:: INTEGRA1 semantic presentation

REAL is non empty non trivial V40() V64() V65() V66() V70() V85() non bounded_below non bounded_above interval set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal V40() cardinal limit_cardinal V64() V65() V66() V67() V68() V69() V70() V85() left_end bounded_below Element of bool REAL
bool REAL is non empty non trivial V40() set
omega is non empty non trivial epsilon-transitive epsilon-connected ordinal V40() cardinal limit_cardinal V64() V65() V66() V67() V68() V69() V70() V85() left_end bounded_below set
bool omega is non empty non trivial V40() set
bool NAT is non empty non trivial V40() set
K147(NAT) is V39() set
{} is set
RAT is non empty non trivial V40() V64() V65() V66() V67() V70() set
the 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 V40() V41() V44() cardinal {} -element FinSequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V64() V65() V66() V67() V68() V69() V70() bounded_below bounded_above real-bounded interval 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 V40() V41() V44() cardinal {} -element FinSequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V64() V65() V66() V67() V68() V69() V70() bounded_below bounded_above real-bounded interval set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
{{},1} is V40() set
COMPLEX is non empty non trivial V40() V64() V70() set
INT is non empty non trivial V40() V64() V65() V66() V67() V68() V70() set
[:COMPLEX,COMPLEX:] is Relation-like non empty non trivial V40() complex-valued set
bool [:COMPLEX,COMPLEX:] is non empty non trivial V40() set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like non empty non trivial V40() complex-valued set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial V40() set
[:REAL,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:REAL,REAL:] is non empty non trivial V40() set
[:[:REAL,REAL:],REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial V40() set
[:RAT,RAT:] is Relation-like RAT -valued non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:RAT,RAT:] is non empty non trivial V40() set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial V40() set
[:INT,INT:] is Relation-like RAT -valued INT -valued non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:INT,INT:] is non empty non trivial V40() set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:[:INT,INT:],INT:] is non empty non trivial V40() set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial V40() complex-valued ext-real-valued real-valued natural-valued set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued non empty non trivial V40() complex-valued ext-real-valued real-valued natural-valued set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial V40() set
K347() is set
ExtREAL is non empty V65() interval set
[:COMPLEX,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:COMPLEX,REAL:] is non empty non trivial V40() set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
0 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Seg 1 is non empty trivial V40() 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
{1} is non empty trivial V40() V44() 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded set
Seg 2 is non empty V40() 2 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
{1,2} is V40() V44() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded set
A is V64() V65() V66() Element of bool REAL
f is V22() real ext-real set
g is V22() real ext-real set
[.f,g.] is V64() V65() V66() interval Element of bool REAL
A is non empty compact V64() V65() V66() interval closed_interval Element of bool REAL
A is V64() V65() V66() Element of bool REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
[.(lower_bound A),(upper_bound A).] is V64() V65() V66() interval Element of bool REAL
f is V22() real ext-real Element of REAL
g is V22() real ext-real Element of REAL
[.f,g.] is V64() V65() V66() interval Element of bool REAL
a1 is V22() real ext-real set
g - a1 is V22() real ext-real Element of REAL
g + a1 is V22() real ext-real Element of REAL
(g + a1) - a1 is V22() real ext-real Element of REAL
a1 is V22() real ext-real set
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( f <= b₁ & b₁ <= g ) } is set
D1 is V22() real ext-real Element of REAL
a1 is V22() real ext-real set
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( f <= b₁ & b₁ <= g ) } is set
D1 is V22() real ext-real Element of REAL
a1 is V22() real ext-real set
f + a1 is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f is V22() real ext-real set
