real Element of REAL
upper_bound (divset (e,((indx (e,T,D)) + 2))) is V22() real ext-real Element of REAL
(vol (divset (e,((indx (e,T,D)) + 2)))) + (e . ((indx (e,T,D)) + 1)) is V22() real ext-real Element of REAL
((vol (divset (e,((indx (e,T,D)) + 2)))) + (e . ((indx (e,T,D)) + 1))) - (e . (indx (e,T,D))) is V22() real ext-real Element of REAL
(upper_bound (divset (e,((indx (e,T,D)) + 1)))) - (lower_bound (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
(vol (divset (e,((indx (e,T,D)) + 2)))) + ((upper_bound (divset (e,((indx (e,T,D)) + 1)))) - (lower_bound (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(vol (divset (e,((indx (e,T,D)) + 1)))) + (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
e | (divset (T,(len T))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (T,(len T)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (e | (divset (T,(len T))))) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (T,(len T)))))) * ((vol (divset (e,((indx (e,T,D)) + 1)))) + (vol (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T))))),K263()) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) - (Sum (mid (H₁(T),(len T),(len T)))) is V22() real ext-real Element of REAL
(vol (divset (e,((indx (e,T,D)) + 2)))) + (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng e)) - (lower_bound (rng e))) * ((vol (divset (e,((indx (e,T,D)) + 2)))) + (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
((lower_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((lower_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(T),(len T),(len T)))) - ((lower_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
((lower_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) + ((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(T),(len T),(len T)))) - ((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
(lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
e | (divset (e,((indx (e,T,D)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (e,((indx (e,T,D)) + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
(upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
e | (divset (e,((indx (e,T,D)) + 2))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (e,((indx (e,T,D)) + 2)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 2)))))) * (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
H₁(e) . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
((lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 2)))))) * (vol (divset (e,((indx (e,T,D)) + 2))))) + (H₁(e) . ((indx (e,T,D)) + 1)) is V22() real ext-real Element of REAL
((lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 2)))))) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) - ((lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((lower_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) - ((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1)))))) - (((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1)))))) is V22() real ext-real Element of REAL
((upper_bound (rng e)) - (lower_bound (rng e))) * (vol (divset (T,(len T)))) is V22() real ext-real Element of REAL
len (lower_volume (e,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (indx (e,T,(len T))) is V22() real ext-real Element of REAL
T . (len T) is V22() real ext-real Element of REAL
e . (len e) is V22() real ext-real Element of REAL
upper_bound f is V22() real ext-real Element of REAL
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
Seg (len (lower_volume (e,e))) is non empty V37() len (lower_volume (e,e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (e,e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
PartSums (lower_volume (e,e)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(PartSums (lower_volume (e,e))) . (indx (e,T,(len T))) is V22() real ext-real Element of REAL
(lower_volume (e,e)) | (indx (e,T,(len T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((lower_volume (e,e)) | (indx (e,T,(len T)))) is V22() real ext-real Element of REAL
K190(REAL,((lower_volume (e,e)) | (indx (e,T,(len T)))),K263()) is V22() real ext-real Element of REAL
Seg (len (lower_volume (e,T))) is non empty V37() len (lower_volume (e,T)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (e,T)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(lower_volume (e,T)) | (len T) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((lower_volume (e,T)) | (len T)) is V22() real ext-real Element of REAL
K190(REAL,((lower_volume (e,T)) | (len T)),K263()) is V22() real ext-real Element of REAL
len (T | D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len ((lower_volume (e,e)) | (indx (e,T,D))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (lower_volume (e,e))) . (indx (e,T,D)) is V22() real ext-real Element of REAL
Sum ((lower_volume (e,e)) | (indx (e,T,D))) is V22() real ext-real Element of REAL
K190(REAL,((lower_volume (e,e)) | (indx (e,T,D))),K263()) is V22() real ext-real Element of REAL
len H₁(e) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom H₁(e) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
H₁(e) | (indx (e,T,D)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | (indx (e,T,D))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | (indx (e,T,D))),K263()) is V22() real ext-real Element of REAL
H₂(e, indx (e,T,D)) + (Sum (mid ((lower_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) is V22() real ext-real Element of REAL
mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(H₁(e) | (indx (e,T,D))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((H₁(e) | (indx (e,T,D))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) is V22() real ext-real Element of REAL
K190(REAL,((H₁(e) | (indx (e,T,D))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,(indx (e,T,D))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (H₁(e),1,(indx (e,T,D)))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(e),1,(indx (e,T,D)))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(e),1,(indx (e,T,D)))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,(indx (e,T,(len T)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),1,(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),1,(indx (e,T,(len T))))),K263()) is V22() real ext-real Element of REAL
H₁(e) | (indx (e,T,(len T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | (indx (e,T,(len T)))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | (indx (e,T,(len T)))),K263()) is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
{A} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(f,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(f,REAL)) is V12() V37() set
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (T,(len T)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng T) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(f,T) is V22() real ext-real Element of REAL
chi (f,f) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
upper_volume ((chi (f,f)),T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (f,f)),T)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (f,f)),T))) is V22() real ext-real set
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
e is Relation-like Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
e | f is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
upper_volume (e,T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (upper_volume (e,T)) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(upper_volume (e,T)),K263()) is V22() real ext-real Element of REAL
upper_volume (e,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (upper_volume (e,e)) is V22() real ext-real Element of REAL
K190(REAL,(upper_volume (e,e)),K263()) is V22() real ext-real Element of REAL
(Sum (upper_volume (e,T))) - (Sum (upper_volume (e,e))) is V22() real ext-real Element of REAL
rng e is non empty V58() V59() V60() Element of K19(REAL)
upper_bound (rng e) is V22() real ext-real Element of REAL
lower_bound (rng e) is V22() real ext-real Element of REAL
(upper_bound (rng e)) - (lower_bound (rng e)) is V22() real ext-real Element of REAL
((upper_bound (rng e)) - (lower_bound (rng e))) * (f,T) is V22() real ext-real Element of REAL
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
indx (e,T,(len T)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume (e,T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len T) - 1 is V22() real ext-real Element of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (e,T,D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
mid (e,1,(indx (e,T,D))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
e | (indx (e,T,D)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(len T) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,D)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e . (indx (e,T,(len T))) is V22() real ext-real Element of REAL
T . (len T) is V22() real ext-real Element of REAL
len (upper_volume (e,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (len e) is V22() real ext-real Element of REAL
upper_bound f is V22() real ext-real Element of REAL
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
Seg (len (upper_volume (e,e))) is non empty V37() len (upper_volume (e,e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume (e,e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
PartSums (upper_volume (e,e)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(PartSums (upper_volume (e,e))) . (indx (e,T,(len T))) is V22() real ext-real Element of REAL
(upper_volume (e,e)) | (indx (e,T,(len T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((upper_volume (e,e)) | (indx (e,T,(len T)))) is V22() real ext-real Element of REAL
K190(REAL,((upper_volume (e,e)) | (indx (e,T,(len T)))),K263()) is V22() real ext-real Element of REAL
len (e | (indx (e,T,D))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
mid ((upper_volume (e,T)),(len T),(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid ((upper_volume (e,T)),(len T),(len T))) is V22() real ext-real Element of REAL
K190(REAL,(mid ((upper_volume (e,T)),(len T),(len T))),K263()) is V22() real ext-real Element of REAL
mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
K190(REAL,(mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))),K263()) is V22() real ext-real Element of REAL
(Sum (mid ((upper_volume (e,T)),(len T),(len T)))) - (Sum (mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) is V22() real ext-real Element of REAL
(indx (e,T,(len T))) - (indx (e,T,D)) is V22() real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,D)) + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . H is V22() real ext-real Element of REAL
T . D is V22() real ext-real Element of REAL
e . (indx (e,T,D)) is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . h is V22() real ext-real Element of REAL
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . h is V22() real ext-real Element of REAL
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid ((upper_volume (e,T)),(len T),(len T))) . 1 is V22() real ext-real Element of REAL
(upper_volume (e,T)) . (len T) is V22() real ext-real Element of REAL
(len T) -' (len T) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((len T) -' (len T)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid ((upper_volume (e,T)),(len T),(len T))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
<*((upper_volume (e,T)) . (len T))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
(indx (e,T,(len T))) -' ((indx (e,T,D)) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,(len T))) - ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
((indx (e,T,(len T))) -' ((indx (e,T,D)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
lower_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
T . D is V22() real ext-real Element of REAL
e . (indx (e,T,D)) is V22() real ext-real Element of REAL
[.(e . (indx (e,T,D))),(e . (indx (e,T,(len T)))).] is V58() V59() V60() interval Element of K19(REAL)
divset (e,(indx (e,T,(len T)))) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (e,(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
lower_bound (divset (e,(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
(indx (e,T,(len T))) - 1 is V22() real ext-real Element of REAL
e . ((indx (e,T,(len T))) - 1) is V22() real ext-real Element of REAL
(mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) . 1 is V22() real ext-real Element of REAL
(upper_volume (e,e)) . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
<*((upper_volume (e,e)) . ((indx (e,T,D)) + 1))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
divset (e,((indx (e,T,D)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e | (divset (e,((indx (e,T,D)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (e,((indx (e,T,D)) + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
vol (divset (e,((indx (e,T,D)) + 1))) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
(mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) . 1 is V22() real ext-real Element of REAL
(upper_volume (e,e)) . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
2 + ((indx (e,T,D)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) . 2 is V22() real ext-real Element of REAL
(2 + ((indx (e,T,D)) + 1)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
H₁(e) . ((2 + ((indx (e,T,D)) + 1)) -' 1) is V22() real ext-real Element of REAL
(2 + ((indx (e,T,D)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(e) . ((2 + ((indx (e,T,D)) + 1)) - 1) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,D)) + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H₁(e) . ((indx (e,T,D)) + (1 + 1)) is V22() real ext-real Element of REAL
(indx (e,T,D)) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(upper_volume (e,e)) . ((indx (e,T,D)) + 2) is V22() real ext-real Element of REAL
<*((upper_volume (e,e)) . ((indx (e,T,D)) + 1)),((upper_volume (e,e)) . ((indx (e,T,D)) + 2))*> is Relation-like NAT -defined Function-like non empty V37() 2 -element FinSequence-like FinSubsequence-like set
((upper_volume (e,e)) . ((indx (e,T,D)) + 1)) + ((upper_volume (e,e)) . ((indx (e,T,D)) + 2)) is V22() real ext-real Element of REAL
divset (e,((indx (e,T,D)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (e,((indx (e,T,D)) + 1))) is V22() real ext-real Element of REAL
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
divset (e,((indx (e,T,D)) + 2)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (e,((indx (e,T,D)) + 2))) is V22() real ext-real Element of REAL
((indx (e,T,(len T))) - ((indx (e,T,D)) + 1)) + 1 is V22() real ext-real Element of REAL
lower_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
T . D is V22() real ext-real Element of REAL
e . (indx (e,T,D)) is V22() real ext-real Element of REAL
vol (divset (T,(len T))) is V22() real ext-real Element of REAL
e . ((indx (e,T,D)) + 2) is V22() real ext-real Element of REAL
e . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
(e . ((indx (e,T,D)) + 2)) - (e . ((indx (e,T,D)) + 1)) is V22() real ext-real Element of REAL
((e . ((indx (e,T,D)) + 2)) - (e . ((indx (e,T,D)) + 1))) + (e . ((indx (e,T,D)) + 1)) is V22() real ext-real Element of REAL
(((e . ((indx (e,T,D)) + 2)) - (e . ((indx (e,T,D)) + 1))) + (e . ((indx (e,T,D)) + 1))) - (e . (indx (e,T,D))) is V22() real ext-real Element of REAL
upper_bound (divset (e,((indx (e,T,D)) + 1))) is V22() real ext-real Element of REAL
((indx (e,T,D)) + 1) - 1 is V22() real ext-real Element of REAL
(indx (e,T,D)) + 0 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
lower_bound (divset (e,((indx (e,T,D)) + 1))) is V22() real ext-real Element of REAL
((indx (e,T,D)) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (e,T,D)) + 2) - 1 is V22() real ext-real Element of REAL
lower_bound (divset (e,((indx (e,T,D)) + 2))) is V22() real ext-real Element of REAL
upper_bound (divset (e,((indx (e,T,D)) + 2))) is V22() real ext-real Element of REAL
(vol (divset (e,((indx (e,T,D)) + 2)))) + (e . ((indx (e,T,D)) + 1)) is V22() real ext-real Element of REAL
((vol (divset (e,((indx (e,T,D)) + 2)))) + (e . ((indx (e,T,D)) + 1))) - (e . (indx (e,T,D))) is V22() real ext-real Element of REAL
(upper_bound (divset (e,((indx (e,T,D)) + 1)))) - (lower_bound (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
(vol (divset (e,((indx (e,T,D)) + 2)))) + ((upper_bound (divset (e,((indx (e,T,D)) + 1)))) - (lower_bound (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(vol (divset (e,((indx (e,T,D)) + 1)))) + (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
e | (divset (T,(len T))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (T,(len T)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (e | (divset (T,(len T))))) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (T,(len T)))))) * ((vol (divset (e,((indx (e,T,D)) + 1)))) + (vol (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
mid (H₁(T),(len T),(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(T),(len T),(len T))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(T),(len T),(len T))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T))))),K263()) is V22() real ext-real Element of REAL
(Sum (mid (H₁(T),(len T),(len T)))) - (Sum (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) is V22() real ext-real Element of REAL
(vol (divset (e,((indx (e,T,D)) + 2)))) + (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng e)) - (lower_bound (rng e))) * ((vol (divset (e,((indx (e,T,D)) + 2)))) + (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(T),(len T),(len T)))) - ((upper_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
(upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
((upper_bound (rng (e | (divset (T,(len T)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) + ((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(T),(len T),(len T)))) - ((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
(upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
e | (divset (e,((indx (e,T,D)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (e,((indx (e,T,D)) + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1)))) is V22() real ext-real Element of REAL
e | (divset (e,((indx (e,T,D)) + 2))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (e,((indx (e,T,D)) + 2)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 2)))))) * (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
H₁(e) . ((indx (e,T,D)) + 1) is V22() real ext-real Element of REAL
((upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 2)))))) * (vol (divset (e,((indx (e,T,D)) + 2))))) + (H₁(e) . ((indx (e,T,D)) + 1)) is V22() real ext-real Element of REAL
((upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 2)))))) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2)))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) - ((upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((upper_bound (rng (e | (divset (e,((indx (e,T,D)) + 1)))))) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) - ((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) is V22() real ext-real Element of REAL
((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1))))) is V22() real ext-real Element of REAL
(((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((upper_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1)))))) - (((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 2))))) + ((lower_bound (rng e)) * (vol (divset (e,((indx (e,T,D)) + 1)))))) is V22() real ext-real Element of REAL
((upper_bound (rng e)) - (lower_bound (rng e))) * (vol (divset (T,(len T)))) is V22() real ext-real Element of REAL
Seg (len (upper_volume (e,T))) is non empty V37() len (upper_volume (e,T)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume (e,T)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
PartSums (upper_volume (e,T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(PartSums (upper_volume (e,T))) . (len T) is V22() real ext-real Element of REAL
(upper_volume (e,T)) | (len T) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((upper_volume (e,T)) | (len T)) is V22() real ext-real Element of REAL
K190(REAL,((upper_volume (e,T)) | (len T)),K263()) is V22() real ext-real Element of REAL
len H₁(T) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (T,1,D) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
T | D is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (e | (indx (e,T,D))) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
rng (T | D) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (e,T,p) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg D is V37() D -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (e,T,D)) is V37() indx (e,T,D) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(e | (indx (e,T,D))) . (indx (e,T,p)) is V22() real ext-real Element of REAL
e . (indx (e,T,p)) is V22() real ext-real Element of REAL
T . p is V22() real ext-real Element of REAL
T . (indx (e,T,p)) is V22() real ext-real Element of REAL
Seg D is V37() D -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
T . p is V22() real ext-real Element of REAL
(T | D) . p is V22() real ext-real Element of REAL
e . (indx (e,T,p)) is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
Seg (indx (e,T,D)) is V37() indx (e,T,D) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume (e,T)) | D is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((upper_volume (e,T)) | D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume (e,e)) | (indx (e,T,D)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
((upper_volume (e,T)) | D) . p is V22() real ext-real Element of REAL
((upper_volume (e,e)) | (indx (e,T,D))) . p is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (e,T,H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg D is V37() D -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (e,T,D)) is V37() indx (e,T,D) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
T . H is V22() real ext-real Element of REAL
e . (indx (e,T,H)) is V22() real ext-real Element of REAL
divset (T,H) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (T,H)) is V22() real ext-real Element of REAL
divset (e,(indx (e,T,H))) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,(indx (e,T,H)))) is V22() real ext-real Element of REAL
upper_bound (divset (T,H)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(indx (e,T,H)))) is V22() real ext-real Element of REAL
lower_bound f is V22() real ext-real Element of REAL
H - 1 is V22() real ext-real Element of REAL
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (e,T,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (e,T,H)) - 1 is V22() real ext-real Element of REAL
e . ((indx (e,T,H)) - 1) is V22() real ext-real Element of REAL
e . (indx (e,T,h)) is V22() real ext-real Element of REAL
T . (H - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (T,H))),(upper_bound (divset (T,H))).] is V58() V59() V60() interval Element of K19(REAL)
((upper_volume (e,T)) | D) . H is V22() real ext-real Element of REAL
(upper_volume (e,T)) . H is V22() real ext-real Element of REAL
e | (divset (e,(indx (e,T,H)))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (e,(indx (e,T,H))))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (e | (divset (e,(indx (e,T,H)))))) is V22() real ext-real Element of REAL
vol (divset (e,(indx (e,T,H)))) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (e,(indx (e,T,H))))))) * (vol (divset (e,(indx (e,T,H))))) is V22() real ext-real Element of REAL
((upper_volume (e,e)) | (indx (e,T,D))) . H is V22() real ext-real Element of REAL
((upper_volume (e,e)) | (indx (e,T,D))) . (indx (e,T,H)) is V22() real ext-real Element of REAL
(upper_volume (e,e)) . (indx (e,T,H)) is V22() real ext-real Element of REAL
len (T | D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len ((upper_volume (e,e)) | (indx (e,T,D))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len H₁(e) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom H₁(e) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len H₁(e)) is non empty V37() len H₁(e) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(PartSums (upper_volume (e,e))) . (indx (e,T,D)) is V22() real ext-real Element of REAL
H₁(e) | (indx (e,T,D)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | (indx (e,T,D))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | (indx (e,T,D))),K263()) is V22() real ext-real Element of REAL
H₂(e, indx (e,T,D)) + (Sum (mid ((upper_volume (e,e)),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) is V22() real ext-real Element of REAL
mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(H₁(e) | (indx (e,T,D))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((H₁(e) | (indx (e,T,D))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) is V22() real ext-real Element of REAL
K190(REAL,((H₁(e) | (indx (e,T,D))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,(indx (e,T,D))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (H₁(e),1,(indx (e,T,D)))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T))))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(e),1,(indx (e,T,D)))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(e),1,(indx (e,T,D)))) ^ (mid (H₁(e),((indx (e,T,D)) + 1),(indx (e,T,(len T)))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,(indx (e,T,(len T)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),1,(indx (e,T,(len T))))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),1,(indx (e,T,(len T))))),K263()) is V22() real ext-real Element of REAL
H₁(e) | (indx (e,T,(len T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | (indx (e,T,(len T)))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | (indx (e,T,(len T)))),K263()) is V22() real ext-real Element of REAL
dom H₁(T) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len H₁(T)) is non empty V37() len H₁(T) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(PartSums (upper_volume (e,T))) . D is V22() real ext-real Element of REAL
H₁(T) | D is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(T) | D) is V22() real ext-real Element of REAL
K190(REAL,(H₁(T) | D),K263()) is V22() real ext-real Element of REAL
mid (H₁(T),(len T),(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(T),(len T),(len T))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(T),(len T),(len T))),K263()) is V22() real ext-real Element of REAL
H₂(T,D) + (Sum (mid (H₁(T),(len T),(len T)))) is V22() real ext-real Element of REAL
(H₁(T) | D) ^ (mid (H₁(T),(len T),(len T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((H₁(T) | D) ^ (mid (H₁(T),(len T),(len T)))) is V22() real ext-real Element of REAL
K190(REAL,((H₁(T) | D) ^ (mid (H₁(T),(len T),(len T)))),K263()) is V22() real ext-real Element of REAL
mid (H₁(T),1,D) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
D + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (H₁(T),(D + 1),(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (H₁(T),1,D)) ^ (mid (H₁(T),(D + 1),(len T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(T),1,D)) ^ (mid (H₁(T),(D + 1),(len T)))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(T),1,D)) ^ (mid (H₁(T),(D + 1),(len T)))),K263()) is V22() real ext-real Element of REAL
mid (H₁(T),1,(len T)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(T),1,(len T))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(T),1,(len T))),K263()) is V22() real ext-real Element of REAL
H₁(T) | (len T) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(T) | (len T)) is V22() real ext-real Element of REAL
K190(REAL,(H₁(T) | (len T)),K263()) is V22() real ext-real Element of REAL
H₂(T,D) + (Sum (mid ((upper_volume (e,T)),(len T),(len T)))) is V22() real ext-real Element of REAL
Sum ((upper_volume (e,e)) | (indx (e,T,D))) is V22() real ext-real Element of REAL