a1 is V22() real ext-real set
[.f,a1.] is V64() V65() V66() interval Element of bool REAL
g is V22() real ext-real set
D1 is V22() real ext-real set
[.g,D1.] is V64() V65() V66() interval Element of bool REAL
A is non empty compact V64() V65() V66() Element of bool REAL
upper_bound A is V22() real ext-real Element of REAL
[:(Seg 1),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(Seg 1),REAL:] is non empty non trivial V40() set
(Seg 1) --> (upper_bound A) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
g is Relation-like Seg 1 -defined REAL -valued Function-like non empty total V30( Seg 1, REAL ) V40() complex-valued ext-real-valued real-valued Element of bool [:(Seg 1),REAL:]
dom g is non empty trivial V40() 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool (Seg 1)
bool (Seg 1) is non empty V40() V44() set
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom a1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a1 . D1 is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a1 . D1 is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
a1 . D1 is V22() real ext-real Element of REAL
rng a1 is V40() V64() V65() V66() bounded_below bounded_above real-bounded Element of bool REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a1 . (len a1) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() Element of bool REAL
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
f is set
g is set
a1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
f is set
g is set
a1 is set
A is non empty compact V64() V65() V66() Element of bool REAL
(A) is set
the Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A) is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
A is non empty compact V64() V65() V66() Element of bool REAL
(A) is non empty set
f is Element of (A)
g is Element of (A)
A is non empty compact V64() V65() V66() Element of bool REAL
(A) is non empty set
f is Relation-like Function-like Element of (A)
g is Relation-like Function-like Element of (A)
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g . A is V22() real ext-real Element of REAL
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
A - 1 is V22() real ext-real Element of REAL
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g . (A - 1) is V22() real ext-real Element of REAL
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
Seg a1 is V40() a1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
2 + D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + 1) - 1 is V22() real ext-real Element of REAL
a1 - 0 is V22() real ext-real Element of REAL
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
D1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
2 - 1 is V22() real ext-real Element of REAL
A - (2 - 1) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
f . g is V22() real ext-real Element of REAL
g - 1 is V22() real ext-real Element of REAL
f . (g - 1) is V22() real ext-real Element of REAL
[.(lower_bound A),(f . g).] is V64() V65() V66() interval Element of bool REAL
a1 is V64() V65() V66() Element of bool REAL
lower_bound a1 is V22() real ext-real Element of REAL
upper_bound a1 is V22() real ext-real Element of REAL
[.(lower_bound a1),(upper_bound a1).] is V64() V65() V66() interval Element of bool REAL
- 1 is V22() real ext-real non positive Element of REAL
1 + (- 1) is V22() real ext-real Element of REAL
g + (1 + (- 1)) is V22() real ext-real Element of REAL
g + (- 1) is V22() real ext-real Element of REAL
D1 is V22() real ext-real Element of REAL
D1 is V22() real ext-real Element of REAL
[.D1,D1.] is V64() V65() V66() interval Element of bool REAL
a2 is V64() V65() V66() Element of bool REAL
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f . a1 is V22() real ext-real Element of REAL
lower_bound a2 is V22() real ext-real Element of REAL
upper_bound a2 is V22() real ext-real Element of REAL
[.(lower_bound a2),(upper_bound a2).] is V64() V65() V66() interval Element of bool REAL
a1 is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound a1 is V22() real ext-real Element of REAL
inf a1 is V22() real ext-real set
upper_bound a1 is V22() real ext-real Element of REAL
sup a1 is V22() real ext-real set
D1 is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound D1 is V22() real ext-real Element of REAL
inf D1 is V22() real ext-real set
upper_bound D1 is V22() real ext-real Element of REAL
sup D1 is V22() real ext-real set
D1 is V22() real ext-real Element of REAL
[.(lower_bound A),D1.] is V64() V65() V66() interval Element of bool REAL