K190(REAL,((upper_volume (e,e)) | (indx (e,T,D))),K263()) is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(f,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(f,REAL)) is V12() V37() set
vol f is V22() real ext-real Element of REAL
lower_bound f is V22() real ext-real Element of REAL
T is Relation-like Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
lower_sum_set T is Relation-like Function-like non empty total V30( divs f, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20((divs f),REAL))
divs f is non empty set
K20((divs f),REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20((divs f),REAL)) is V12() V37() set
rng (lower_sum_set T) is non empty V58() V59() V60() Element of K19(REAL)
dom (lower_sum_set T) is non empty set
e is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of divs f
(lower_sum_set T) . e is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e . 1 is V22() real ext-real Element of REAL
e . 1 is V22() real ext-real Element of REAL
lower_volume (T,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(lower_volume (T,e)) . 1 is V22() real ext-real Element of REAL
divset (e,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T | (divset (e,1)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (T | (divset (e,1))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (T | (divset (e,1)))) is V22() real ext-real Element of REAL
vol (divset (e,1)) is V22() real ext-real Element of REAL
(lower_bound (rng (T | (divset (e,1))))) * (vol (divset (e,1))) is V22() real ext-real Element of REAL
upper_bound f is V22() real ext-real Element of REAL
(upper_bound f) - (lower_bound f) is V22() real ext-real Element of REAL
lower_sum (T,e) is V22() real ext-real Element of REAL
Sum (lower_volume (T,e)) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(lower_volume (T,e)),K263()) is V22() real ext-real Element of REAL
(lower_volume (T,e)) | 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(lower_volume (T,e)) /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((lower_volume (T,e)) | 1) ^ ((lower_volume (T,e)) /^ 1) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (((lower_volume (T,e)) | 1) ^ ((lower_volume (T,e)) /^ 1)) is V22() real ext-real Element of REAL
K190(REAL,(((lower_volume (T,e)) | 1) ^ ((lower_volume (T,e)) /^ 1)),K263()) is V22() real ext-real Element of REAL
e . (len e) is V22() real ext-real Element of REAL
e /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(len e) - 1 is V22() real ext-real Element of REAL
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom D is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len D) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D . (len D) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (lower_volume (T,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . 2 is V22() real ext-real Element of REAL
rng D is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len ((lower_volume (T,e)) | 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
mid ((lower_volume (T,e)),2,(len (lower_volume (T,e)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (mid ((lower_volume (T,e)),2,(len (lower_volume (T,e))))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(len (lower_volume (T,e))) -' 2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((len (lower_volume (T,e))) -' 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len e) -' 2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((len e) -' 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len e) - 2 is V22() real ext-real Element of REAL
((len e) - 2) + 1 is V22() real ext-real Element of REAL
p is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
lower_volume (T,p) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(mid ((lower_volume (T,e)),2,(len (lower_volume (T,e))))) . H is V22() real ext-real Element of REAL
(lower_volume (T,p)) . H is V22() real ext-real Element of REAL
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (e,(H + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
divset (p,H) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (e,(H + 1))) is V22() real ext-real Element of REAL
e . (H + 1) is V22() real ext-real Element of REAL
dom p is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
p . H is V22() real ext-real Element of REAL
lower_bound (divset (e,(H + 1))) is V22() real ext-real Element of REAL
(H + 1) - 1 is V22() real ext-real Element of REAL
e . ((H + 1) - 1) is V22() real ext-real Element of REAL
upper_bound (divset (p,H)) is V22() real ext-real Element of REAL
lower_bound (divset (p,H)) is V22() real ext-real Element of REAL
[.(lower_bound f),(p . H).] is V58() V59() V60() interval Element of K19(REAL)
H - 1 is V22() real ext-real Element of REAL
p . (H - 1) is V22() real ext-real Element of REAL
(H - 1) + 1 is V22() real ext-real Element of REAL
e . ((H - 1) + 1) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
upper_bound (divset (p,H)) is V22() real ext-real Element of REAL
lower_bound (divset (p,H)) is V22() real ext-real Element of REAL
[.(lower_bound (divset (p,H))),(upper_bound (divset (p,H))).] is V58() V59() V60() interval Element of K19(REAL)
(len (lower_volume (T,e))) - 1 is V22() real ext-real Element of REAL
(len (lower_volume (T,e))) - 2 is V22() real ext-real Element of REAL
((len (lower_volume (T,e))) - 2) + 1 is V22() real ext-real Element of REAL
H + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(H + 2) - 1 is V22() real ext-real Element of REAL
(lower_volume (T,e)) . ((H + 2) - 1) is V22() real ext-real Element of REAL
(lower_volume (T,e)) . (H + 1) is V22() real ext-real Element of REAL
T | (divset (e,(H + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (T | (divset (e,(H + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (T | (divset (e,(H + 1))))) is V22() real ext-real Element of REAL
vol (divset (e,(H + 1))) is V22() real ext-real Element of REAL
(lower_bound (rng (T | (divset (e,(H + 1)))))) * (vol (divset (e,(H + 1)))) is V22() real ext-real Element of REAL
len p is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len p) is non empty V37() len p -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom p is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (T,e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
((lower_volume (T,e)) | 1) . 1 is V22() real ext-real Element of REAL
<*((lower_volume (T,e)) . 1)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
2 -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
2 - 1 is V22() real ext-real Element of REAL
rng p is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
dom p is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
p . 1 is V22() real ext-real Element of REAL
e . (1 + 1) is V22() real ext-real Element of REAL
len (lower_volume (T,p)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (e,1)) is V22() real ext-real Element of REAL
lower_bound (divset (e,1)) is V22() real ext-real Element of REAL
(upper_bound (divset (e,1))) - (lower_bound (divset (e,1))) is V22() real ext-real Element of REAL
(upper_bound (divset (e,1))) - (lower_bound f) is V22() real ext-real Element of REAL
(e . 1) - (lower_bound f) is V22() real ext-real Element of REAL
Sum (lower_volume (T,p)) is V22() real ext-real Element of REAL
K190(REAL,(lower_volume (T,p)),K263()) is V22() real ext-real Element of REAL
0 + (Sum (lower_volume (T,p))) is V22() real ext-real Element of REAL
lower_sum (T,p) is V22() real ext-real Element of REAL
(lower_sum_set T) . p is V22() real ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
(lower_sum_set T) . y is V22() real ext-real Element of REAL
y . 1 is V22() real ext-real Element of REAL
e . 1 is V22() real ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
(lower_sum_set T) . y is V22() real ext-real Element of REAL
y . 1 is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of e
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
mid (e,f,T) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (e,f,T)) . 1 is V22() real ext-real Element of REAL
len (mid (e,f,T)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(mid (e,f,T)) . (len (mid (e,f,T))) is V22() real ext-real Element of REAL
y is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound y is V22() real ext-real Element of REAL
upper_bound y is V22() real ext-real Element of REAL
p is V22() real ext-real Element of REAL
H is V22() real ext-real Element of REAL
[.A,(upper_bound y).] is V58() V59() V60() interval Element of K19(REAL)
p is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound p is V22() real ext-real Element of REAL
upper_bound p is V22() real ext-real Element of REAL
[.(lower_bound p),(upper_bound p).] is V58() V59() V60() interval Element of K19(REAL)
H is V22() real ext-real Element of REAL
rng (mid (e,f,T)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol A is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
<*(lower_bound A)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
<*(lower_bound A)*> ^ f is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f . 1 is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
(f . 1) - (lower_bound A) is V22() real ext-real Element of REAL
dom (<*(lower_bound A)*> ^ f) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(<*(lower_bound A)*> ^ f) . e is V22() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(<*(lower_bound A)*> ^ f) . e is V22() real ext-real Element of REAL
dom <*(lower_bound A)*> is non empty V12() V37() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len <*(lower_bound A)*>) is non empty V37() len <*(lower_bound A)*> -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
{1} is non empty V12() V37() V41() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len (<*(lower_bound A)*> ^ f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (<*(lower_bound A)*> ^ f)) is non empty V37() len (<*(lower_bound A)*> ^ f) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
rng <*(lower_bound A)*> is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
{(lower_bound A)} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng f is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(len <*(lower_bound A)*>) + y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
f . y is V22() real ext-real Element of REAL
rng (<*(lower_bound A)*> ^ f) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng f is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng <*(lower_bound A)*>) \/ (rng f) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len <*(lower_bound A)*>) is non empty V37() len <*(lower_bound A)*> -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
{1} is non empty V12() V37() V41() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
<*(lower_bound A)*> . e is V22() real ext-real Element of REAL
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
f . y is V22() real ext-real Element of REAL
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(len <*(lower_bound A)*>) + y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound A)*>) + y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f . y is V22() real ext-real Element of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(len <*(lower_bound A)*>) + D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f . D is V22() real ext-real Element of REAL
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound A is V22() real ext-real Element of REAL
<*(lower_bound A)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
f . 1 is V22() real ext-real Element of REAL
<*(lower_bound A)*> ^ f is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom (<*(lower_bound A)*> ^ f) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(<*(lower_bound A)*> ^ f) . e is V22() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(<*(lower_bound A)*> ^ f) . e is V22() real ext-real Element of REAL
dom <*(lower_bound A)*> is non empty V12() V37() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len <*(lower_bound A)*>) is non empty V37() len <*(lower_bound A)*> -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
{1} is non empty V12() V37() V41() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len (<*(lower_bound A)*> ^ f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (<*(lower_bound A)*> ^ f)) is non empty V37() len (<*(lower_bound A)*> ^ f) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
rng <*(lower_bound A)*> is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
{(lower_bound A)} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng f is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(len <*(lower_bound A)*>) + y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
f . y is V22() real ext-real Element of REAL
rng (<*(lower_bound A)*> ^ f) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng f is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng <*(lower_bound A)*>) \/ (rng f) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len <*(lower_bound A)*>) is non empty V37() len <*(lower_bound A)*> -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
{1} is non empty V12() V37() V41() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
<*(lower_bound A)*> . e is V22() real ext-real Element of REAL
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
f . y is V22() real ext-real Element of REAL
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(len <*(lower_bound A)*>) + y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound A)*>) + y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f . y is V22() real ext-real Element of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(len <*(lower_bound A)*>) + D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f . D is V22() real ext-real Element of REAL
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
lower_bound A is V22() real ext-real Element of REAL
<*(lower_bound A)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
<*(lower_bound A)*> ^ f is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
T is Relation-like Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
upper_volume (T,f) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
lower_volume (T,f) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
upper_volume (T,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(upper_volume (T,e)) /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
lower_volume (T,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(lower_volume (T,e)) /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (e,(e + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
divset (f,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len f) + (len <*(lower_bound A)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
lower_bound (divset (f,e)) is V22() real ext-real Element of REAL
lower_bound (divset (e,(e + 1))) is V22() real ext-real Element of REAL
upper_bound (divset (f,e)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(e + 1))) is V22() real ext-real Element of REAL
(e + 1) - 1 is V22() real ext-real Element of REAL
e . ((e + 1) - 1) is V22() real ext-real Element of REAL
e . (e + 1) is V22() real ext-real Element of REAL
e + (len <*(lower_bound A)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (e + (len <*(lower_bound A)*>)) is V22() real ext-real Element of REAL
f . e is V22() real ext-real Element of REAL
e . (e + 1) is V22() real ext-real Element of REAL
e + (len <*(lower_bound A)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (e + (len <*(lower_bound A)*>)) is V22() real ext-real Element of REAL
f . e is V22() real ext-real Element of REAL
e - 1 is V22() real ext-real Element of REAL
f . (e - 1) is V22() real ext-real Element of REAL
(e - 1) + (len <*(lower_bound A)*>) is V22() real ext-real Element of REAL
e . ((e - 1) + (len <*(lower_bound A)*>)) is V22() real ext-real Element of REAL
(e - 1) + 1 is V22() real ext-real Element of REAL
e . ((e - 1) + 1) is V22() real ext-real Element of REAL
(e + 1) - 1 is V22() real ext-real Element of REAL
e . ((e + 1) - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (f,e))),(upper_bound (divset (f,e))).] is V58() V59() V60() interval Element of K19(REAL)
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound A)*>) + (len f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 + (len f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
rng (upper_volume (T,e)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
dom (upper_volume (T,e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume (T,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((upper_volume (T,e)) /^ 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(len (upper_volume (T,e))) - 1 is V22() real ext-real Element of REAL
(len e) - 1 is V22() real ext-real Element of REAL
len (upper_volume (T,f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(upper_volume (T,f)) . D is V22() real ext-real Element of REAL
((upper_volume (T,e)) /^ 1) . D is V22() real ext-real Element of REAL
D + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len (upper_volume (T,f))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume (T,f))) is non empty V37() len (upper_volume (T,f)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (f,D) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T | (divset (f,D)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (T | (divset (f,D))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (T | (divset (f,D)))) is V22() real ext-real Element of REAL
vol (divset (f,D)) is V22() real ext-real Element of REAL
(upper_bound (rng (T | (divset (f,D))))) * (vol (divset (f,D))) is V22() real ext-real Element of REAL
divset (e,(D + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T | (divset (e,(D + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (T | (divset (e,(D + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (T | (divset (e,(D + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng (T | (divset (e,(D + 1)))))) * (vol (divset (f,D))) is V22() real ext-real Element of REAL
vol (divset (e,(D + 1))) is V22() real ext-real Element of REAL
(upper_bound (rng (T | (divset (e,(D + 1)))))) * (vol (divset (e,(D + 1)))) is V22() real ext-real Element of REAL
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom ((upper_volume (T,e)) /^ 1) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume (T,e)) . (D + 1) is V22() real ext-real Element of REAL
rng (lower_volume (T,e)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
dom (lower_volume (T,e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (lower_volume (T,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((lower_volume (T,e)) /^ 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(len (lower_volume (T,e))) - 1 is V22() real ext-real Element of REAL
(len e) - 1 is V22() real ext-real Element of REAL
len (lower_volume (T,f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(lower_volume (T,f)) . e is V22() real ext-real Element of REAL
((lower_volume (T,e)) /^ 1) . e is V22() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (lower_volume (T,f))) is non empty V37() len (lower_volume (T,f)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (f,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T | (divset (f,e)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (T | (divset (f,e))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (T | (divset (f,e)))) is V22() real ext-real Element of REAL
vol (divset (f,e)) is V22() real ext-real Element of REAL
(lower_bound (rng (T | (divset (f,e))))) * (vol (divset (f,e))) is V22() real ext-real Element of REAL
divset (e,(e + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T | (divset (e,(e + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (T | (divset (e,(e + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (T | (divset (e,(e + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng (T | (divset (e,(e + 1)))))) * (vol (divset (f,e))) is V22() real ext-real Element of REAL
vol (divset (e,(e + 1))) is V22() real ext-real Element of REAL
(lower_bound (rng (T | (divset (e,(e + 1)))))) * (vol (divset (e,(e + 1)))) is V22() real ext-real Element of REAL
(len (lower_volume (T,f))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom ((lower_volume (T,e)) /^ 1) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(lower_volume (T,e)) . (e + 1) is V22() real ext-real Element of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound A is V22() real ext-real Element of REAL
<*(lower_bound A)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
<*(lower_bound A)*> ^ f is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
divset (T,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (T,1)) is V22() real ext-real Element of REAL
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
upper_bound (divset (T,1)) is V22() real ext-real Element of REAL
T . 1 is V22() real ext-real Element of REAL
lower_bound (divset (T,1)) is V22() real ext-real Element of REAL
(T . 1) - (lower_bound A) is V22() real ext-real Element of REAL
(lower_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound A is V22() real ext-real Element of REAL
<*(lower_bound A)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
<*(lower_bound A)*> ^ f is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,f) is V22() real ext-real Element of REAL
chi (A,A) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
upper_volume ((chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),f))) is V22() real ext-real set
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,T) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),T)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),T))) is V22() real ext-real set
divset (T,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (T,1)) is V22() real ext-real Element of REAL
dom (upper_volume ((chi (A,A)),f)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (A,A)),f)) . e is V22() real ext-real Element of REAL
dom (upper_volume ((chi (A,A)),T)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (A,A)),T)) . e is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),T)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),T))) is non empty V37() len (upper_volume ((chi (A,A)),T)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (T,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(chi (A,A)) | (divset (T,e)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng ((chi (A,A)) | (divset (T,e))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng ((chi (A,A)) | (divset (T,e)))) is V22() real ext-real Element of REAL
vol (divset (T,e)) is V22() real ext-real Element of REAL
(upper_bound (rng ((chi (A,A)) | (divset (T,e))))) * (vol (divset (T,e))) is V22() real ext-real Element of REAL
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len <*(lower_bound A)*>) is non empty V37() len <*(lower_bound A)*> -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom <*(lower_bound A)*> is non empty V12() V37() 1 -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(len <*(lower_bound A)*>) + y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (f,y) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
y + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (T,(y + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(chi (A,A)) | (divset (f,y)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng ((chi (A,A)) | (divset (f,y))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng ((chi (A,A)) | (divset (f,y)))) is V22() real ext-real Element of REAL
vol (divset (f,y)) is V22() real ext-real Element of REAL
(upper_bound (rng ((chi (A,A)) | (divset (f,y))))) * (vol (divset (f,y))) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),f)) . y is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),f))) is non empty V37() len (upper_volume ((chi (A,A)),f)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume ((chi (A,A)),f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),f))) is non empty V37() len (upper_volume ((chi (A,A)),f)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V37() len f -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom f is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound A)*>) + e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (f,e) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(chi (A,A)) | (divset (f,e)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng ((chi (A,A)) | (divset (f,e))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng ((chi (A,A)) | (divset (f,e)))) is V22() real ext-real Element of REAL
vol (divset (f,e)) is V22() real ext-real Element of REAL
(upper_bound (rng ((chi (A,A)) | (divset (f,e))))) * (vol (divset (f,e))) is V22() real ext-real Element of REAL
divset (T,(e + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(chi (A,A)) | (divset (T,(e + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng ((chi (A,A)) | (divset (T,(e + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng ((chi (A,A)) | (divset (T,(e + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng ((chi (A,A)) | (divset (T,(e + 1)))))) * (vol (divset (f,e))) is V22() real ext-real Element of REAL
vol (divset (T,(e + 1))) is V22() real ext-real Element of REAL
(upper_bound (rng ((chi (A,A)) | (divset (T,(e + 1)))))) * (vol (divset (T,(e + 1)))) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),T)) . (e + 1) is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
{A} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(f,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(f,REAL)) is V12() V37() set
vol f is V22() real ext-real Element of REAL
lower_bound f is V22() real ext-real Element of REAL
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (T,(len T)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng T) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(f,T) is V22() real ext-real Element of REAL
chi (f,f) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
upper_volume ((chi (f,f)),T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (f,f)),T)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (f,f)),T))) is V22() real ext-real set
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
e is Relation-like Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
e | f is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
lower_volume (e,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (lower_volume (e,e)) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(lower_volume (e,e)),K263()) is V22() real ext-real Element of REAL
lower_volume (e,T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (lower_volume (e,T)) is V22() real ext-real Element of REAL
K190(REAL,(lower_volume (e,T)),K263()) is V22() real ext-real Element of REAL
(Sum (lower_volume (e,e))) - (Sum (lower_volume (e,T))) is V22() real ext-real Element of REAL
rng e is non empty V58() V59() V60() Element of K19(REAL)
upper_bound (rng e) is V22() real ext-real Element of REAL
lower_bound (rng e) is V22() real ext-real Element of REAL
(upper_bound (rng e)) - (lower_bound (rng e)) is V22() real ext-real Element of REAL
((upper_bound (rng e)) - (lower_bound (rng e))) * (f,T) is V22() real ext-real Element of REAL
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
<*(lower_bound f)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
<*(lower_bound f)*> ^ T is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len <*(lower_bound f)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound f)*>) + (len T) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound f)*>) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y . (len y) is V22() real ext-real Element of REAL
((len <*(lower_bound f)*>) + (len T)) - (len <*(lower_bound f)*>) is V22() real ext-real Element of REAL
T . (((len <*(lower_bound f)*>) + (len T)) - (len <*(lower_bound f)*>)) is V22() real ext-real Element of REAL
T . (len T) is V22() real ext-real Element of REAL
upper_bound f is V22() real ext-real Element of REAL
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
D is V22() real ext-real Element of REAL
rng <*(lower_bound f)*> is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng <*(lower_bound f)*>) \/ (rng T) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
{(lower_bound f)} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
p is V22() real ext-real Element of REAL
H is V22() real ext-real Element of REAL
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
len D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 + (len T) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
T . 1 is V22() real ext-real Element of REAL
(T . 1) - (lower_bound f) is V22() real ext-real Element of REAL
lower_volume (e,D) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(lower_volume (e,D)) /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(lower_volume (e,D)) /. 1 is V22() real ext-real Element of REAL
<*((lower_volume (e,D)) /. 1)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
<*((lower_volume (e,D)) /. 1)*> ^ (lower_volume (e,T)) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (lower_volume (e,D)) is V22() real ext-real Element of REAL
K190(REAL,(lower_volume (e,D)),K263()) is V22() real ext-real Element of REAL
((lower_volume (e,D)) /. 1) + (Sum (lower_volume (e,T))) is V22() real ext-real Element of REAL
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom D is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len D) - 1 is V22() real ext-real Element of REAL
divset (D,(len D)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (D,(len D))) is V22() real ext-real Element of REAL
D . (len T) is V22() real ext-real Element of REAL
lower_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
<*(lower_bound f)*> ^ e is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng D is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(lower_volume (e,D)) . 1 is V22() real ext-real Element of REAL
divset (D,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e | (divset (D,1)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (D,1))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (e | (divset (D,1)))) is V22() real ext-real Element of REAL
vol (divset (D,1)) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (D,1))))) * (vol (divset (D,1))) is V22() real ext-real Element of REAL
Seg (len D) is non empty V37() len D -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (lower_volume (e,D)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (lower_volume (e,D))) is non empty V37() len (lower_volume (e,D)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (e,D)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound f)*>) + (len e) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . 1 is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . H is V22() real ext-real Element of REAL
H is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len H is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H . (len H) is V22() real ext-real Element of REAL
((len <*(lower_bound f)*>) + (len e)) - (len <*(lower_bound f)*>) is V22() real ext-real Element of REAL
e . (((len <*(lower_bound f)*>) + (len e)) - (len <*(lower_bound f)*>)) is V22() real ext-real Element of REAL
e . (len e) is V22() real ext-real Element of REAL
rng H is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
h is V22() real ext-real Element of REAL
rng <*(lower_bound f)*> is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng <*(lower_bound f)*>) \/ (rng e) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
{(lower_bound f)} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
v is V22() real ext-real Element of REAL
v1 is V22() real ext-real Element of REAL
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
h is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng h is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng <*(lower_bound f)*> is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng e) \/ (rng <*(lower_bound f)*>) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng T) \/ (rng <*(lower_bound f)*>) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
((rng T) \/ (rng <*(lower_bound f)*>)) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng D) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
D . (len D) is V22() real ext-real Element of REAL
D . ((len <*(lower_bound f)*>) + (len T)) is V22() real ext-real Element of REAL
upper_bound (divset (D,(len D))) is V22() real ext-real Element of REAL
(len T) + (len <*(lower_bound f)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len e) + (len <*(lower_bound f)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len h is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
lower_volume (e,h) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (lower_volume (e,h)) is V22() real ext-real Element of REAL
K190(REAL,(lower_volume (e,h)),K263()) is V22() real ext-real Element of REAL
(Sum (lower_volume (e,h))) - (Sum (lower_volume (e,D))) is V22() real ext-real Element of REAL
(f,D) is V22() real ext-real Element of REAL
upper_volume ((chi (f,f)),D) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (f,f)),D)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (f,f)),D))) is V22() real ext-real set
((upper_bound (rng e)) - (lower_bound (rng e))) * (f,D) is V22() real ext-real Element of REAL
dom h is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(lower_volume (e,h)) . 1 is V22() real ext-real Element of REAL
divset (h,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e | (divset (h,1)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (h,1))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (e | (divset (h,1)))) is V22() real ext-real Element of REAL
vol (divset (h,1)) is V22() real ext-real Element of REAL
(lower_bound (rng (e | (divset (h,1))))) * (vol (divset (h,1))) is V22() real ext-real Element of REAL
Seg (len h) is non empty V37() len h -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (lower_volume (e,h)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (lower_volume (e,h))) is non empty V37() len (lower_volume (e,h)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (e,h)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(lower_volume (e,h)) /. 1 is V22() real ext-real Element of REAL
(lower_volume (e,h)) /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
<*((lower_volume (e,h)) /. 1)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
<*((lower_volume (e,h)) /. 1)*> ^ (lower_volume (e,e)) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((lower_volume (e,h)) /. 1) + (Sum (lower_volume (e,e))) is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
{A} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(f,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(f,REAL)) is V12() V37() set
vol f is V22() real ext-real Element of REAL
lower_bound f is V22() real ext-real Element of REAL
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (T,(len T)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng T) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(f,T) is V22() real ext-real Element of REAL
chi (f,f) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
upper_volume ((chi (f,f)),T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (f,f)),T)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (f,f)),T))) is V22() real ext-real set
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
e is Relation-like Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
e | f is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
upper_volume (e,T) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (upper_volume (e,T)) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(upper_volume (e,T)),K263()) is V22() real ext-real Element of REAL
upper_volume (e,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (upper_volume (e,e)) is V22() real ext-real Element of REAL
K190(REAL,(upper_volume (e,e)),K263()) is V22() real ext-real Element of REAL
(Sum (upper_volume (e,T))) - (Sum (upper_volume (e,e))) is V22() real ext-real Element of REAL
rng e is non empty V58() V59() V60() Element of K19(REAL)
upper_bound (rng e) is V22() real ext-real Element of REAL
lower_bound (rng e) is V22() real ext-real Element of REAL
(upper_bound (rng e)) - (lower_bound (rng e)) is V22() real ext-real Element of REAL
((upper_bound (rng e)) - (lower_bound (rng e))) * (f,T) is V22() real ext-real Element of REAL
Seg (len T) is non empty V37() len T -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
<*(lower_bound f)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
<*(lower_bound f)*> ^ T is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len <*(lower_bound f)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound f)*>) + (len T) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound f)*>) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y . (len y) is V22() real ext-real Element of REAL
((len <*(lower_bound f)*>) + (len T)) - (len <*(lower_bound f)*>) is V22() real ext-real Element of REAL
T . (((len <*(lower_bound f)*>) + (len T)) - (len <*(lower_bound f)*>)) is V22() real ext-real Element of REAL
T . (len T) is V22() real ext-real Element of REAL
upper_bound f is V22() real ext-real Element of REAL
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
D is V22() real ext-real Element of REAL
rng <*(lower_bound f)*> is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng <*(lower_bound f)*>) \/ (rng T) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
{(lower_bound f)} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
p is V22() real ext-real Element of REAL
H is V22() real ext-real Element of REAL
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
len D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 + (len T) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
T . 1 is V22() real ext-real Element of REAL
(T . 1) - (lower_bound f) is V22() real ext-real Element of REAL
upper_volume (e,D) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(upper_volume (e,D)) /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(upper_volume (e,D)) /. 1 is V22() real ext-real Element of REAL
<*((upper_volume (e,D)) /. 1)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
<*((upper_volume (e,D)) /. 1)*> ^ (upper_volume (e,T)) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (upper_volume (e,D)) is V22() real ext-real Element of REAL
K190(REAL,(upper_volume (e,D)),K263()) is V22() real ext-real Element of REAL
((upper_volume (e,D)) /. 1) + (Sum (upper_volume (e,T))) is V22() real ext-real Element of REAL
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom D is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len D) - 1 is V22() real ext-real Element of REAL
divset (D,(len D)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (D,(len D))) is V22() real ext-real Element of REAL
D . (len T) is V22() real ext-real Element of REAL
lower_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
<*(lower_bound f)*> ^ e is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng D is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(upper_volume (e,D)) . 1 is V22() real ext-real Element of REAL
divset (D,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e | (divset (D,1)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (D,1))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (e | (divset (D,1)))) is V22() real ext-real Element of REAL
vol (divset (D,1)) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (D,1))))) * (vol (divset (D,1))) is V22() real ext-real Element of REAL
Seg (len D) is non empty V37() len D -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume (e,D)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume (e,D))) is non empty V37() len (upper_volume (e,D)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume (e,D)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound f)*>) + (len e) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . 1 is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . H is V22() real ext-real Element of REAL
H is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len H is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H . (len H) is V22() real ext-real Element of REAL
((len <*(lower_bound f)*>) + (len e)) - (len <*(lower_bound f)*>) is V22() real ext-real Element of REAL
e . (((len <*(lower_bound f)*>) + (len e)) - (len <*(lower_bound f)*>)) is V22() real ext-real Element of REAL
e . (len e) is V22() real ext-real Element of REAL
rng H is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
h is V22() real ext-real Element of REAL
rng <*(lower_bound f)*> is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng <*(lower_bound f)*>) \/ (rng e) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
{(lower_bound f)} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
v is V22() real ext-real Element of REAL
v1 is V22() real ext-real Element of REAL
upper_bound (divset (T,(len T))) is V22() real ext-real Element of REAL
h is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng h is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng <*(lower_bound f)*> is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng e) \/ (rng <*(lower_bound f)*>) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng T) \/ (rng <*(lower_bound f)*>) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
((rng T) \/ (rng <*(lower_bound f)*>)) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng D) \/ {A} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
D . (len D) is V22() real ext-real Element of REAL
D . ((len <*(lower_bound f)*>) + (len T)) is V22() real ext-real Element of REAL
upper_bound (divset (D,(len D))) is V22() real ext-real Element of REAL
(len T) + (len <*(lower_bound f)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len e) + (len <*(lower_bound f)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len h is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_volume (e,h) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (upper_volume (e,h)) is V22() real ext-real Element of REAL
K190(REAL,(upper_volume (e,h)),K263()) is V22() real ext-real Element of REAL
(Sum (upper_volume (e,D))) - (Sum (upper_volume (e,h))) is V22() real ext-real Element of REAL
(f,D) is V22() real ext-real Element of REAL
upper_volume ((chi (f,f)),D) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (f,f)),D)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (f,f)),D))) is V22() real ext-real set
((upper_bound (rng e)) - (lower_bound (rng e))) * (f,D) is V22() real ext-real Element of REAL
dom h is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume (e,h)) . 1 is V22() real ext-real Element of REAL
divset (h,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e | (divset (h,1)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (h,1))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (e | (divset (h,1)))) is V22() real ext-real Element of REAL
vol (divset (h,1)) is V22() real ext-real Element of REAL
(upper_bound (rng (e | (divset (h,1))))) * (vol (divset (h,1))) is V22() real ext-real Element of REAL
Seg (len h) is non empty V37() len h -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume (e,h)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume (e,h))) is non empty V37() len (upper_volume (e,h)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume (e,h)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume (e,h)) /. 1 is V22() real ext-real Element of REAL
(upper_volume (e,h)) /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
<*((upper_volume (e,h)) /. 1)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
<*((upper_volume (e,h)) /. 1)*> ^ (upper_volume (e,e)) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((upper_volume (e,h)) /. 1) + (Sum (upper_volume (e,e))) is V22() real ext-real Element of REAL
A is V22() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of e
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
mid (e,f,T) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of e
indx (y,e,f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (y,e,T) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
mid (y,(indx (y,e,f)),(indx (y,e,T))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (y,(indx (y,e,f)),(indx (y,e,T)))) . 1 is V22() real ext-real Element of REAL
T - f is V22() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(T - f) + 1 is V22() real ext-real Element of REAL
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid (e,f,T)) . 1 is V22() real ext-real Element of REAL
1 + f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + f) - 1 is V22() real ext-real Element of REAL
e . ((1 + f) - 1) is V22() real ext-real Element of REAL
e . f is V22() real ext-real Element of REAL
y . (indx (y,e,f)) is V22() real ext-real Element of REAL
y . (indx (y,e,T)) is V22() real ext-real Element of REAL
e . T is V22() real ext-real Element of REAL
dom y is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid (y,(indx (y,e,f)),(indx (y,e,T)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(mid (y,(indx (y,e,f)),(indx (y,e,T)))) . (len (mid (y,(indx (y,e,f)),(indx (y,e,T))))) is V22() real ext-real Element of REAL
H is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound H is V22() real ext-real Element of REAL
upper_bound H is V22() real ext-real Element of REAL
(indx (y,e,T)) - (indx (y,e,f)) is V22() real ext-real Element of REAL
((indx (y,e,T)) - (indx (y,e,f))) + 1 is V22() real ext-real Element of REAL
h is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of H
len h is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h . (len h) is V22() real ext-real Element of REAL
(((indx (y,e,T)) - (indx (y,e,f))) + 1) - 1 is V22() real ext-real Element of REAL
((((indx (y,e,T)) - (indx (y,e,f))) + 1) - 1) + (indx (y,e,f)) is V22() real ext-real Element of REAL
y . (((((indx (y,e,T)) - (indx (y,e,f))) + 1) - 1) + (indx (y,e,f))) is V22() real ext-real Element of REAL
h . 1 is V22() real ext-real Element of REAL
1 + (indx (y,e,f)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + (indx (y,e,f))) - 1 is V22() real ext-real Element of REAL
y . ((1 + (indx (y,e,f))) - 1) is V22() real ext-real Element of REAL
len (mid (e,f,T)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(mid (e,f,T)) . (len (mid (e,f,T))) is V22() real ext-real Element of REAL
v is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound v is V22() real ext-real Element of REAL
upper_bound v is V22() real ext-real Element of REAL
((T - f) + 1) - 1 is V22() real ext-real Element of REAL
(((T - f) + 1) - 1) + f is V22() real ext-real Element of REAL
e . ((((T - f) + 1) - 1) + f) is V22() real ext-real Element of REAL
[.(lower_bound H),(upper_bound H).] is V58() V59() V60() interval Element of K19(REAL)
v1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of H
rng v1 is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng h is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
k is set
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
dom v1 is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v1 . n is V22() real ext-real Element of REAL
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . s is V22() real ext-real Element of REAL
len v1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v1 . 1 is V22() real ext-real Element of REAL
s + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(indx (y,e,f)) + sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v1 . (len v1) is V22() real ext-real Element of REAL
(indx (y,e,T)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(s + 1) - (indx (y,e,f)) is V22() real ext-real Element of REAL
((indx (y,e,T)) + 1) - (indx (y,e,f)) is V22() real ext-real Element of REAL
(indx (y,e,f)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom h is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
h . sD is V22() real ext-real Element of REAL
sD + (indx (y,e,f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(sD + (indx (y,e,f))) - 1 is V22() real ext-real Element of REAL
y . ((sD + (indx (y,e,f))) - 1) is V22() real ext-real Element of REAL
card (rng h) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of omega
card (rng v1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of omega
len v1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
A is V22() real ext-real Element of REAL
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
rng T is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
T . 1 is V22() real ext-real Element of REAL
len T is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
T . (len T) is V22() real ext-real Element of REAL
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T . e is V22() real ext-real Element of REAL
A is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom A is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
A . f is V22() real ext-real Element of REAL
T is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
A . T is V22() real ext-real Element of REAL
mid (A,f,T) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (mid (A,f,T)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
A . e is V22() real ext-real Element of REAL
len A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
f + e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e - f is V22() real ext-real Element of REAL
T - f is V22() real ext-real Element of REAL
(e - f) + 1 is V22() real ext-real Element of REAL
(T - f) + 1 is V22() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid (A,f,T)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
T -' f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(T -' f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom (mid (A,f,T)) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(mid (A,f,T)) . e is V22() real ext-real Element of REAL
e + f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(e + f) - 1 is V22() real ext-real Element of REAL
A . ((e + f) - 1) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(f,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(f,REAL)) is V12() V37() set
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (T,A) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e is Relation-like Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
e | f is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
e | (divset (T,A)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (T,A))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (e | (divset (T,A)))) is V22() real ext-real Element of REAL
rng e is non empty V58() V59() V60() Element of K19(REAL)
upper_bound (rng e) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
f is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(f,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(f,REAL)) is V12() V37() set
T is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of f
dom T is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (T,A) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
e is Relation-like Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
e | f is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng e is non empty V58() V59() V60() Element of K19(REAL)
lower_bound (rng e) is V22() real ext-real Element of REAL
e | (divset (T,A)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(f,REAL))
rng (e | (divset (T,A))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (e | (divset (T,A)))) is V22() real ext-real Element of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
divs A is non empty set
K20(NAT,(divs A)) is Relation-like V12() V37() set
K19(K20(NAT,(divs A))) is V12() V37() set
vol A is V22() real ext-real Element of REAL
f is Relation-like Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
f | A is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
lower_integral f is V22() real ext-real Element of REAL
T is Relation-like Function-like non empty total V30( NAT , divs A) Element of K19(K20(NAT,(divs A)))
(A,T) is Relation-like Function-like non empty total V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lower_sum (f,T) is Relation-like Function-like non empty total V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (lower_sum (f,T)) is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,e) is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng e) \/ (rng e) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,e) is V22() real ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,y) is V22() real ext-real Element of REAL
(lower_sum (f,y)) - (lower_sum (f,e)) is V22() real ext-real Element of REAL
(lower_sum (f,y)) - (lower_sum (f,e)) is V22() real ext-real Element of REAL
chi (A,A) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng f is non empty V58() V59() V60() Element of K19(REAL)
upper_bound (rng f) is V22() real ext-real Element of REAL
lower_bound (rng f) is V22() real ext-real Element of REAL
(upper_bound (rng f)) - (lower_bound (rng f)) is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
upper_volume ((chi (A,A)),e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
inf (rng (upper_volume ((chi (A,A)),e))) is V22() real ext-real set
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len e) * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,e) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),e))) is V22() real ext-real set
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng e) \/ (rng e) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,e) is V22() real ext-real Element of REAL
((len e) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
lower_sum (f,e) is V22() real ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,y) is V22() real ext-real Element of REAL
(lower_sum (f,y)) - (lower_sum (f,e)) is V22() real ext-real Element of REAL
(lower_sum (f,y)) - (lower_sum (f,e)) is V22() real ext-real Element of REAL
len y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom y is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
lower_volume (f,y) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
PartSums (lower_volume (f,y)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
lower_volume (f,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
PartSums (lower_volume (f,e)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . D is V22() real ext-real Element of REAL
D * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(D * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . p is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,H) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
lower_bound (divset (e,H)) is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
upper_bound (divset (e,H)) is V22() real ext-real Element of REAL
divset (e,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,1)) is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),e))) is non empty V37() len (upper_volume ((chi (A,A)),e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
vol (divset (e,1)) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),e)) . 1 is V22() real ext-real Element of REAL
upper_bound (divset (e,1)) is V22() real ext-real Element of REAL
e . 1 is V22() real ext-real Element of REAL
(e . 1) - (lower_bound A) is V22() real ext-real Element of REAL
(upper_bound (divset (e,H))) - (lower_bound A) is V22() real ext-real Element of REAL
(upper_bound (divset (e,1))) - (lower_bound (divset (e,1))) is V22() real ext-real Element of REAL
vol (divset (e,H)) is V22() real ext-real Element of REAL
p - 1 is V22() real ext-real Element of REAL
e . (p - 1) is V22() real ext-real Element of REAL
divset (e,p) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
upper_bound (divset (e,p)) is V22() real ext-real Element of REAL
(upper_bound (divset (e,p))) - (lower_bound (divset (e,p))) is V22() real ext-real Element of REAL
(upper_bound (divset (e,p))) - (lower_bound A) is V22() real ext-real Element of REAL
(upper_bound (divset (e,H))) - (lower_bound (divset (e,H))) is V22() real ext-real Element of REAL
vol (divset (e,H)) is V22() real ext-real Element of REAL
vol (divset (e,p)) is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),e))) is non empty V37() len (upper_volume ((chi (A,A)),e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),e)) . p is V22() real ext-real Element of REAL
e . 1 is V22() real ext-real Element of REAL
1 * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(1 * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,p) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
indx (y,e,p) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (lower_volume (f,y))) . (indx (y,e,p)) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,e))) . p is V22() real ext-real Element of REAL
H₂(y, indx (y,e,p)) - H₂(e,p) is V22() real ext-real Element of REAL
1 * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(1 * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
p - 1 is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (lower_volume (f,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (lower_volume (f,e))) is non empty V37() len (lower_volume (f,e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (f,e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
indx (y,e,H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len y) is non empty V37() len y -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
mid (y,1,(indx (y,e,H))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y | (indx (y,e,H)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,H)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))),K263()) is V22() real ext-real Element of REAL
mid ((lower_volume (f,e)),p,p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid ((lower_volume (f,e)),p,p)) is V22() real ext-real Element of REAL
K190(REAL,(mid ((lower_volume (f,e)),p,p)),K263()) is V22() real ext-real Element of REAL
(Sum (mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p))))) - (Sum (mid ((lower_volume (f,e)),p,p))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (A,e) is V22() real ext-real Element of REAL
(indx (y,e,p)) - (indx (y,e,H)) is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,H)) + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,p)) is V22() real ext-real Element of REAL
y . v is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
y . h is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
v1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
v1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (e,p)) is V22() real ext-real Element of REAL
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
(rng e) \/ (rng e) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng e) /\ (divset (e,p)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
(mid ((lower_volume (f,e)),p,p)) . 1 is V22() real ext-real Element of REAL
(lower_volume (f,e)) . p is V22() real ext-real Element of REAL
p -' p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(p -' p) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid ((lower_volume (f,e)),p,p)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
<*((lower_volume (f,e)) . p)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
len (lower_volume (f,y)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (lower_volume (f,y))) is non empty V37() len (lower_volume (f,y)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(indx (y,e,p)) -' ((indx (y,e,H)) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,p)) - ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
((indx (y,e,p)) -' ((indx (y,e,H)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
y . (indx (y,e,p)) is V22() real ext-real Element of REAL
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
[.(y . (indx (y,e,H))),(y . (indx (y,e,p))).] is V58() V59() V60() interval Element of K19(REAL)
divset (y,(indx (y,e,p))) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (y,(indx (y,e,p)))) is V22() real ext-real Element of REAL
(indx (y,e,p)) - 1 is V22() real ext-real Element of REAL
lower_bound (divset (y,(indx (y,e,p)))) is V22() real ext-real Element of REAL
(mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) . 1 is V22() real ext-real Element of REAL
(lower_volume (f,y)) . ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
<*((lower_volume (f,y)) . ((indx (y,e,H)) + 1))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
divset (y,((indx (y,e,H)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
f | (divset (y,((indx (y,e,H)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,H)) + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
vol (divset (y,((indx (y,e,H)) + 1))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
(mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) . 1 is V22() real ext-real Element of REAL
(lower_volume (f,y)) . ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
2 + ((indx (y,e,H)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) . 2 is V22() real ext-real Element of REAL
(2 + ((indx (y,e,H)) + 1)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
H₁(y) . ((2 + ((indx (y,e,H)) + 1)) -' 1) is V22() real ext-real Element of REAL
(2 + ((indx (y,e,H)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((2 + ((indx (y,e,H)) + 1)) - 1) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,H)) + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H₁(y) . ((indx (y,e,H)) + (1 + 1)) is V22() real ext-real Element of REAL
(indx (y,e,H)) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(lower_volume (f,y)) . ((indx (y,e,H)) + 2) is V22() real ext-real Element of REAL
<*((lower_volume (f,y)) . ((indx (y,e,H)) + 1)),((lower_volume (f,y)) . ((indx (y,e,H)) + 2))*> is Relation-like NAT -defined Function-like non empty V37() 2 -element FinSequence-like FinSubsequence-like set
((lower_volume (f,y)) . ((indx (y,e,H)) + 1)) + ((lower_volume (f,y)) . ((indx (y,e,H)) + 2)) is V22() real ext-real Element of REAL
divset (y,((indx (y,e,H)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (y,((indx (y,e,H)) + 1))) is V22() real ext-real Element of REAL
upper_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
y . (indx (y,e,p)) is V22() real ext-real Element of REAL
divset (y,((indx (y,e,H)) + 2)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (y,((indx (y,e,H)) + 2))) is V22() real ext-real Element of REAL
((indx (y,e,p)) - ((indx (y,e,H)) + 1)) + 1 is V22() real ext-real Element of REAL
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
vol (divset (e,p)) is V22() real ext-real Element of REAL
y . ((indx (y,e,H)) + 2) is V22() real ext-real Element of REAL
y . ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
(y . ((indx (y,e,H)) + 2)) - (y . ((indx (y,e,H)) + 1)) is V22() real ext-real Element of REAL
((y . ((indx (y,e,H)) + 2)) - (y . ((indx (y,e,H)) + 1))) + (y . ((indx (y,e,H)) + 1)) is V22() real ext-real Element of REAL
(((y . ((indx (y,e,H)) + 2)) - (y . ((indx (y,e,H)) + 1))) + (y . ((indx (y,e,H)) + 1))) - (y . (indx (y,e,H))) is V22() real ext-real Element of REAL
upper_bound (divset (y,((indx (y,e,H)) + 1))) is V22() real ext-real Element of REAL
((indx (y,e,H)) + 1) - 1 is V22() real ext-real Element of REAL
(indx (y,e,H)) + 0 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
lower_bound (divset (y,((indx (y,e,H)) + 1))) is V22() real ext-real Element of REAL
((indx (y,e,H)) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,H)) + 2) - 1 is V22() real ext-real Element of REAL
lower_bound (divset (y,((indx (y,e,H)) + 2))) is V22() real ext-real Element of REAL
upper_bound (divset (y,((indx (y,e,H)) + 2))) is V22() real ext-real Element of REAL
(vol (divset (y,((indx (y,e,H)) + 2)))) + (y . ((indx (y,e,H)) + 1)) is V22() real ext-real Element of REAL
((vol (divset (y,((indx (y,e,H)) + 2)))) + (y . ((indx (y,e,H)) + 1))) - (y . (indx (y,e,H))) is V22() real ext-real Element of REAL
(upper_bound (divset (y,((indx (y,e,H)) + 1)))) - (lower_bound (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
(vol (divset (y,((indx (y,e,H)) + 2)))) + ((upper_bound (divset (y,((indx (y,e,H)) + 1)))) - (lower_bound (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(vol (divset (y,((indx (y,e,H)) + 1)))) + (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
f | (divset (e,p)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (e,p))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (e,p)))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (e,p))))) * ((vol (divset (y,((indx (y,e,H)) + 1)))) + (vol (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p)))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),p,p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),p,p)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),p,p)),K263()) is V22() real ext-real Element of REAL
(Sum (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) - (Sum (mid (H₁(e),p,p))) is V22() real ext-real Element of REAL
(vol (divset (y,((indx (y,e,H)) + 2)))) + (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * ((vol (divset (y,((indx (y,e,H)) + 2)))) + (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
f | (divset (y,((indx (y,e,H)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,H)) + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
f | (divset (y,((indx (y,e,H)) + 2))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,H)) + 2)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
n is V22() real ext-real set
n * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
((lower_volume (f,e)) . p) - (n * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
n * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (e,p))))) * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (e,p))))) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
((lower_bound (rng (f | (divset (e,p))))) * (vol (divset (y,((indx (y,e,H)) + 1))))) + ((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),p,p))) - ((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 2)))))) * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
H₁(y) . ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
((lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 2)))))) * (vol (divset (y,((indx (y,e,H)) + 2))))) + (H₁(y) . ((indx (y,e,H)) + 1)) is V22() real ext-real Element of REAL
((lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 2)))))) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) - ((lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((lower_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) - ((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1)))))) - (((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1)))))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (vol (divset (e,p))) is V22() real ext-real Element of REAL
len (y | (indx (y,e,H))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e | H is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (e | H) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
rng (y | (indx (y,e,H))) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
h is set
dom (e | H) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(e | H) . v is V22() real ext-real Element of REAL
len (e | H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len (e | H)) is V37() len (e | H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
e . v is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
indx (y,e,v) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,v)) is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . v1 is V22() real ext-real Element of REAL
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len (y | (indx (y,e,H)))) is V37() len (y | (indx (y,e,H))) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
dom (y | (indx (y,e,H))) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(y | (indx (y,e,H))) . v1 is V22() real ext-real Element of REAL
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
h is set
dom (y | (indx (y,e,H))) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(y | (indx (y,e,H))) . v is V22() real ext-real Element of REAL
Seg (len (y | (indx (y,e,H)))) is V37() len (y | (indx (y,e,H))) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
len (e | H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . v is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(e | H) . v1 is V22() real ext-real Element of REAL
dom (e | H) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
Seg (len (e | H)) is V37() len (e | H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
dom (e | H) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(e | H) . H is V22() real ext-real Element of REAL
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
mid (e,1,H) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (y,e,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(y | (indx (y,e,H))) . (indx (y,e,h)) is V22() real ext-real Element of REAL
y . (indx (y,e,h)) is V22() real ext-real Element of REAL
e . h is V22() real ext-real Element of REAL
e . (indx (y,e,h)) is V22() real ext-real Element of REAL
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
e . h is V22() real ext-real Element of REAL
(e | H) . h is V22() real ext-real Element of REAL
y . (indx (y,e,h)) is V22() real ext-real Element of REAL
y . h is V22() real ext-real Element of REAL
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(lower_volume (f,e)) | H is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((lower_volume (f,e)) | H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(lower_volume (f,y)) | (indx (y,e,H)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (lower_volume (f,y)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (lower_volume (f,y))) is non empty V37() len (lower_volume (f,y)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (f,y)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
((lower_volume (f,e)) | H) . h is V22() real ext-real Element of REAL
((lower_volume (f,y)) | (indx (y,e,H))) . h is V22() real ext-real Element of REAL
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (y,e,v) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
e . v is V22() real ext-real Element of REAL
y . (indx (y,e,v)) is V22() real ext-real Element of REAL
divset (e,v) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,v)) is V22() real ext-real Element of REAL
divset (y,(indx (y,e,v))) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (y,(indx (y,e,v)))) is V22() real ext-real Element of REAL
upper_bound (divset (e,v)) is V22() real ext-real Element of REAL
upper_bound (divset (y,(indx (y,e,v)))) is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
v - 1 is V22() real ext-real Element of REAL
v + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (y,e,v1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,v)) - 1 is V22() real ext-real Element of REAL
y . ((indx (y,e,v)) - 1) is V22() real ext-real Element of REAL
y . (indx (y,e,v1)) is V22() real ext-real Element of REAL
e . (v - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (y,(indx (y,e,v))))),(upper_bound (divset (y,(indx (y,e,v))))).] is V58() V59() V60() interval Element of K19(REAL)
((lower_volume (f,e)) | H) . v is V22() real ext-real Element of REAL
(lower_volume (f,e)) . v is V22() real ext-real Element of REAL
f | (divset (y,(indx (y,e,v)))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,(indx (y,e,v))))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (y,(indx (y,e,v)))))) is V22() real ext-real Element of REAL
vol (divset (y,(indx (y,e,v)))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (y,(indx (y,e,v))))))) * (vol (divset (y,(indx (y,e,v))))) is V22() real ext-real Element of REAL
((lower_volume (f,y)) | (indx (y,e,H))) . v is V22() real ext-real Element of REAL
((lower_volume (f,y)) | (indx (y,e,H))) . (indx (y,e,v)) is V22() real ext-real Element of REAL
(lower_volume (f,y)) . (indx (y,e,v)) is V22() real ext-real Element of REAL
len (lower_volume (f,y)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (e | H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len ((lower_volume (f,y)) | (indx (y,e,H))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len H₁(y) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len H₁(y)) is non empty V37() len H₁(y) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(y) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len H₁(e) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len H₁(e)) is non empty V37() len H₁(e) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(e) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(PartSums (lower_volume (f,e))) . H is V22() real ext-real Element of REAL
H₁(e) | H is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | H) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | H),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),p,p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),p,p)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),p,p)),K263()) is V22() real ext-real Element of REAL
H₂(e,H) + (Sum (mid (H₁(e),p,p))) is V22() real ext-real Element of REAL
(H₁(e) | H) ^ (mid (H₁(e),p,p)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((H₁(e) | H) ^ (mid (H₁(e),p,p))) is V22() real ext-real Element of REAL
K190(REAL,((H₁(e) | H) ^ (mid (H₁(e),p,p))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,H) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (H₁(e),(H + 1),p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (H₁(e),1,H)) ^ (mid (H₁(e),(H + 1),p)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(e),1,H)) ^ (mid (H₁(e),(H + 1),p))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(e),1,H)) ^ (mid (H₁(e),(H + 1),p))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),1,p)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),1,p)),K263()) is V22() real ext-real Element of REAL
H₁(e) | p is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | p) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | p),K263()) is V22() real ext-real Element of REAL
H₂(e,H) + (Sum (mid ((lower_volume (f,e)),p,p))) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,y))) . (indx (y,e,H)) is V22() real ext-real Element of REAL
H₁(y) | (indx (y,e,H)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(y) | (indx (y,e,H))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(y) | (indx (y,e,H))),K263()) is V22() real ext-real Element of REAL
H₂(y, indx (y,e,H)) + (Sum (mid ((lower_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p))))) is V22() real ext-real Element of REAL
mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(H₁(y) | (indx (y,e,H))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((H₁(y) | (indx (y,e,H))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) is V22() real ext-real Element of REAL
K190(REAL,((H₁(y) | (indx (y,e,H))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),1,(indx (y,e,H))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (H₁(y),1,(indx (y,e,H)))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(y),1,(indx (y,e,H)))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(y),1,(indx (y,e,H)))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),1,(indx (y,e,p))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),1,(indx (y,e,p)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),1,(indx (y,e,p)))),K263()) is V22() real ext-real Element of REAL
H₁(y) | (indx (y,e,p)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(y) | (indx (y,e,p))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(y) | (indx (y,e,p))),K263()) is V22() real ext-real Element of REAL
Seg (len (lower_volume (f,y))) is non empty V37() len (lower_volume (f,y)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (lower_volume (f,y)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Sum ((lower_volume (f,y)) | (indx (y,e,H))) is V22() real ext-real Element of REAL
K190(REAL,((lower_volume (f,y)) | (indx (y,e,H))),K263()) is V22() real ext-real Element of REAL
H is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal set
e . H is V22() real ext-real Element of REAL
H * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(H * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (H + 1) is V22() real ext-real Element of REAL
(H + 1) * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
((H + 1) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,h) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
indx (y,e,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,h)) is V22() real ext-real Element of REAL
e . h is V22() real ext-real Element of REAL
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,v) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
indx (y,e,v) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (lower_volume (f,y))) . (indx (y,e,v)) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,e))) . v is V22() real ext-real Element of REAL
((PartSums (lower_volume (f,y))) . (indx (y,e,v))) - ((PartSums (lower_volume (f,e))) . v) is V22() real ext-real Element of REAL
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,v) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
indx (y,e,v) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (lower_volume (f,y))) . (indx (y,e,v)) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,e))) . v is V22() real ext-real Element of REAL
H₂(y, indx (y,e,v)) - H₂(e,v) is V22() real ext-real Element of REAL
v + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(rng e) /\ (divset (e,h)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v - 1 is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
lower_bound (divset (e,v)) is V22() real ext-real Element of REAL
e . (v - 1) is V22() real ext-real Element of REAL
upper_bound (divset (e,h)) is V22() real ext-real Element of REAL
len H₁(y) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
indx (y,e,(v + 1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
e . (v + 1) is V22() real ext-real Element of REAL
1 + (indx (y,e,(v + 1))) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,h)) + 1) - (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
mid (y,(indx (y,e,(v + 1))),(indx (y,e,h))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (y,(indx (y,e,(v + 1))),(indx (y,e,h)))) . 1 is V22() real ext-real Element of REAL
1 - 1 is V22() real ext-real Element of REAL
(1 - 1) + (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
y . ((1 - 1) + (indx (y,e,(v + 1)))) is V22() real ext-real Element of REAL
y . (indx (y,e,v)) is V22() real ext-real Element of REAL
e . v is V22() real ext-real Element of REAL
len H₁(e) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len H₁(e)) is non empty V37() len H₁(e) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(e) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
H₁(e) | v is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | v) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(H₁(e) | v),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,v) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),1,v)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),1,v)),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),(v + 1),h) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),(v + 1),h)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),(v + 1),h)),K263()) is V22() real ext-real Element of REAL
H₂(e,v) + (Sum (mid (H₁(e),(v + 1),h))) is V22() real ext-real Element of REAL
(mid (H₁(e),1,v)) ^ (mid (H₁(e),(v + 1),h)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(e),1,v)) ^ (mid (H₁(e),(v + 1),h))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(e),1,v)) ^ (mid (H₁(e),(v + 1),h))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,h) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),1,h)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),1,h)),K263()) is V22() real ext-real Element of REAL
H₁(e) | h is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | h) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | h),K263()) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,e))) . h is V22() real ext-real Element of REAL
Seg (len y) is non empty V37() len y -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len H₁(y)) is non empty V37() len H₁(y) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(y) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
h - v is V22() real ext-real Element of REAL
(indx (y,e,v)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))),K263()) is V22() real ext-real Element of REAL
(Sum (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))) - (Sum (mid (H₁(e),(v + 1),h))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (A,e) is V22() real ext-real Element of REAL
(indx (y,e,h)) - (indx (y,e,v)) is V22() real ext-real Element of REAL
upper_bound (divset (e,h)) is V22() real ext-real Element of REAL
lower_bound (divset (e,h)) is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
(indx (y,e,v)) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y . ((indx (y,e,v)) + 2) is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
(rng e) /\ (divset (e,h)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
y . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
(indx (y,e,v)) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,v)) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) -' ((indx (y,e,v)) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,h)) -' ((indx (y,e,v)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) - ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
((indx (y,e,h)) - ((indx (y,e,v)) + 1)) + 1 is V22() real ext-real Element of REAL
divset (e,(v + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (e,(v + 1))) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (vol (divset (e,(v + 1)))) is V22() real ext-real Element of REAL
divset (y,((indx (y,e,v)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
y . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
lower_bound (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
((indx (y,e,v)) + 1) - 1 is V22() real ext-real Element of REAL
y . (((indx (y,e,v)) + 1) - 1) is V22() real ext-real Element of REAL
lower_bound (divset (e,(v + 1))) is V22() real ext-real Element of REAL
(v + 1) - 1 is V22() real ext-real Element of REAL
e . ((v + 1) - 1) is V22() real ext-real Element of REAL
vol (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) . 1 is V22() real ext-real Element of REAL
1 + ((indx (y,e,v)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((1 + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
H₁(y) . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
<*(H₁(y) . ((indx (y,e,v)) + 1))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
f | (divset (y,((indx (y,e,v)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,v)) + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (y,((indx (y,e,v)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (y,((indx (y,e,v)) + 1)))))) * (vol (divset (y,((indx (y,e,v)) + 1)))) is V22() real ext-real Element of REAL
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (y,((indx (y,e,v)) + 2)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (y,((indx (y,e,v)) + 2))) is V22() real ext-real Element of REAL
((indx (y,e,v)) + 2) - 1 is V22() real ext-real Element of REAL
lower_bound (divset (y,((indx (y,e,v)) + 2))) is V22() real ext-real Element of REAL
y . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
vol (divset (y,((indx (y,e,v)) + 2))) is V22() real ext-real Element of REAL
(e . h) - (y . ((indx (y,e,v)) + 1)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(v + 1))) is V22() real ext-real Element of REAL
lower_bound (divset (e,(v + 1))) is V22() real ext-real Element of REAL
(v + 1) - 1 is V22() real ext-real Element of REAL
e . ((v + 1) - 1) is V22() real ext-real Element of REAL
(e . (v + 1)) - (e . v) is V22() real ext-real Element of REAL
divset (y,((indx (y,e,v)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
H₁(y) . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
f | (divset (y,((indx (y,e,v)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,v)) + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (y,((indx (y,e,v)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (y,((indx (y,e,v)) + 1)))))) * (vol (divset (y,((indx (y,e,v)) + 1)))) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (vol (divset (y,((indx (y,e,v)) + 1)))) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) . 2 is V22() real ext-real Element of REAL
2 + ((indx (y,e,v)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(2 + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((2 + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
(indx (y,e,v)) + 0 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,v)) + 0) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H₁(y) . (((indx (y,e,v)) + 0) + 2) is V22() real ext-real Element of REAL
(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) . 1 is V22() real ext-real Element of REAL
1 + ((indx (y,e,v)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((1 + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
H₁(y) . ((indx (y,e,v)) + 2) is V22() real ext-real Element of REAL
<*(H₁(y) . ((indx (y,e,v)) + 1)),(H₁(y) . ((indx (y,e,v)) + 2))*> is Relation-like NAT -defined Function-like non empty V37() 2 -element FinSequence-like FinSubsequence-like set
(H₁(y) . ((indx (y,e,v)) + 1)) + (H₁(y) . ((indx (y,e,v)) + 2)) is V22() real ext-real Element of REAL
upper_bound (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
lower_bound (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
((indx (y,e,v)) + 1) - 1 is V22() real ext-real Element of REAL
y . (((indx (y,e,v)) + 1) - 1) is V22() real ext-real Element of REAL
(y . ((indx (y,e,v)) + 1)) - (e . v) is V22() real ext-real Element of REAL
f | (divset (y,((indx (y,e,v)) + 2))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,v)) + 2)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (y,((indx (y,e,v)) + 2))))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (y,((indx (y,e,v)) + 2)))))) * (vol (divset (y,((indx (y,e,v)) + 2)))) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (vol (divset (y,((indx (y,e,v)) + 2)))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,v)) + 1))))) + ((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,v)) + 2))))) is V22() real ext-real Element of REAL
len (mid (H₁(e),(v + 1),h)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
h -' (v + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(h -' (v + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h - h is V22() real ext-real Element of REAL
(h - h) + 1 is V22() real ext-real Element of REAL
(v + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(h + 1) - (v + 1) is V22() real ext-real Element of REAL
(mid (H₁(e),(v + 1),h)) . 1 is V22() real ext-real Element of REAL
(1 - 1) + (v + 1) is V22() real ext-real Element of REAL
H₁(e) . ((1 - 1) + (v + 1)) is V22() real ext-real Element of REAL
f | (divset (e,(v + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (e,(v + 1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (e,(v + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (e,(v + 1)))))) * (vol (divset (e,(v + 1)))) is V22() real ext-real Element of REAL
<*((lower_bound (rng (f | (divset (e,(v + 1)))))) * (vol (divset (e,(v + 1)))))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
len (upper_volume ((chi (A,A)),e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),e))) is non empty V37() len (upper_volume ((chi (A,A)),e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),e)) . (v + 1) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (vol (divset (e,(v + 1)))) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (vol (divset (e,(v + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) * (vol (divset (e,(v + 1))))) - ((lower_bound (rng f)) * (vol (divset (e,(v + 1))))) is V22() real ext-real Element of REAL
mid (e,(v + 1),h) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
v1 is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound v1 is V22() real ext-real Element of REAL
upper_bound v1 is V22() real ext-real Element of REAL
n is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of v1
len n is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
n . (len n) is V22() real ext-real Element of REAL
k is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of v1
len k is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
k . (len k) is V22() real ext-real Element of REAL
(v1,k) is V22() real ext-real Element of REAL
chi (v1,v1) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
K20(v1,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(v1,REAL)) is V12() V37() set
upper_volume ((chi (v1,v1)),k) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (v1,v1)),k)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (v1,v1)),k))) is V22() real ext-real set
h -' (v + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(h -' (v + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len k) + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((len k) + (v + 1)) - 1 is V22() real ext-real Element of REAL
h - (v + 1) is V22() real ext-real Element of REAL
(h - (v + 1)) + 1 is V22() real ext-real Element of REAL
((h - (v + 1)) + 1) + (v + 1) is V22() real ext-real Element of REAL
(((h - (v + 1)) + 1) + (v + 1)) - 1 is V22() real ext-real Element of REAL
s is set
lower_bound A is V22() real ext-real Element of REAL
sD is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
(h - v) - 1 is V22() real ext-real Element of REAL
((h - v) - 1) + (v + 1) is V22() real ext-real Element of REAL
e . (((h - v) - 1) + (v + 1)) is V22() real ext-real Element of REAL
f | v1 is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
s is Relation-like Function-like non empty total V30(v1, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
lower_volume (s,k) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (lower_volume (s,k)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom k is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (k,(len k)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (k,(len k))) is V22() real ext-real Element of REAL
lower_bound (divset (e,h)) is V22() real ext-real Element of REAL
upper_bound (divset (k,(len k))) is V22() real ext-real Element of REAL
upper_bound (divset (e,h)) is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
(len k) - 1 is V22() real ext-real Element of REAL
(len k) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((len k) - 1) + (v + 1) is V22() real ext-real Element of REAL
(((len k) - 1) + (v + 1)) - 1 is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
k . ((len k) - 1) is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
vol v1 is V22() real ext-real Element of REAL
(upper_bound v1) - (e . v) is V22() real ext-real Element of REAL
(e . h) - (e . v) is V22() real ext-real Element of REAL
y . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . sD is V22() real ext-real Element of REAL
upper_bound (divset (e,v)) is V22() real ext-real Element of REAL
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . sD is V22() real ext-real Element of REAL
(v + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h - 1 is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(lower_volume (s,k)) . sD is V22() real ext-real Element of REAL
(mid (H₁(e),(v + 1),h)) . sD is V22() real ext-real Element of REAL
Seg (len (lower_volume (s,k))) is non empty V37() len (lower_volume (s,k)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len k) is non empty V37() len k -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (k,sD) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
s | (divset (k,sD)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
rng (s | (divset (k,sD))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (s | (divset (k,sD)))) is V22() real ext-real Element of REAL
vol (divset (k,sD)) is V22() real ext-real Element of REAL
(lower_bound (rng (s | (divset (k,sD))))) * (vol (divset (k,sD))) is V22() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
1 + m is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
sD + m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(v + 1) - 1 is V22() real ext-real Element of REAL
h - ((v + 1) - 1) is V22() real ext-real Element of REAL
sD + ((v + 1) - 1) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (e,(sD + m)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,(sD + m))) is V22() real ext-real Element of REAL
lower_bound (divset (k,sD)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(sD + m))) is V22() real ext-real Element of REAL
upper_bound (divset (k,sD)) is V22() real ext-real Element of REAL
k . sD is V22() real ext-real Element of REAL
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
k . sD is V22() real ext-real Element of REAL
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
sD - 1 is V22() real ext-real Element of REAL
k . (sD - 1) is V22() real ext-real Element of REAL
(sD - 1) + (v + 1) is V22() real ext-real Element of REAL
((sD - 1) + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . (((sD - 1) + (v + 1)) - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (k,sD))),(upper_bound (divset (k,sD))).] is V58() V59() V60() interval Element of K19(REAL)
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
H₁(e) . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
f | (divset (e,(sD + m))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (e,(sD + m)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (e,(sD + m))))) is V22() real ext-real Element of REAL
vol (divset (e,(sD + m))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (e,(sD + m)))))) * (vol (divset (e,(sD + m)))) is V22() real ext-real Element of REAL
s | v1 is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
dom f is non empty set
A /\ (dom f) is V58() V59() V60() Element of K19(REAL)
sD is V22() real ext-real set
dom s is non empty set
v1 /\ (dom s) is V58() V59() V60() Element of K19(REAL)
m is set
s . m is V22() real ext-real Element of REAL
(dom f) /\ v1 is V58() V59() V60() Element of K19(REAL)
(dom f) /\ A is V58() V59() V60() Element of K19(REAL)
f . m is V22() real ext-real Element of REAL
m is V22() real ext-real set
D1 is set
s . D1 is V22() real ext-real Element of REAL
(dom f) /\ v1 is V58() V59() V60() Element of K19(REAL)
(dom f) /\ A is V58() V59() V60() Element of K19(REAL)
f . D1 is V22() real ext-real Element of REAL
rng s is non empty V58() V59() V60() Element of K19(REAL)
lower_bound (rng s) is V22() real ext-real Element of REAL
upper_bound (rng s) is V22() real ext-real Element of REAL
(upper_bound (rng s)) - (lower_bound (rng s)) is V22() real ext-real Element of REAL
((upper_bound (rng s)) - (lower_bound (rng s))) * (v1,k) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (v1,k) is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
(indx (y,e,h)) -' (indx (y,e,(v + 1))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,h)) -' (indx (y,e,(v + 1)))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) - (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
((indx (y,e,h)) - (indx (y,e,(v + 1)))) + 1 is V22() real ext-real Element of REAL
lower_volume (s,n) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (lower_volume (s,n)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) - ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
((indx (y,e,h)) - ((indx (y,e,v)) + 1)) + 1 is V22() real ext-real Element of REAL
rng k is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
{(e . (H + 1))} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng k) \/ {(e . (H + 1))} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng n is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sD is set
m is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . D1 is V22() real ext-real Element of REAL
k . 1 is V22() real ext-real Element of REAL
D1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(indx (y,e,(v + 1))) + D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
D1 - (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
(D1 - (indx (y,e,(v + 1)))) + 1 is V22() real ext-real Element of REAL
(indx (y,e,(v + 1))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(D1 + 1) - (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
dom n is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
n . D2 is V22() real ext-real Element of REAL
D2 + (indx (y,e,(v + 1))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(D2 + (indx (y,e,(v + 1)))) - 1 is V22() real ext-real Element of REAL
y . ((D2 + (indx (y,e,(v + 1)))) - 1) is V22() real ext-real Element of REAL
m is V22() real ext-real Element of REAL
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . D2 is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . (h - 1) is V22() real ext-real Element of REAL
m is V22() real ext-real Element of REAL
m is V22() real ext-real Element of REAL
dom n is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len n) - 1 is V22() real ext-real Element of REAL
((len n) - 1) + (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
sD is set
n . 1 is V22() real ext-real Element of REAL
m is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . D1 is V22() real ext-real Element of REAL
D1 - v is V22() real ext-real Element of REAL
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
v + D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
k . D2 is V22() real ext-real Element of REAL
(D1 - v) - 1 is V22() real ext-real Element of REAL
((D1 - v) - 1) + (v + 1) is V22() real ext-real Element of REAL
e . (((D1 - v) - 1) + (v + 1)) is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . D1 is V22() real ext-real Element of REAL
D2 is V22() real ext-real Element of REAL
(rng e) /\ (divset (e,h)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
upper_bound (divset (e,v)) is V22() real ext-real Element of REAL
dom (upper_volume ((chi (v1,v1)),k)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (v1,v1)),k)) . sD is V22() real ext-real Element of REAL
len (upper_volume ((chi (v1,v1)),k)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (v1,v1)),k))) is non empty V37() len (upper_volume ((chi (v1,v1)),k)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len k) is non empty V37() len k -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
sD + v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (k,sD) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (k,sD)) is V22() real ext-real Element of REAL
divset (e,(sD + v)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,(sD + v))) is V22() real ext-real Element of REAL
upper_bound (divset (k,sD)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(sD + v))) is V22() real ext-real Element of REAL
k . sD is V22() real ext-real Element of REAL
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
(sD + v) - 1 is V22() real ext-real Element of REAL
e . ((sD + v) - 1) is V22() real ext-real Element of REAL
k . sD is V22() real ext-real Element of REAL
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
sD - 1 is V22() real ext-real Element of REAL
k . (sD - 1) is V22() real ext-real Element of REAL
(sD - 1) + (v + 1) is V22() real ext-real Element of REAL
((sD - 1) + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . (((sD - 1) + (v + 1)) - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (e,(sD + v)))),(upper_bound (divset (e,(sD + v)))).] is V58() V59() V60() interval Element of K19(REAL)
len (upper_volume ((chi (A,A)),e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),e))) is non empty V37() len (upper_volume ((chi (A,A)),e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
vol (divset (k,sD)) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),e)) . (sD + v) is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(lower_volume (s,n)) . m is V22() real ext-real Element of REAL
(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) . m is V22() real ext-real Element of REAL
Seg (len (lower_volume (s,n))) is non empty V37() len (lower_volume (s,n)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
m + ((indx (y,e,v)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(m + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((m + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
1 + D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,v)) + 1) - 1 is V22() real ext-real Element of REAL
(indx (y,e,h)) - (((indx (y,e,v)) + 1) - 1) is V22() real ext-real Element of REAL
m + (((indx (y,e,v)) + 1) - 1) is V22() real ext-real Element of REAL
m + D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (y,(m + D1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
f | (divset (y,(m + D1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,(m + D1)))) is V58() V59() V60() Element of K19(REAL)
lower_bound (rng (f | (divset (y,(m + D1))))) is V22() real ext-real Element of REAL
vol (divset (y,(m + D1))) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (divset (y,(m + D1)))))) * (vol (divset (y,(m + D1)))) is V22() real ext-real Element of REAL
Seg (len n) is non empty V37() len n -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (n,m) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (n,m)) is V22() real ext-real Element of REAL
lower_bound (divset (y,(m + D1))) is V22() real ext-real Element of REAL
upper_bound (divset (n,m)) is V22() real ext-real Element of REAL
upper_bound (divset (y,(m + D1))) is V22() real ext-real Element of REAL
n . m is V22() real ext-real Element of REAL
y . ((m + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
(m + D1) - 1 is V22() real ext-real Element of REAL
y . ((m + D1) - 1) is V22() real ext-real Element of REAL
n . m is V22() real ext-real Element of REAL
y . ((m + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
m - 1 is V22() real ext-real Element of REAL
n . (m - 1) is V22() real ext-real Element of REAL
(m - 1) + ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
((m - 1) + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
y . (((m - 1) + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (n,m))),(upper_bound (divset (n,m))).] is V58() V59() V60() interval Element of K19(REAL)
s | (divset (y,(m + D1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
rng (s | (divset (y,(m + D1)))) is V58() V59() V60() Element of K19(REAL)
Sum (lower_volume (s,n)) is V22() real ext-real Element of REAL
K190(REAL,(lower_volume (s,n)),K263()) is V22() real ext-real Element of REAL
Sum (lower_volume (s,k)) is V22() real ext-real Element of REAL
K190(REAL,(lower_volume (s,k)),K263()) is V22() real ext-real Element of REAL
(Sum (lower_volume (s,n))) - (Sum (lower_volume (s,k))) is V22() real ext-real Element of REAL
len (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) -' ((indx (y,e,v)) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,h)) -' ((indx (y,e,v)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid (H₁(e),(v + 1),h)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(H₂(y, indx (y,e,v)) - H₂(e,v)) + ((Sum (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))) - (Sum (mid (H₁(e),(v + 1),h)))) is V22() real ext-real Element of REAL
((H * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e)) + (((upper_bound (rng f)) - (lower_bound (rng f))) * (A,e)) is V22() real ext-real Element of REAL
H₁(y) | (indx (y,e,v)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(y) | (indx (y,e,v))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(y) | (indx (y,e,v))),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),1,(indx (y,e,v))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),1,(indx (y,e,v)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),1,(indx (y,e,v)))),K263()) is V22() real ext-real Element of REAL
H₂(y, indx (y,e,v)) + (Sum (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))) is V22() real ext-real Element of REAL
(mid (H₁(y),1,(indx (y,e,v)))) ^ (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(y),1,(indx (y,e,v)))) ^ (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(y),1,(indx (y,e,v)))) ^ (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),1,(indx (y,e,h))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),1,(indx (y,e,h)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),1,(indx (y,e,h)))),K263()) is V22() real ext-real Element of REAL
H₁(y) | (indx (y,e,h)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(y) | (indx (y,e,h))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(y) | (indx (y,e,h))),K263()) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,y))) . (indx (y,e,h)) is V22() real ext-real Element of REAL
H₂(y, indx (y,e,h)) - H₂(e,h) is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,h) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
indx (y,e,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (lower_volume (f,y))) . (indx (y,e,h)) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,e))) . h is V22() real ext-real Element of REAL
((PartSums (lower_volume (f,y))) . (indx (y,e,h))) - ((PartSums (lower_volume (f,e))) . h) is V22() real ext-real Element of REAL
p is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . p is V22() real ext-real Element of REAL
p * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(p * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,H) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
indx (y,e,H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (lower_volume (f,y))) . (indx (y,e,H)) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,e))) . H is V22() real ext-real Element of REAL
((PartSums (lower_volume (f,y))) . (indx (y,e,H))) - ((PartSums (lower_volume (f,e))) . H) is V22() real ext-real Element of REAL
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (len e) is V22() real ext-real Element of REAL
indx (y,e,(len e)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,(len e))) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
y . (len y) is V22() real ext-real Element of REAL
e . (len e) is V22() real ext-real Element of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,D) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
indx (y,e,D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (lower_volume (f,y))) . (indx (y,e,D)) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,e))) . D is V22() real ext-real Element of REAL
H₂(y, indx (y,e,D)) - H₂(e,D) is V22() real ext-real Element of REAL
e . D is V22() real ext-real Element of REAL
upper_bound (divset (e,D)) is V22() real ext-real Element of REAL
(PartSums (lower_volume (f,y))) . (len y) is V22() real ext-real Element of REAL
H₂(y, len y) - (lower_sum (f,e)) is V22() real ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,y) is V22() real ext-real Element of REAL
(lower_sum (f,y)) - (lower_sum (f,e)) is V22() real ext-real Element of REAL
lim (A,T) is V22() real ext-real Element of REAL
e is V22() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(A,T) . y is V22() real ext-real Element of REAL
T . y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,(T . y)) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),(T . y)) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),(T . y))) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),(T . y)))) is V22() real ext-real set
dom (upper_volume ((chi (A,A)),(T . y))) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (A,A)),(T . y))) . D is V22() real ext-real Element of REAL
p is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
len (upper_volume ((chi (A,A)),(T . y))) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),(T . y)))) is non empty V37() len (upper_volume ((chi (A,A)),(T . y))) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len p is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len p) is non empty V37() len p -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom p is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset ((T . y),D) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset ((T . y),D)) is V22() real ext-real Element of REAL
((A,T) . y) - 0 is V22() real ext-real Element of REAL
abs (((A,T) . y) - 0) is V22() real ext-real Element of REAL
e + (abs (((A,T) . y) - 0)) is V22() real ext-real Element of REAL
((A,T) . y) + (abs (((A,T) . y) - 0)) is V22() real ext-real Element of REAL
y is V22() real ext-real set
y / 2 is V22() real ext-real Element of REAL
lower_sum_set f is Relation-like Function-like non empty total V30( divs A, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20((divs A),REAL))
K20((divs A),REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20((divs A),REAL)) is V12() V37() set
rng (lower_sum_set f) is non empty V58() V59() V60() Element of K19(REAL)
upper_bound (rng (lower_sum_set f)) is V22() real ext-real Element of REAL
D is V22() real ext-real Element of REAL
D / 2 is V22() real ext-real Element of REAL
(lower_integral f) - (D / 2) is V22() real ext-real Element of REAL
p is V22() real ext-real set
dom (lower_sum_set f) is non empty set
lower_bound A is V22() real ext-real Element of REAL
H is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(lower_sum_set f) . H is V22() real ext-real Element of REAL
H . 1 is V22() real ext-real Element of REAL
len H is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
dom v is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
v1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued non-decreasing FinSequence of REAL
dom v1 is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len H) is non empty V37() len H -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
k is Relation-like Function-like set
dom k is set
rng k is set
k * v1 is Relation-like REAL -valued complex-valued ext-real-valued real-valued set
dom H is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (H,n) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (H,n)) is V22() real ext-real Element of REAL
Seg (len v) is V37() len v -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
k . n is set
v . n is V22() real ext-real Element of REAL
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
v1 . s is V22() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
v1 . k is V22() real ext-real Element of REAL
2 * (len H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((upper_bound (rng f)) - (lower_bound (rng f))) + 1 is V22() real ext-real Element of REAL
(2 * (len H)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) is V22() real ext-real Element of REAL
D / ((2 * (len H)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) is V22() real ext-real Element of REAL
min ((v1 . k),(D / ((2 * (len H)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) is V22() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
lower_sum (f,H) is V22() real ext-real Element of REAL
v1 . 1 is V22() real ext-real Element of REAL
dom H is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (H,s) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (H,s)) is V22() real ext-real Element of REAL
upper_bound (divset (H,s)) is V22() real ext-real Element of REAL
H . s is V22() real ext-real Element of REAL
lower_bound (divset (H,s)) is V22() real ext-real Element of REAL
(H . s) - (lower_bound A) is V22() real ext-real Element of REAL
upper_bound (divset (H,s)) is V22() real ext-real Element of REAL
H . s is V22() real ext-real Element of REAL
lower_bound (divset (H,s)) is V22() real ext-real Element of REAL
s - 1 is V22() real ext-real Element of REAL
H . (s - 1) is V22() real ext-real Element of REAL
(H . s) - (H . (s - 1)) is V22() real ext-real Element of REAL
s + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
rng v1 is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
rng v is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v . s is V22() real ext-real Element of REAL
divset (H,s) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (H,s)) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),H) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),H)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
inf (rng (upper_volume ((chi (A,A)),H))) is V22() real ext-real set
len v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len v) is V37() len v -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len v1) is V37() len v1 -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),H)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (A,A)),H)) . s is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),H)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),H))) is non empty V37() len (upper_volume ((chi (A,A)),H)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (H,s) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (H,s)) is V22() real ext-real Element of REAL
v . s is V22() real ext-real Element of REAL
rng v is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
rng v1 is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v1 . sD is V22() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v . m is V22() real ext-real Element of REAL
divset (H,m) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (H,m)) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),H)) . m is V22() real ext-real Element of REAL
(len H) * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(len H) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) is V22() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(lower_sum (f,T)) . m is V22() real ext-real Element of REAL
((lower_sum (f,T)) . m) - (lower_integral f) is V22() real ext-real Element of REAL
abs (((lower_sum (f,T)) . m) - (lower_integral f)) is V22() real ext-real Element of REAL
T . m is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,T) . m is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,D1) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),D1) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),D1)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),D1))) is V22() real ext-real set
((A,T) . m) * ((2 * (len H)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) is V22() real ext-real Element of REAL
((A,T) . m) * ((len H) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) is V22() real ext-real Element of REAL
(((A,T) . m) * ((len H) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1))) * 2 is V22() real ext-real Element of REAL
((len H) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((A,T) . m) is V22() real ext-real Element of REAL
lower_sum (f,(T . m)) is V22() real ext-real Element of REAL
(lower_sum_set f) . (T . m) is V22() real ext-real Element of REAL
(lower_integral f) - ((lower_sum (f,T)) . m) is V22() real ext-real Element of REAL
((len H) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * ((A,T) . m) is V22() real ext-real Element of REAL
rng D1 is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng H is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng D1) \/ (rng H) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,D1) is V22() real ext-real Element of REAL
sD1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng sD1 is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,sD1) is V22() real ext-real Element of REAL
(lower_sum (f,sD1)) - (lower_sum (f,H)) is V22() real ext-real Element of REAL
(lower_sum (f,sD1)) - (lower_sum (f,D1)) is V22() real ext-real Element of REAL
(lower_sum (f,H)) - (lower_sum (f,(T . m))) is V22() real ext-real Element of REAL
(lower_sum (f,sD1)) - (lower_sum (f,(T . m))) is V22() real ext-real Element of REAL
((lower_sum (f,H)) - (lower_sum (f,(T . m)))) - ((lower_sum (f,sD1)) - (lower_sum (f,(T . m)))) is V22() real ext-real Element of REAL
(lower_sum (f,H)) - (lower_sum (f,sD1)) is V22() real ext-real Element of REAL
((len H) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,D1) is V22() real ext-real Element of REAL
c₂₂ is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng c₂₂ is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
lower_sum (f,c₂₂) is V22() real ext-real Element of REAL
(lower_sum (f,c₂₂)) - (lower_sum (f,D1)) is V22() real ext-real Element of REAL
(D / 2) + (D / 2) is V22() real ext-real Element of REAL
((lower_sum (f,H)) - (lower_sum (f,(T . m)))) + (D / 2) is V22() real ext-real Element of REAL
(lower_sum (f,H)) + (D / 2) is V22() real ext-real Element of REAL
(lower_integral f) - (lower_sum (f,(T . m))) is V22() real ext-real Element of REAL
((lower_integral f) - (lower_sum (f,(T . m)))) + (lower_sum (f,(T . m))) is V22() real ext-real Element of REAL
((lower_sum (f,H)) + (D / 2)) - (lower_sum (f,(T . m))) is V22() real ext-real Element of REAL
abs ((lower_integral f) - ((lower_sum (f,T)) . m)) is V22() real ext-real Element of REAL
- ((lower_integral f) - ((lower_sum (f,T)) . m)) is V22() real ext-real Element of REAL
abs (- ((lower_integral f) - ((lower_sum (f,T)) . m))) is V22() real ext-real Element of REAL
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(lower_sum (f,T)) . s is V22() real ext-real Element of REAL
((lower_sum (f,T)) . s) - (lower_integral f) is V22() real ext-real Element of REAL
abs (((lower_sum (f,T)) . s) - (lower_integral f)) is V22() real ext-real Element of REAL
A is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
K20(A,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(A,REAL)) is V12() V37() set
divs A is non empty set
K20(NAT,(divs A)) is Relation-like V12() V37() set
K19(K20(NAT,(divs A))) is V12() V37() set
vol A is V22() real ext-real Element of REAL
f is Relation-like Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
f | A is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
upper_integral f is V22() real ext-real Element of REAL
T is Relation-like Function-like non empty total V30( NAT , divs A) Element of K19(K20(NAT,(divs A)))
(A,T) is Relation-like Function-like non empty total V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
upper_sum (f,T) is Relation-like Function-like non empty total V30( NAT , REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
lim (upper_sum (f,T)) is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,e) is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng e) \/ (rng e) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,e) is V22() real ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,y) is V22() real ext-real Element of REAL
(upper_sum (f,e)) - (upper_sum (f,y)) is V22() real ext-real Element of REAL
(upper_sum (f,e)) - (upper_sum (f,y)) is V22() real ext-real Element of REAL
chi (A,A) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng f is non empty V58() V59() V60() Element of K19(REAL)
upper_bound (rng f) is V22() real ext-real Element of REAL
lower_bound (rng f) is V22() real ext-real Element of REAL
(upper_bound (rng f)) - (lower_bound (rng f)) is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
upper_volume ((chi (A,A)),e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
inf (rng (upper_volume ((chi (A,A)),e))) is V22() real ext-real set
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len e) * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
e is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,e) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),e))) is V22() real ext-real set
rng e is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng e) \/ (rng e) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,e) is V22() real ext-real Element of REAL
((len e) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
upper_sum (f,e) is V22() real ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,y) is V22() real ext-real Element of REAL
(upper_sum (f,e)) - (upper_sum (f,y)) is V22() real ext-real Element of REAL
(upper_sum (f,e)) - (upper_sum (f,y)) is V22() real ext-real Element of REAL
len y is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom y is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom e is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
upper_volume (f,e) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
PartSums (upper_volume (f,e)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
upper_volume (f,y) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
PartSums (upper_volume (f,y)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . D is V22() real ext-real Element of REAL
D * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(D * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . p is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,H) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
lower_bound (divset (e,H)) is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
upper_bound (divset (e,H)) is V22() real ext-real Element of REAL
divset (e,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,1)) is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),e))) is non empty V37() len (upper_volume ((chi (A,A)),e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
vol (divset (e,1)) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),e)) . 1 is V22() real ext-real Element of REAL
upper_bound (divset (e,1)) is V22() real ext-real Element of REAL
e . 1 is V22() real ext-real Element of REAL
(e . 1) - (lower_bound A) is V22() real ext-real Element of REAL
(upper_bound (divset (e,H))) - (lower_bound A) is V22() real ext-real Element of REAL
(upper_bound (divset (e,1))) - (lower_bound (divset (e,1))) is V22() real ext-real Element of REAL
vol (divset (e,H)) is V22() real ext-real Element of REAL
p - 1 is V22() real ext-real Element of REAL
e . (p - 1) is V22() real ext-real Element of REAL
divset (e,p) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
upper_bound (divset (e,p)) is V22() real ext-real Element of REAL
(upper_bound (divset (e,p))) - (lower_bound (divset (e,p))) is V22() real ext-real Element of REAL
(upper_bound (divset (e,p))) - (lower_bound A) is V22() real ext-real Element of REAL
(upper_bound (divset (e,H))) - (lower_bound (divset (e,H))) is V22() real ext-real Element of REAL
vol (divset (e,H)) is V22() real ext-real Element of REAL
vol (divset (e,p)) is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),e))) is non empty V37() len (upper_volume ((chi (A,A)),e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),e)) . p is V22() real ext-real Element of REAL
e . 1 is V22() real ext-real Element of REAL
1 * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(1 * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,p) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(PartSums (upper_volume (f,e))) . p is V22() real ext-real Element of REAL
indx (y,e,p) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (upper_volume (f,y))) . (indx (y,e,p)) is V22() real ext-real Element of REAL
H₂(e,p) - H₂(y, indx (y,e,p)) is V22() real ext-real Element of REAL
1 * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(1 * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
p - 1 is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume (f,e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume (f,e))) is non empty V37() len (upper_volume (f,e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume (f,e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
indx (y,e,H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len y) is non empty V37() len y -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
mid (y,1,(indx (y,e,H))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y | (indx (y,e,H)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,H)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid ((upper_volume (f,e)),p,p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid ((upper_volume (f,e)),p,p)) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(mid ((upper_volume (f,e)),p,p)),K263()) is V22() real ext-real Element of REAL
mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) is V22() real ext-real Element of REAL
K190(REAL,(mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))),K263()) is V22() real ext-real Element of REAL
(Sum (mid ((upper_volume (f,e)),p,p))) - (Sum (mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p))))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (A,e) is V22() real ext-real Element of REAL
(indx (y,e,p)) - (indx (y,e,H)) is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,H)) + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,p)) is V22() real ext-real Element of REAL
y . v is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
y . h is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
v1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
v1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (e,p)) is V22() real ext-real Element of REAL
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
(rng e) /\ (divset (e,p)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
(mid ((upper_volume (f,e)),p,p)) . 1 is V22() real ext-real Element of REAL
(upper_volume (f,e)) . p is V22() real ext-real Element of REAL
p -' p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(p -' p) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid ((upper_volume (f,e)),p,p)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
<*((upper_volume (f,e)) . p)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
len (upper_volume (f,y)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume (f,y))) is non empty V37() len (upper_volume (f,y)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(indx (y,e,p)) -' ((indx (y,e,H)) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,p)) - ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
((indx (y,e,p)) -' ((indx (y,e,H)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
y . (indx (y,e,p)) is V22() real ext-real Element of REAL
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
[.(y . (indx (y,e,H))),(y . (indx (y,e,p))).] is V58() V59() V60() interval Element of K19(REAL)
divset (y,(indx (y,e,p))) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (y,(indx (y,e,p)))) is V22() real ext-real Element of REAL
(indx (y,e,p)) - 1 is V22() real ext-real Element of REAL
lower_bound (divset (y,(indx (y,e,p)))) is V22() real ext-real Element of REAL
(mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) . 1 is V22() real ext-real Element of REAL
(upper_volume (f,y)) . ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
<*((upper_volume (f,y)) . ((indx (y,e,H)) + 1))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
divset (y,((indx (y,e,H)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
f | (divset (y,((indx (y,e,H)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,H)) + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
vol (divset (y,((indx (y,e,H)) + 1))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
(mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) . 1 is V22() real ext-real Element of REAL
(upper_volume (f,y)) . ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
2 + ((indx (y,e,H)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p)))) . 2 is V22() real ext-real Element of REAL
(2 + ((indx (y,e,H)) + 1)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
H₁(y) . ((2 + ((indx (y,e,H)) + 1)) -' 1) is V22() real ext-real Element of REAL
(2 + ((indx (y,e,H)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((2 + ((indx (y,e,H)) + 1)) - 1) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,H)) + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H₁(y) . ((indx (y,e,H)) + (1 + 1)) is V22() real ext-real Element of REAL
(indx (y,e,H)) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(upper_volume (f,y)) . ((indx (y,e,H)) + 2) is V22() real ext-real Element of REAL
<*((upper_volume (f,y)) . ((indx (y,e,H)) + 1)),((upper_volume (f,y)) . ((indx (y,e,H)) + 2))*> is Relation-like NAT -defined Function-like non empty V37() 2 -element FinSequence-like FinSubsequence-like set
((upper_volume (f,y)) . ((indx (y,e,H)) + 1)) + ((upper_volume (f,y)) . ((indx (y,e,H)) + 2)) is V22() real ext-real Element of REAL
divset (y,((indx (y,e,H)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (y,((indx (y,e,H)) + 1))) is V22() real ext-real Element of REAL
upper_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . p is V22() real ext-real Element of REAL
y . (indx (y,e,p)) is V22() real ext-real Element of REAL
divset (y,((indx (y,e,H)) + 2)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (y,((indx (y,e,H)) + 2))) is V22() real ext-real Element of REAL
((indx (y,e,p)) - ((indx (y,e,H)) + 1)) + 1 is V22() real ext-real Element of REAL
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
vol (divset (e,p)) is V22() real ext-real Element of REAL
y . ((indx (y,e,H)) + 2) is V22() real ext-real Element of REAL
y . ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
(y . ((indx (y,e,H)) + 2)) - (y . ((indx (y,e,H)) + 1)) is V22() real ext-real Element of REAL
((y . ((indx (y,e,H)) + 2)) - (y . ((indx (y,e,H)) + 1))) + (y . ((indx (y,e,H)) + 1)) is V22() real ext-real Element of REAL
(((y . ((indx (y,e,H)) + 2)) - (y . ((indx (y,e,H)) + 1))) + (y . ((indx (y,e,H)) + 1))) - (y . (indx (y,e,H))) is V22() real ext-real Element of REAL
upper_bound (divset (y,((indx (y,e,H)) + 1))) is V22() real ext-real Element of REAL
((indx (y,e,H)) + 1) - 1 is V22() real ext-real Element of REAL
(indx (y,e,H)) + 0 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
lower_bound (divset (y,((indx (y,e,H)) + 1))) is V22() real ext-real Element of REAL
((indx (y,e,H)) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,H)) + 2) - 1 is V22() real ext-real Element of REAL
lower_bound (divset (y,((indx (y,e,H)) + 2))) is V22() real ext-real Element of REAL
upper_bound (divset (y,((indx (y,e,H)) + 2))) is V22() real ext-real Element of REAL
(vol (divset (y,((indx (y,e,H)) + 2)))) + (y . ((indx (y,e,H)) + 1)) is V22() real ext-real Element of REAL
((vol (divset (y,((indx (y,e,H)) + 2)))) + (y . ((indx (y,e,H)) + 1))) - (y . (indx (y,e,H))) is V22() real ext-real Element of REAL
(upper_bound (divset (y,((indx (y,e,H)) + 1)))) - (lower_bound (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
(vol (divset (y,((indx (y,e,H)) + 2)))) + ((upper_bound (divset (y,((indx (y,e,H)) + 1)))) - (lower_bound (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(vol (divset (y,((indx (y,e,H)) + 1)))) + (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
f | (divset (e,p)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (e,p))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (e,p)))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (e,p))))) * ((vol (divset (y,((indx (y,e,H)) + 1)))) + (vol (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
mid (H₁(e),p,p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),p,p)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),p,p)),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p)))),K263()) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),p,p))) - (Sum (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) is V22() real ext-real Element of REAL
(vol (divset (y,((indx (y,e,H)) + 2)))) + (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * ((vol (divset (y,((indx (y,e,H)) + 2)))) + (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
f | (divset (y,((indx (y,e,H)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,H)) + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
f | (divset (y,((indx (y,e,H)) + 2))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,H)) + 2)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
sD is V22() real ext-real set
sD * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
((upper_volume (f,e)) . p) - (sD * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
sD * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (e,p))))) * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (e,p))))) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng (f | (divset (e,p))))) * (vol (divset (y,((indx (y,e,H)) + 1))))) + ((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),p,p))) - ((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 2)))))) * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
H₁(y) . ((indx (y,e,H)) + 1) is V22() real ext-real Element of REAL
((upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 2)))))) * (vol (divset (y,((indx (y,e,H)) + 2))))) + (H₁(y) . ((indx (y,e,H)) + 1)) is V22() real ext-real Element of REAL
((upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 2)))))) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2)))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) - ((upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((upper_bound (rng (f | (divset (y,((indx (y,e,H)) + 1)))))) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(Sum (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) - ((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) is V22() real ext-real Element of REAL
((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1))))) is V22() real ext-real Element of REAL
(((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((upper_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1)))))) - (((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 2))))) + ((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,H)) + 1)))))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (vol (divset (e,p))) is V22() real ext-real Element of REAL
len (y | (indx (y,e,H))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e | H is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (e | H) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
rng (y | (indx (y,e,H))) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
h is set
dom (e | H) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(e | H) . v is V22() real ext-real Element of REAL
len (e | H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len (e | H)) is V37() len (e | H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
e . v is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
indx (y,e,v) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,v)) is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . v1 is V22() real ext-real Element of REAL
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len (y | (indx (y,e,H)))) is V37() len (y | (indx (y,e,H))) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
dom (y | (indx (y,e,H))) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(y | (indx (y,e,H))) . v1 is V22() real ext-real Element of REAL
lower_bound (divset (e,p)) is V22() real ext-real Element of REAL
e . H is V22() real ext-real Element of REAL
h is set
dom (y | (indx (y,e,H))) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(y | (indx (y,e,H))) . v is V22() real ext-real Element of REAL
Seg (len (y | (indx (y,e,H)))) is V37() len (y | (indx (y,e,H))) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
len (e | H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . v is V22() real ext-real Element of REAL
y . (indx (y,e,H)) is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
Seg (len (e | H)) is V37() len (e | H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
dom (e | H) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(e | H) . v1 is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
Seg (len (e | H)) is V37() len (e | H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
dom (e | H) is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(e | H) . H is V22() real ext-real Element of REAL
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
mid (e,1,H) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (y,e,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(y | (indx (y,e,H))) . (indx (y,e,h)) is V22() real ext-real Element of REAL
y . (indx (y,e,h)) is V22() real ext-real Element of REAL
e . h is V22() real ext-real Element of REAL
e . (indx (y,e,h)) is V22() real ext-real Element of REAL
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
e . h is V22() real ext-real Element of REAL
(e | H) . h is V22() real ext-real Element of REAL
y . (indx (y,e,h)) is V22() real ext-real Element of REAL
y . h is V22() real ext-real Element of REAL
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume (f,e)) | H is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((upper_volume (f,e)) | H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume (f,y)) | (indx (y,e,H)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (upper_volume (f,y)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume (f,y))) is non empty V37() len (upper_volume (f,y)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume (f,y)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
((upper_volume (f,e)) | H) . h is V22() real ext-real Element of REAL
((upper_volume (f,y)) | (indx (y,e,H))) . h is V22() real ext-real Element of REAL
Seg H is V37() H -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
indx (y,e,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (indx (y,e,H)) is V37() indx (y,e,H) -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
e . h is V22() real ext-real Element of REAL
y . (indx (y,e,h)) is V22() real ext-real Element of REAL
divset (e,h) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,h)) is V22() real ext-real Element of REAL
divset (y,(indx (y,e,h))) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (y,(indx (y,e,h)))) is V22() real ext-real Element of REAL
upper_bound (divset (e,h)) is V22() real ext-real Element of REAL
upper_bound (divset (y,(indx (y,e,h)))) is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
indx (y,e,v) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) - 1 is V22() real ext-real Element of REAL
y . ((indx (y,e,h)) - 1) is V22() real ext-real Element of REAL
y . (indx (y,e,v)) is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (y,(indx (y,e,h))))),(upper_bound (divset (y,(indx (y,e,h))))).] is V58() V59() V60() interval Element of K19(REAL)
(upper_volume (f,e)) . h is V22() real ext-real Element of REAL
f | (divset (y,(indx (y,e,h)))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,(indx (y,e,h))))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (y,(indx (y,e,h)))))) is V22() real ext-real Element of REAL
vol (divset (y,(indx (y,e,h)))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (y,(indx (y,e,h))))))) * (vol (divset (y,(indx (y,e,h))))) is V22() real ext-real Element of REAL
((upper_volume (f,y)) | (indx (y,e,H))) . (indx (y,e,h)) is V22() real ext-real Element of REAL
(upper_volume (f,y)) . (indx (y,e,h)) is V22() real ext-real Element of REAL
len (upper_volume (f,y)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (e | H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len ((upper_volume (f,y)) | (indx (y,e,H))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
len H₁(y) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len H₁(y)) is non empty V37() len H₁(y) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(y) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len H₁(e) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len H₁(e)) is non empty V37() len H₁(e) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(e) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(PartSums (upper_volume (f,e))) . H is V22() real ext-real Element of REAL
H₁(e) | H is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | H) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | H),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),p,p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),p,p)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),p,p)),K263()) is V22() real ext-real Element of REAL
H₂(e,H) + (Sum (mid (H₁(e),p,p))) is V22() real ext-real Element of REAL
(H₁(e) | H) ^ (mid (H₁(e),p,p)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((H₁(e) | H) ^ (mid (H₁(e),p,p))) is V22() real ext-real Element of REAL
K190(REAL,((H₁(e) | H) ^ (mid (H₁(e),p,p))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,H) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (H₁(e),(H + 1),p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (H₁(e),1,H)) ^ (mid (H₁(e),(H + 1),p)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(e),1,H)) ^ (mid (H₁(e),(H + 1),p))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(e),1,H)) ^ (mid (H₁(e),(H + 1),p))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,p) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),1,p)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),1,p)),K263()) is V22() real ext-real Element of REAL
H₁(e) | p is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | p) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | p),K263()) is V22() real ext-real Element of REAL
H₂(e,H) + (Sum (mid ((upper_volume (f,e)),p,p))) is V22() real ext-real Element of REAL
(PartSums (upper_volume (f,y))) . (indx (y,e,H)) is V22() real ext-real Element of REAL
H₁(y) | (indx (y,e,H)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(y) | (indx (y,e,H))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(y) | (indx (y,e,H))),K263()) is V22() real ext-real Element of REAL
H₂(y, indx (y,e,H)) + (Sum (mid ((upper_volume (f,y)),((indx (y,e,H)) + 1),(indx (y,e,p))))) is V22() real ext-real Element of REAL
mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(H₁(y) | (indx (y,e,H))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((H₁(y) | (indx (y,e,H))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) is V22() real ext-real Element of REAL
K190(REAL,((H₁(y) | (indx (y,e,H))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),1,(indx (y,e,H))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (H₁(y),1,(indx (y,e,H)))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(y),1,(indx (y,e,H)))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(y),1,(indx (y,e,H)))) ^ (mid (H₁(y),((indx (y,e,H)) + 1),(indx (y,e,p))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),1,(indx (y,e,p))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),1,(indx (y,e,p)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),1,(indx (y,e,p)))),K263()) is V22() real ext-real Element of REAL
H₁(y) | (indx (y,e,p)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(y) | (indx (y,e,p))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(y) | (indx (y,e,p))),K263()) is V22() real ext-real Element of REAL
Seg (len (upper_volume (f,y))) is non empty V37() len (upper_volume (f,y)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume (f,y)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Sum ((upper_volume (f,y)) | (indx (y,e,H))) is V22() real ext-real Element of REAL
K190(REAL,((upper_volume (f,y)) | (indx (y,e,H))),K263()) is V22() real ext-real Element of REAL
H is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal set
e . H is V22() real ext-real Element of REAL
H * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(H * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
H + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (H + 1) is V22() real ext-real Element of REAL
(H + 1) * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
((H + 1) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,h) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
indx (y,e,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,h)) is V22() real ext-real Element of REAL
e . h is V22() real ext-real Element of REAL
len H₁(y) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,v) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(PartSums (upper_volume (f,e))) . v is V22() real ext-real Element of REAL
indx (y,e,v) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (upper_volume (f,y))) . (indx (y,e,v)) is V22() real ext-real Element of REAL
((PartSums (upper_volume (f,e))) . v) - ((PartSums (upper_volume (f,y))) . (indx (y,e,v))) is V22() real ext-real Element of REAL
v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,v) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(PartSums (upper_volume (f,e))) . v is V22() real ext-real Element of REAL
indx (y,e,v) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (upper_volume (f,y))) . (indx (y,e,v)) is V22() real ext-real Element of REAL
H₂(e,v) - H₂(y, indx (y,e,v)) is V22() real ext-real Element of REAL
v + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(rng e) /\ (divset (e,h)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v - 1 is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (v - 1) is V22() real ext-real Element of REAL
upper_bound (divset (e,h)) is V22() real ext-real Element of REAL
lower_bound (divset (e,v)) is V22() real ext-real Element of REAL
y . (indx (y,e,v)) is V22() real ext-real Element of REAL
e . v is V22() real ext-real Element of REAL
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
indx (y,e,(v + 1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
e . (v + 1) is V22() real ext-real Element of REAL
len H₁(e) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len H₁(e)) is non empty V37() len H₁(e) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(e) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
H₁(e) | v is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | v) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(H₁(e) | v),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,v) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),1,v)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),1,v)),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),(v + 1),h) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),(v + 1),h)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),(v + 1),h)),K263()) is V22() real ext-real Element of REAL
H₂(e,v) + (Sum (mid (H₁(e),(v + 1),h))) is V22() real ext-real Element of REAL
(mid (H₁(e),1,v)) ^ (mid (H₁(e),(v + 1),h)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(e),1,v)) ^ (mid (H₁(e),(v + 1),h))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(e),1,v)) ^ (mid (H₁(e),(v + 1),h))),K263()) is V22() real ext-real Element of REAL
mid (H₁(e),1,h) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(e),1,h)) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(e),1,h)),K263()) is V22() real ext-real Element of REAL
H₁(e) | h is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(e) | h) is V22() real ext-real Element of REAL
K190(REAL,(H₁(e) | h),K263()) is V22() real ext-real Element of REAL
(PartSums (upper_volume (f,e))) . h is V22() real ext-real Element of REAL
Seg (len y) is non empty V37() len y -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len H₁(y)) is non empty V37() len H₁(y) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom H₁(y) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
1 + (indx (y,e,(v + 1))) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,h)) + 1) - (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
mid (y,(indx (y,e,(v + 1))),(indx (y,e,h))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (y,(indx (y,e,(v + 1))),(indx (y,e,h)))) . 1 is V22() real ext-real Element of REAL
1 - 1 is V22() real ext-real Element of REAL
(1 - 1) + (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
y . ((1 - 1) + (indx (y,e,(v + 1)))) is V22() real ext-real Element of REAL
h - v is V22() real ext-real Element of REAL
(indx (y,e,v)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))),K263()) is V22() real ext-real Element of REAL
(Sum (mid (H₁(e),(v + 1),h))) - (Sum (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (A,e) is V22() real ext-real Element of REAL
(indx (y,e,h)) - (indx (y,e,v)) is V22() real ext-real Element of REAL
upper_bound (divset (e,h)) is V22() real ext-real Element of REAL
lower_bound (divset (e,h)) is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
(indx (y,e,v)) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
y . ((indx (y,e,v)) + 2) is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
(rng e) /\ (divset (e,h)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
y . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . v1 is V22() real ext-real Element of REAL
(indx (y,e,v)) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,v)) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) -' ((indx (y,e,v)) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,h)) -' ((indx (y,e,v)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) - ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
((indx (y,e,h)) - ((indx (y,e,v)) + 1)) + 1 is V22() real ext-real Element of REAL
divset (e,(v + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (e,(v + 1))) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (vol (divset (e,(v + 1)))) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (y,((indx (y,e,v)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
y . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
lower_bound (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
((indx (y,e,v)) + 1) - 1 is V22() real ext-real Element of REAL
y . (((indx (y,e,v)) + 1) - 1) is V22() real ext-real Element of REAL
lower_bound (divset (e,(v + 1))) is V22() real ext-real Element of REAL
(v + 1) - 1 is V22() real ext-real Element of REAL
e . ((v + 1) - 1) is V22() real ext-real Element of REAL
vol (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) . 1 is V22() real ext-real Element of REAL
1 + ((indx (y,e,v)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((1 + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
H₁(y) . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
<*(H₁(y) . ((indx (y,e,v)) + 1))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
f | (divset (y,((indx (y,e,v)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,v)) + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (y,((indx (y,e,v)) + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (y,((indx (y,e,v)) + 1)))))) * (vol (divset (y,((indx (y,e,v)) + 1)))) is V22() real ext-real Element of REAL
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (y,((indx (y,e,v)) + 2)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (y,((indx (y,e,v)) + 2))) is V22() real ext-real Element of REAL
((indx (y,e,v)) + 2) - 1 is V22() real ext-real Element of REAL
lower_bound (divset (y,((indx (y,e,v)) + 2))) is V22() real ext-real Element of REAL
y . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
vol (divset (y,((indx (y,e,v)) + 2))) is V22() real ext-real Element of REAL
(e . h) - (y . ((indx (y,e,v)) + 1)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(v + 1))) is V22() real ext-real Element of REAL
lower_bound (divset (e,(v + 1))) is V22() real ext-real Element of REAL
(v + 1) - 1 is V22() real ext-real Element of REAL
e . ((v + 1) - 1) is V22() real ext-real Element of REAL
(e . (v + 1)) - (e . v) is V22() real ext-real Element of REAL
divset (y,((indx (y,e,v)) + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
H₁(y) . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
f | (divset (y,((indx (y,e,v)) + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,v)) + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (y,((indx (y,e,v)) + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (y,((indx (y,e,v)) + 1)))))) * (vol (divset (y,((indx (y,e,v)) + 1)))) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (vol (divset (y,((indx (y,e,v)) + 1)))) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) . 2 is V22() real ext-real Element of REAL
2 + ((indx (y,e,v)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(2 + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((2 + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
(indx (y,e,v)) + 0 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,v)) + 0) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
H₁(y) . (((indx (y,e,v)) + 0) + 2) is V22() real ext-real Element of REAL
(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) . 1 is V22() real ext-real Element of REAL
1 + ((indx (y,e,v)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((1 + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
H₁(y) . ((indx (y,e,v)) + 2) is V22() real ext-real Element of REAL
<*(H₁(y) . ((indx (y,e,v)) + 1)),(H₁(y) . ((indx (y,e,v)) + 2))*> is Relation-like NAT -defined Function-like non empty V37() 2 -element FinSequence-like FinSubsequence-like set
(H₁(y) . ((indx (y,e,v)) + 1)) + (H₁(y) . ((indx (y,e,v)) + 2)) is V22() real ext-real Element of REAL
upper_bound (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
lower_bound (divset (y,((indx (y,e,v)) + 1))) is V22() real ext-real Element of REAL
((indx (y,e,v)) + 1) - 1 is V22() real ext-real Element of REAL
y . (((indx (y,e,v)) + 1) - 1) is V22() real ext-real Element of REAL
(y . ((indx (y,e,v)) + 1)) - (e . v) is V22() real ext-real Element of REAL
f | (divset (y,((indx (y,e,v)) + 2))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,((indx (y,e,v)) + 2)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (y,((indx (y,e,v)) + 2))))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (y,((indx (y,e,v)) + 2)))))) * (vol (divset (y,((indx (y,e,v)) + 2)))) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (vol (divset (y,((indx (y,e,v)) + 2)))) is V22() real ext-real Element of REAL
((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,v)) + 1))))) + ((lower_bound (rng f)) * (vol (divset (y,((indx (y,e,v)) + 2))))) is V22() real ext-real Element of REAL
h -' (v + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(h -' (v + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h - (v + 1) is V22() real ext-real Element of REAL
(h - (v + 1)) + 1 is V22() real ext-real Element of REAL
len (mid (H₁(e),(v + 1),h)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(v + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(h + 1) - (v + 1) is V22() real ext-real Element of REAL
(mid (H₁(e),(v + 1),h)) . 1 is V22() real ext-real Element of REAL
1 + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + (v + 1)) - 1 is V22() real ext-real Element of REAL
H₁(e) . ((1 + (v + 1)) - 1) is V22() real ext-real Element of REAL
f | (divset (e,(v + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (e,(v + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (e,(v + 1))))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (e,(v + 1)))))) * (vol (divset (e,(v + 1)))) is V22() real ext-real Element of REAL
<*((upper_bound (rng (f | (divset (e,(v + 1)))))) * (vol (divset (e,(v + 1)))))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
len (upper_volume ((chi (A,A)),e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),e))) is non empty V37() len (upper_volume ((chi (A,A)),e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(upper_volume ((chi (A,A)),e)) . (v + 1) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (vol (divset (e,(v + 1)))) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (vol (divset (e,(v + 1)))) is V22() real ext-real Element of REAL
((upper_bound (rng f)) * (vol (divset (e,(v + 1))))) - ((lower_bound (rng f)) * (vol (divset (e,(v + 1))))) is V22() real ext-real Element of REAL
h -' (v + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
h - (v + 1) is V22() real ext-real Element of REAL
(h -' (v + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (e,(v + 1),h) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
v1 is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound v1 is V22() real ext-real Element of REAL
upper_bound v1 is V22() real ext-real Element of REAL
n is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of v1
len n is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
n . (len n) is V22() real ext-real Element of REAL
k is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of v1
len k is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
k . (len k) is V22() real ext-real Element of REAL
(h - (v + 1)) + 1 is V22() real ext-real Element of REAL
(len k) + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((len k) + (v + 1)) - 1 is V22() real ext-real Element of REAL
dom k is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (k,(len k)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (k,(len k))) is V22() real ext-real Element of REAL
lower_bound (divset (e,h)) is V22() real ext-real Element of REAL
upper_bound (divset (k,(len k))) is V22() real ext-real Element of REAL
upper_bound (divset (e,h)) is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
(len k) - 1 is V22() real ext-real Element of REAL
(len k) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((len k) - 1) + (v + 1) is V22() real ext-real Element of REAL
(((len k) - 1) + (v + 1)) - 1 is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
k . ((len k) - 1) is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
s is set
lower_bound A is V22() real ext-real Element of REAL
sD is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
(h - v) - 1 is V22() real ext-real Element of REAL
((h - v) - 1) + (v + 1) is V22() real ext-real Element of REAL
e . (((h - v) - 1) + (v + 1)) is V22() real ext-real Element of REAL
K20(v1,REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20(v1,REAL)) is V12() V37() set
f | v1 is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
(v1,k) is V22() real ext-real Element of REAL
chi (v1,v1) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
upper_volume ((chi (v1,v1)),k) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (v1,v1)),k)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (v1,v1)),k))) is V22() real ext-real set
s is Relation-like Function-like non empty total V30(v1, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
s | v1 is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
dom f is non empty set
A /\ (dom f) is V58() V59() V60() Element of K19(REAL)
sD is V22() real ext-real set
dom s is non empty set
v1 /\ (dom s) is V58() V59() V60() Element of K19(REAL)
m is set
s . m is V22() real ext-real Element of REAL
(dom f) /\ v1 is V58() V59() V60() Element of K19(REAL)
(dom f) /\ A is V58() V59() V60() Element of K19(REAL)
f . m is V22() real ext-real Element of REAL
m is V22() real ext-real set
D1 is set
s . D1 is V22() real ext-real Element of REAL
(dom f) /\ v1 is V58() V59() V60() Element of K19(REAL)
(dom f) /\ A is V58() V59() V60() Element of K19(REAL)
f . D1 is V22() real ext-real Element of REAL
h - 1 is V22() real ext-real Element of REAL
e . (h - 1) is V22() real ext-real Element of REAL
upper_volume (s,k) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (upper_volume (s,k)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(upper_volume (s,k)) . sD is V22() real ext-real Element of REAL
(mid (H₁(e),(v + 1),h)) . sD is V22() real ext-real Element of REAL
Seg (len (upper_volume (s,k))) is non empty V37() len (upper_volume (s,k)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len k) is non empty V37() len k -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (k,sD) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
s | (divset (k,sD)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
rng (s | (divset (k,sD))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (s | (divset (k,sD)))) is V22() real ext-real Element of REAL
vol (divset (k,sD)) is V22() real ext-real Element of REAL
(upper_bound (rng (s | (divset (k,sD))))) * (vol (divset (k,sD))) is V22() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
1 + m is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
sD + m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(v + 1) - 1 is V22() real ext-real Element of REAL
h - ((v + 1) - 1) is V22() real ext-real Element of REAL
sD + ((v + 1) - 1) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (e,(sD + m)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,(sD + m))) is V22() real ext-real Element of REAL
lower_bound (divset (k,sD)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(sD + m))) is V22() real ext-real Element of REAL
upper_bound (divset (k,sD)) is V22() real ext-real Element of REAL
k . sD is V22() real ext-real Element of REAL
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
k . sD is V22() real ext-real Element of REAL
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
sD - 1 is V22() real ext-real Element of REAL
k . (sD - 1) is V22() real ext-real Element of REAL
(sD - 1) + (v + 1) is V22() real ext-real Element of REAL
((sD - 1) + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . (((sD - 1) + (v + 1)) - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (k,sD))),(upper_bound (divset (k,sD))).] is V58() V59() V60() interval Element of K19(REAL)
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
H₁(e) . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
f | (divset (e,(sD + m))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (e,(sD + m)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (e,(sD + m))))) is V22() real ext-real Element of REAL
vol (divset (e,(sD + m))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (e,(sD + m)))))) * (vol (divset (e,(sD + m)))) is V22() real ext-real Element of REAL
len (mid (H₁(e),(v + 1),h)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Sum (upper_volume (s,k)) is V22() real ext-real Element of REAL
K190(REAL,(upper_volume (s,k)),K263()) is V22() real ext-real Element of REAL
vol v1 is V22() real ext-real Element of REAL
(upper_bound v1) - (e . v) is V22() real ext-real Element of REAL
(e . h) - (e . v) is V22() real ext-real Element of REAL
rng s is non empty V58() V59() V60() Element of K19(REAL)
lower_bound (rng s) is V22() real ext-real Element of REAL
upper_bound (rng s) is V22() real ext-real Element of REAL
(upper_bound (rng s)) - (lower_bound (rng s)) is V22() real ext-real Element of REAL
((upper_bound (rng s)) - (lower_bound (rng s))) * (v1,k) is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) * (v1,k) is V22() real ext-real Element of REAL
y . ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . sD is V22() real ext-real Element of REAL
upper_bound (divset (e,v)) is V22() real ext-real Element of REAL
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . sD is V22() real ext-real Element of REAL
(v + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) -' (indx (y,e,(v + 1))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,h)) -' (indx (y,e,(v + 1)))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) - (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
((indx (y,e,h)) - (indx (y,e,(v + 1)))) + 1 is V22() real ext-real Element of REAL
upper_volume (s,n) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (upper_volume (s,n)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) - ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
((indx (y,e,h)) - ((indx (y,e,v)) + 1)) + 1 is V22() real ext-real Element of REAL
rng k is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
{(e . (H + 1))} is non empty V12() V37() 1 -element V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng k) \/ {(e . (H + 1))} is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng n is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sD is set
m is V22() real ext-real Element of REAL
((h - (v + 1)) + 1) + (v + 1) is V22() real ext-real Element of REAL
(((h - (v + 1)) + 1) + (v + 1)) - 1 is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . D1 is V22() real ext-real Element of REAL
D1 - (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
(D1 - (indx (y,e,(v + 1)))) + 1 is V22() real ext-real Element of REAL
k . 1 is V22() real ext-real Element of REAL
D1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(indx (y,e,(v + 1))) + D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
D2 + (indx (y,e,(v + 1))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(D2 + (indx (y,e,(v + 1)))) - 1 is V22() real ext-real Element of REAL
(indx (y,e,(v + 1))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(D1 + 1) - (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
dom n is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
n . D2 is V22() real ext-real Element of REAL
m is V22() real ext-real Element of REAL
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . D2 is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
m is V22() real ext-real Element of REAL
m is V22() real ext-real Element of REAL
dom n is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len n) - 1 is V22() real ext-real Element of REAL
((len n) - 1) + (indx (y,e,(v + 1))) is V22() real ext-real Element of REAL
sD is set
n . 1 is V22() real ext-real Element of REAL
m is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . D1 is V22() real ext-real Element of REAL
D1 - v is V22() real ext-real Element of REAL
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
v + D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
k . D2 is V22() real ext-real Element of REAL
(D1 - v) - 1 is V22() real ext-real Element of REAL
((D1 - v) - 1) + (v + 1) is V22() real ext-real Element of REAL
e . (((D1 - v) - 1) + (v + 1)) is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
e . D1 is V22() real ext-real Element of REAL
D2 is V22() real ext-real Element of REAL
(rng e) /\ (divset (e,h)) is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
upper_bound (divset (e,v)) is V22() real ext-real Element of REAL
dom (upper_volume ((chi (v1,v1)),k)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (v1,v1)),k)) . sD is V22() real ext-real Element of REAL
len (upper_volume ((chi (v1,v1)),k)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (v1,v1)),k))) is non empty V37() len (upper_volume ((chi (v1,v1)),k)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len k) is non empty V37() len k -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
sD + v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (k,sD) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (k,sD)) is V22() real ext-real Element of REAL
divset (e,(sD + v)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
lower_bound (divset (e,(sD + v))) is V22() real ext-real Element of REAL
upper_bound (divset (k,sD)) is V22() real ext-real Element of REAL
upper_bound (divset (e,(sD + v))) is V22() real ext-real Element of REAL
k . sD is V22() real ext-real Element of REAL
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
(sD + v) - 1 is V22() real ext-real Element of REAL
e . ((sD + v) - 1) is V22() real ext-real Element of REAL
k . sD is V22() real ext-real Element of REAL
sD + (v + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(sD + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . ((sD + (v + 1)) - 1) is V22() real ext-real Element of REAL
sD - 1 is V22() real ext-real Element of REAL
k . (sD - 1) is V22() real ext-real Element of REAL
(sD - 1) + (v + 1) is V22() real ext-real Element of REAL
((sD - 1) + (v + 1)) - 1 is V22() real ext-real Element of REAL
e . (((sD - 1) + (v + 1)) - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (e,(sD + v)))),(upper_bound (divset (e,(sD + v)))).] is V58() V59() V60() interval Element of K19(REAL)
len (upper_volume ((chi (A,A)),e)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),e))) is non empty V37() len (upper_volume ((chi (A,A)),e)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),e)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
vol (divset (k,sD)) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),e)) . (sD + v) is V22() real ext-real Element of REAL
Sum (upper_volume (s,n)) is V22() real ext-real Element of REAL
K190(REAL,(upper_volume (s,n)),K263()) is V22() real ext-real Element of REAL
(Sum (upper_volume (s,k))) - (Sum (upper_volume (s,n))) is V22() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(upper_volume (s,n)) . m is V22() real ext-real Element of REAL
(mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) . m is V22() real ext-real Element of REAL
Seg (len (upper_volume (s,n))) is non empty V37() len (upper_volume (s,n)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
m + ((indx (y,e,v)) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(m + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
H₁(y) . ((m + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
Seg (len n) is non empty V37() len n -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (n,m) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
s | (divset (n,m)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(v1,REAL))
rng (s | (divset (n,m))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (s | (divset (n,m)))) is V22() real ext-real Element of REAL
vol (divset (n,m)) is V22() real ext-real Element of REAL
(upper_bound (rng (s | (divset (n,m))))) * (vol (divset (n,m))) is V22() real ext-real Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
1 + D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,v)) + 1) - 1 is V22() real ext-real Element of REAL
(indx (y,e,h)) - (((indx (y,e,v)) + 1) - 1) is V22() real ext-real Element of REAL
m + (((indx (y,e,v)) + 1) - 1) is V22() real ext-real Element of REAL
m + D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (y,(m + D1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
f | (divset (y,(m + D1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (y,(m + D1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (y,(m + D1))))) is V22() real ext-real Element of REAL
vol (divset (y,(m + D1))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (y,(m + D1)))))) * (vol (divset (y,(m + D1)))) is V22() real ext-real Element of REAL
lower_bound (divset (n,m)) is V22() real ext-real Element of REAL
lower_bound (divset (y,(m + D1))) is V22() real ext-real Element of REAL
upper_bound (divset (n,m)) is V22() real ext-real Element of REAL
upper_bound (divset (y,(m + D1))) is V22() real ext-real Element of REAL
y . (1 + D1) is V22() real ext-real Element of REAL
n . m is V22() real ext-real Element of REAL
(1 + (indx (y,e,(v + 1)))) - 1 is V22() real ext-real Element of REAL
y . ((1 + (indx (y,e,(v + 1)))) - 1) is V22() real ext-real Element of REAL
(1 + D1) - 1 is V22() real ext-real Element of REAL
y . ((1 + D1) - 1) is V22() real ext-real Element of REAL
n . m is V22() real ext-real Element of REAL
y . ((m + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
m - 1 is V22() real ext-real Element of REAL
n . (m - 1) is V22() real ext-real Element of REAL
(m - 1) + ((indx (y,e,v)) + 1) is V22() real ext-real Element of REAL
((m - 1) + ((indx (y,e,v)) + 1)) - 1 is V22() real ext-real Element of REAL
y . (((m - 1) + ((indx (y,e,v)) + 1)) - 1) is V22() real ext-real Element of REAL
[.(lower_bound (divset (n,m))),(upper_bound (divset (n,m))).] is V58() V59() V60() interval Element of K19(REAL)
len (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(indx (y,e,h)) -' ((indx (y,e,v)) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((indx (y,e,h)) -' ((indx (y,e,v)) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(H₂(e,v) - H₂(y, indx (y,e,v))) + ((Sum (mid (H₁(e),(v + 1),h))) - (Sum (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))))) is V22() real ext-real Element of REAL
((H * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e)) + (((upper_bound (rng f)) - (lower_bound (rng f))) * (A,e)) is V22() real ext-real Element of REAL
H₁(y) | (indx (y,e,v)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(y) | (indx (y,e,v))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(y) | (indx (y,e,v))),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),1,(indx (y,e,v))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),1,(indx (y,e,v)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),1,(indx (y,e,v)))),K263()) is V22() real ext-real Element of REAL
H₂(y, indx (y,e,v)) + (Sum (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))) is V22() real ext-real Element of REAL
(mid (H₁(y),1,(indx (y,e,v)))) ^ (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((mid (H₁(y),1,(indx (y,e,v)))) ^ (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))) is V22() real ext-real Element of REAL
K190(REAL,((mid (H₁(y),1,(indx (y,e,v)))) ^ (mid (H₁(y),((indx (y,e,v)) + 1),(indx (y,e,h))))),K263()) is V22() real ext-real Element of REAL
mid (H₁(y),1,(indx (y,e,h))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (mid (H₁(y),1,(indx (y,e,h)))) is V22() real ext-real Element of REAL
K190(REAL,(mid (H₁(y),1,(indx (y,e,h)))),K263()) is V22() real ext-real Element of REAL
H₁(y) | (indx (y,e,h)) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (H₁(y) | (indx (y,e,h))) is V22() real ext-real Element of REAL
K190(REAL,(H₁(y) | (indx (y,e,h))),K263()) is V22() real ext-real Element of REAL
(PartSums (upper_volume (f,y))) . (indx (y,e,h)) is V22() real ext-real Element of REAL
H₂(e,h) - H₂(y, indx (y,e,h)) is V22() real ext-real Element of REAL
h is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,h) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(PartSums (upper_volume (f,e))) . h is V22() real ext-real Element of REAL
indx (y,e,h) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (upper_volume (f,y))) . (indx (y,e,h)) is V22() real ext-real Element of REAL
((PartSums (upper_volume (f,e))) . h) - ((PartSums (upper_volume (f,y))) . (indx (y,e,h))) is V22() real ext-real Element of REAL
p is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . p is V22() real ext-real Element of REAL
p * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(p * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,e) is V22() real ext-real Element of REAL
H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,H) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(PartSums (upper_volume (f,e))) . H is V22() real ext-real Element of REAL
indx (y,e,H) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (upper_volume (f,y))) . (indx (y,e,H)) is V22() real ext-real Element of REAL
((PartSums (upper_volume (f,e))) . H) - ((PartSums (upper_volume (f,y))) . (indx (y,e,H))) is V22() real ext-real Element of REAL
len e is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
e . (len e) is V22() real ext-real Element of REAL
indx (y,e,(len e)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y . (indx (y,e,(len e))) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
y . (len y) is V22() real ext-real Element of REAL
e . (len e) is V22() real ext-real Element of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (e,D) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
(PartSums (upper_volume (f,e))) . D is V22() real ext-real Element of REAL
indx (y,e,D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(PartSums (upper_volume (f,y))) . (indx (y,e,D)) is V22() real ext-real Element of REAL
H₂(e,D) - H₂(y, indx (y,e,D)) is V22() real ext-real Element of REAL
Seg (len e) is non empty V37() len e -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
e . D is V22() real ext-real Element of REAL
upper_bound (divset (e,D)) is V22() real ext-real Element of REAL
(PartSums (upper_volume (f,y))) . (len y) is V22() real ext-real Element of REAL
(upper_sum (f,e)) - H₂(y, len y) is V22() real ext-real Element of REAL
y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng y is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,y) is V22() real ext-real Element of REAL
(upper_sum (f,e)) - (upper_sum (f,y)) is V22() real ext-real Element of REAL
lim (A,T) is V22() real ext-real Element of REAL
e is V22() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
y is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(A,T) . y is V22() real ext-real Element of REAL
((A,T) . y) - 0 is V22() real ext-real Element of REAL
abs (((A,T) . y) - 0) is V22() real ext-real Element of REAL
e + (abs (((A,T) . y) - 0)) is V22() real ext-real Element of REAL
((A,T) . y) + (abs (((A,T) . y) - 0)) is V22() real ext-real Element of REAL
T . y is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,(T . y)) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),(T . y)) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),(T . y))) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),(T . y)))) is V22() real ext-real set
dom (upper_volume ((chi (A,A)),(T . y))) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
p is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (A,A)),(T . y))) . p is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),(T . y))) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),(T . y)))) is non empty V37() len (upper_volume ((chi (A,A)),(T . y))) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
len D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len D) is non empty V37() len D -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom D is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset ((T . y),p) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset ((T . y),p)) is V22() real ext-real Element of REAL
e is V22() real ext-real set
e / 2 is V22() real ext-real Element of REAL
upper_sum_set f is Relation-like Function-like non empty total V30( divs A, REAL ) complex-valued ext-real-valued real-valued Element of K19(K20((divs A),REAL))
K20((divs A),REAL) is Relation-like V12() V37() complex-valued ext-real-valued real-valued set
K19(K20((divs A),REAL)) is V12() V37() set
rng (upper_sum_set f) is non empty V58() V59() V60() Element of K19(REAL)
lower_bound (rng (upper_sum_set f)) is V22() real ext-real Element of REAL
e is V22() real ext-real Element of REAL
e / 2 is V22() real ext-real Element of REAL
(upper_integral f) + (e / 2) is V22() real ext-real Element of REAL
y is V22() real ext-real set
dom (upper_sum_set f) is non empty set
lower_bound A is V22() real ext-real Element of REAL
D is Relation-like NAT -defined Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of divs A
(upper_sum_set f) . D is V22() real ext-real Element of REAL
p is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
len p is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len p) is non empty V37() len p -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom p is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
p . 1 is V22() real ext-real Element of REAL
p . 1 is V22() real ext-real Element of REAL
upper_volume (f,p) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(upper_volume (f,p)) . 1 is V22() real ext-real Element of REAL
divset (p,1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
f | (divset (p,1)) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (p,1))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (p,1)))) is V22() real ext-real Element of REAL
vol (divset (p,1)) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (p,1))))) * (vol (divset (p,1))) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
upper_sum (f,p) is V22() real ext-real Element of REAL
Sum (upper_volume (f,p)) is V22() real ext-real Element of REAL
K263() is Relation-like Function-like total V30(K20(REAL,REAL), REAL ) complex-valued ext-real-valued real-valued Element of K19(K20(K20(REAL,REAL),REAL))
K190(REAL,(upper_volume (f,p)),K263()) is V22() real ext-real Element of REAL
(upper_volume (f,p)) | 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(upper_volume (f,p)) /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((upper_volume (f,p)) | 1) ^ ((upper_volume (f,p)) /^ 1) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (((upper_volume (f,p)) | 1) ^ ((upper_volume (f,p)) /^ 1)) is V22() real ext-real Element of REAL
K190(REAL,(((upper_volume (f,p)) | 1) ^ ((upper_volume (f,p)) /^ 1)),K263()) is V22() real ext-real Element of REAL
p . (len p) is V22() real ext-real Element of REAL
p /^ 1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
H is Relation-like NAT -defined REAL -valued Function-like one-to-one V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len H is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(len p) - 1 is V22() real ext-real Element of REAL
h is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len h is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom h is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
(len h) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
h . (len h) is V22() real ext-real Element of REAL
Seg (len h) is non empty V37() len h -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
p . 2 is V22() real ext-real Element of REAL
rng p is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng h is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
len (upper_volume (f,p)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid ((upper_volume (f,p)),2,(len (upper_volume (f,p)))) is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (mid ((upper_volume (f,p)),2,(len (upper_volume (f,p))))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(len (upper_volume (f,p))) -' 2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((len (upper_volume (f,p))) -' 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len p) -' 2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
((len p) -' 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len p) - 2 is V22() real ext-real Element of REAL
((len p) - 2) + 1 is V22() real ext-real Element of REAL
v is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
upper_volume (f,v) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
(mid ((upper_volume (f,p)),2,(len (upper_volume (f,p))))) . v1 is V22() real ext-real Element of REAL
(upper_volume (f,v)) . v1 is V22() real ext-real Element of REAL
v1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
divset (p,(v1 + 1)) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
divset (v,v1) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
upper_bound (divset (p,(v1 + 1))) is V22() real ext-real Element of REAL
p . (v1 + 1) is V22() real ext-real Element of REAL
dom v is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
v . v1 is V22() real ext-real Element of REAL
lower_bound (divset (p,(v1 + 1))) is V22() real ext-real Element of REAL
(v1 + 1) - 1 is V22() real ext-real Element of REAL
p . ((v1 + 1) - 1) is V22() real ext-real Element of REAL
upper_bound (divset (v,v1)) is V22() real ext-real Element of REAL
lower_bound (divset (v,v1)) is V22() real ext-real Element of REAL
[.(lower_bound A),(v . v1).] is V58() V59() V60() interval Element of K19(REAL)
v1 - 1 is V22() real ext-real Element of REAL
v . (v1 - 1) is V22() real ext-real Element of REAL
(v1 - 1) + 1 is V22() real ext-real Element of REAL
p . ((v1 - 1) + 1) is V22() real ext-real Element of REAL
p . v1 is V22() real ext-real Element of REAL
upper_bound (divset (v,v1)) is V22() real ext-real Element of REAL
lower_bound (divset (v,v1)) is V22() real ext-real Element of REAL
[.(lower_bound (divset (v,v1))),(upper_bound (divset (v,v1))).] is V58() V59() V60() interval Element of K19(REAL)
(len (upper_volume (f,p))) - 1 is V22() real ext-real Element of REAL
(len (upper_volume (f,p))) - 2 is V22() real ext-real Element of REAL
((len (upper_volume (f,p))) - 2) + 1 is V22() real ext-real Element of REAL
v1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(v1 + 2) - 1 is V22() real ext-real Element of REAL
(upper_volume (f,p)) . ((v1 + 2) - 1) is V22() real ext-real Element of REAL
(upper_volume (f,p)) . (v1 + 1) is V22() real ext-real Element of REAL
f | (divset (p,(v1 + 1))) is Relation-like Function-like complex-valued ext-real-valued real-valued Element of K19(K20(A,REAL))
rng (f | (divset (p,(v1 + 1)))) is V58() V59() V60() Element of K19(REAL)
upper_bound (rng (f | (divset (p,(v1 + 1))))) is V22() real ext-real Element of REAL
vol (divset (p,(v1 + 1))) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (divset (p,(v1 + 1)))))) * (vol (divset (p,(v1 + 1)))) is V22() real ext-real Element of REAL
len v is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len v) is non empty V37() len v -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom v is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len ((upper_volume (f,p)) | 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
dom (upper_volume (f,p)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
((upper_volume (f,p)) | 1) . 1 is V22() real ext-real Element of REAL
<*((upper_volume (f,p)) . 1)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty V12() V37() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
2 -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
2 - 1 is V22() real ext-real Element of REAL
len v is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom v is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
v . 1 is V22() real ext-real Element of REAL
p . (1 + 1) is V22() real ext-real Element of REAL
len (upper_volume (f,v)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
upper_bound (divset (p,1)) is V22() real ext-real Element of REAL
lower_bound (divset (p,1)) is V22() real ext-real Element of REAL
(upper_bound (divset (p,1))) - (lower_bound (divset (p,1))) is V22() real ext-real Element of REAL
(upper_bound (divset (p,1))) - (lower_bound A) is V22() real ext-real Element of REAL
(p . 1) - (lower_bound A) is V22() real ext-real Element of REAL
Sum (upper_volume (f,v)) is V22() real ext-real Element of REAL
K190(REAL,(upper_volume (f,v)),K263()) is V22() real ext-real Element of REAL
0 + (Sum (upper_volume (f,v))) is V22() real ext-real Element of REAL
upper_sum (f,v) is V22() real ext-real Element of REAL
(upper_sum_set f) . v is V22() real ext-real Element of REAL
H is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(upper_sum_set f) . H is V22() real ext-real Element of REAL
H . 1 is V22() real ext-real Element of REAL
p . 1 is V22() real ext-real Element of REAL
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(upper_sum_set f) . D is V22() real ext-real Element of REAL
D . 1 is V22() real ext-real Element of REAL
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(upper_sum_set f) . D is V22() real ext-real Element of REAL
D . 1 is V22() real ext-real Element of REAL
len D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
v is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len v is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
dom v is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
2 * (len D) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v1 is Relation-like NAT -defined REAL -valued Function-like V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued non-decreasing FinSequence of REAL
dom v1 is V37() V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len D) is non empty V37() len D -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
k is Relation-like Function-like set
dom k is set
rng k is set
k * v1 is Relation-like REAL -valued complex-valued ext-real-valued real-valued set
dom D is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (D,n) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (D,n)) is V22() real ext-real Element of REAL
k . n is set
v . n is V22() real ext-real Element of REAL
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v1 . s is V22() real ext-real Element of REAL
k is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal set
v1 . k is V22() real ext-real Element of REAL
((upper_bound (rng f)) - (lower_bound (rng f))) + 1 is V22() real ext-real Element of REAL
(2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) is V22() real ext-real Element of REAL
e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) is V22() real ext-real Element of REAL
min ((v1 . k),(e / ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)))) is V22() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
upper_sum (f,D) is V22() real ext-real Element of REAL
v1 . 1 is V22() real ext-real Element of REAL
dom D is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
divset (D,s) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (D,s)) is V22() real ext-real Element of REAL
upper_bound (divset (D,s)) is V22() real ext-real Element of REAL
D . s is V22() real ext-real Element of REAL
lower_bound (divset (D,s)) is V22() real ext-real Element of REAL
(D . s) - (lower_bound A) is V22() real ext-real Element of REAL
upper_bound (divset (D,s)) is V22() real ext-real Element of REAL
D . s is V22() real ext-real Element of REAL
lower_bound (divset (D,s)) is V22() real ext-real Element of REAL
s - 1 is V22() real ext-real Element of REAL
D . (s - 1) is V22() real ext-real Element of REAL
(D . s) - (D . (s - 1)) is V22() real ext-real Element of REAL
s + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
rng v1 is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
rng v is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v . s is V22() real ext-real Element of REAL
divset (D,s) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (D,s)) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),D) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),D)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
inf (rng (upper_volume ((chi (A,A)),D))) is V22() real ext-real set
len v1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
Seg (len v) is V37() len v -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
Seg (len v1) is V37() len v1 -element V58() V59() V60() V61() V62() V63() bounded_below bounded_above real-bounded Element of K19(NAT)
rng v is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
rng v1 is V37() V58() V59() V60() bounded_below bounded_above real-bounded Element of K19(REAL)
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v . s is V22() real ext-real Element of REAL
dom D is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
len (upper_volume ((chi (A,A)),D)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),D))) is non empty V37() len (upper_volume ((chi (A,A)),D)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom (upper_volume ((chi (A,A)),D)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (D,s) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (D,s)) is V22() real ext-real Element of REAL
(upper_volume ((chi (A,A)),D)) . s is V22() real ext-real Element of REAL
dom (upper_volume ((chi (A,A)),D)) is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_volume ((chi (A,A)),D)) . s is V22() real ext-real Element of REAL
len (upper_volume ((chi (A,A)),D)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (upper_volume ((chi (A,A)),D))) is non empty V37() len (upper_volume ((chi (A,A)),D)) -element V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
dom D is non empty V37() V58() V59() V60() V61() V62() V63() left_end right_end bounded_below bounded_above real-bounded Element of K19(NAT)
divset (D,s) is non empty V58() V59() V60() V100() closed_interval bounded_below bounded_above real-bounded interval Element of K19(REAL)
vol (divset (D,s)) is V22() real ext-real Element of REAL
v . s is V22() real ext-real Element of REAL
sD is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
v1 . sD is V22() real ext-real Element of REAL
m is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_sum (f,T)) . m is V22() real ext-real Element of REAL
((upper_sum (f,T)) . m) - (upper_integral f) is V22() real ext-real Element of REAL
abs (((upper_sum (f,T)) . m) - (upper_integral f)) is V22() real ext-real Element of REAL
T . m is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
(A,D1) is V22() real ext-real Element of REAL
upper_volume ((chi (A,A)),D1) is Relation-like NAT -defined REAL -valued Function-like non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (upper_volume ((chi (A,A)),D1)) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
sup (rng (upper_volume ((chi (A,A)),D1))) is V22() real ext-real set
(A,T) . m is V22() real ext-real Element of REAL
rng D1 is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
rng D is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
(rng D1) \/ (rng D) is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,D1) is V22() real ext-real Element of REAL
D2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng D2 is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,D2) is V22() real ext-real Element of REAL
(upper_sum (f,D)) - (upper_sum (f,D2)) is V22() real ext-real Element of REAL
(upper_sum (f,D1)) - (upper_sum (f,D2)) is V22() real ext-real Element of REAL
upper_sum (f,(T . m)) is V22() real ext-real Element of REAL
(upper_sum (f,(T . m))) - (upper_sum (f,D)) is V22() real ext-real Element of REAL
(upper_sum (f,(T . m))) - (upper_sum (f,D2)) is V22() real ext-real Element of REAL
(upper_sum (f,D)) + (upper_sum (f,(T . m))) is V22() real ext-real Element of REAL
((upper_sum (f,D)) + (upper_sum (f,(T . m)))) - (upper_sum (f,(T . m))) is V22() real ext-real Element of REAL
(((upper_sum (f,D)) + (upper_sum (f,(T . m)))) - (upper_sum (f,(T . m)))) - (upper_integral f) is V22() real ext-real Element of REAL
(upper_sum (f,(T . m))) - (upper_integral f) is V22() real ext-real Element of REAL
((upper_sum (f,(T . m))) - (upper_integral f)) + (upper_sum (f,D)) is V22() real ext-real Element of REAL
(((upper_sum (f,(T . m))) - (upper_integral f)) + (upper_sum (f,D))) - (upper_sum (f,(T . m))) is V22() real ext-real Element of REAL
(upper_sum (f,(T . m))) + (e / 2) is V22() real ext-real Element of REAL
((upper_sum (f,(T . m))) + (e / 2)) - (upper_sum (f,D)) is V22() real ext-real Element of REAL
(upper_sum_set f) . (T . m) is V22() real ext-real Element of REAL
(len D) * ((upper_bound (rng f)) - (lower_bound (rng f))) is V22() real ext-real Element of REAL
(len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1) is V22() real ext-real Element of REAL
((A,T) . m) * ((2 * (len D)) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) is V22() real ext-real Element of REAL
((A,T) . m) * ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) is V22() real ext-real Element of REAL
(((A,T) . m) * ((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1))) * 2 is V22() real ext-real Element of REAL
((len D) * (((upper_bound (rng f)) - (lower_bound (rng f))) + 1)) * ((A,T) . m) is V22() real ext-real Element of REAL
((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * (A,D1) is V22() real ext-real Element of REAL
c₂₂ is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V37() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing Division of A
rng c₂₂ is non empty V37() V58() V59() V60() left_end right_end bounded_below bounded_above real-bounded Element of K19(REAL)
upper_sum (f,c₂₂) is V22() real ext-real Element of REAL
(upper_sum (f,D1)) - (upper_sum (f,c₂₂)) is V22() real ext-real Element of REAL
((len D) * ((upper_bound (rng f)) - (lower_bound (rng f)))) * ((A,T) . m) is V22() real ext-real Element of REAL
(e / 2) + (e / 2) is V22() real ext-real Element of REAL
((upper_sum (f,(T . m))) - (upper_sum (f,D))) + (e / 2) is V22() real ext-real Element of REAL
s is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V37() cardinal V58() V59() V60() V61() V62() V63() V69() V70() bounded_below bounded_above real-bounded Element of NAT
(upper_sum (f,T)) . s is V22() real ext-real Element of REAL
((upper_sum (f,T)) . s) - (upper_integral f) is V22() real ext-real Element of REAL
abs (((upper_sum (f,T)) . s) - (upper_integral f)) is V22() real ext-real Element of REAL