D1 is V22() real ext-real Element of REAL
a2 is V22() real ext-real Element of REAL
[.D1,a2.] is V64() V65() V66() interval Element of bool REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(f,g,A) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound f is V22() real ext-real Element of REAL
inf f is V22() real ext-real set
upper_bound f is V22() real ext-real Element of REAL
sup f is V22() real ext-real set
[.(lower_bound f),(upper_bound f).] is V64() V65() V66() interval Element of bool REAL
lower_bound (f,g,A) is V22() real ext-real Element of REAL
inf (f,g,A) is V22() real ext-real set
g . A is V22() real ext-real Element of REAL
a1 is V22() real ext-real Element of REAL
upper_bound (f,g,A) is V22() real ext-real Element of REAL
sup (f,g,A) is V22() real ext-real set
[.(lower_bound f),a1.] is V64() V65() V66() interval Element of bool REAL
g . A is V22() real ext-real Element of REAL
lower_bound f is V22() real ext-real Element of REAL
inf f is V22() real ext-real set
upper_bound f is V22() real ext-real Element of REAL
sup f is V22() real ext-real set
[.(lower_bound f),(upper_bound f).] is V64() V65() V66() interval Element of bool REAL
A - 1 is V22() real ext-real Element of REAL
g . (A - 1) is V22() real ext-real Element of REAL
[.(g . (A - 1)),(g . A).] is V64() V65() V66() interval Element of bool REAL
upper_bound (f,g,A) is V22() real ext-real Element of REAL
sup (f,g,A) is V22() real ext-real set
lower_bound (f,g,A) is V22() real ext-real Element of REAL
inf (f,g,A) is V22() real ext-real set
A is V64() V65() V66() Element of bool REAL
upper_bound A is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
A is non empty V64() V65() V66() bounded_below bounded_above real-bounded Element of bool REAL
(A) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom a1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
a1 . D1 is V22() real ext-real Element of REAL
(A,g,D1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f | (A,g,D1) is Relation-like A -defined (A,g,D1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (f | (A,g,D1)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (f | (A,g,D1))) is V22() real ext-real Element of REAL
((A,g,D1)) is V22() real ext-real Element of REAL
upper_bound (A,g,D1) is V22() real ext-real Element of REAL
sup (A,g,D1) is V22() real ext-real set
lower_bound (A,g,D1) is V22() real ext-real Element of REAL
inf (A,g,D1) is V22() real ext-real set
(upper_bound (A,g,D1)) - (lower_bound (A,g,D1)) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (A,g,D1)))) * ((A,g,D1)) is V22() real ext-real Element of REAL
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom a1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
a1 . D1 is V22() real ext-real Element of REAL
D1 . D1 is V22() real ext-real Element of REAL
(A,g,D1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f | (A,g,D1) is Relation-like A -defined (A,g,D1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (f | (A,g,D1)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (f | (A,g,D1))) is V22() real ext-real Element of REAL
((A,g,D1)) is V22() real ext-real Element of REAL
upper_bound (A,g,D1) is V22() real ext-real Element of REAL
sup (A,g,D1) is V22() real ext-real set
lower_bound (A,g,D1) is V22() real ext-real Element of REAL
inf (A,g,D1) is V22() real ext-real set
(upper_bound (A,g,D1)) - (lower_bound (A,g,D1)) is V22() real ext-real Element of REAL
(upper_bound (rng (f | (A,g,D1)))) * ((A,g,D1)) is V22() real ext-real Element of REAL
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom a1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
a1 . D1 is V22() real ext-real Element of REAL
(A,g,D1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f | (A,g,D1) is Relation-like A -defined (A,g,D1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (f | (A,g,D1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (f | (A,g,D1))) is V22() real ext-real Element of REAL
((A,g,D1)) is V22() real ext-real Element of REAL
upper_bound (A,g,D1) is V22() real ext-real Element of REAL
sup (A,g,D1) is V22() real ext-real set
lower_bound (A,g,D1) is V22() real ext-real Element of REAL
inf (A,g,D1) is V22() real ext-real set
(upper_bound (A,g,D1)) - (lower_bound (A,g,D1)) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (A,g,D1)))) * ((A,g,D1)) is V22() real ext-real Element of REAL
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom a1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
a1 . D1 is V22() real ext-real Element of REAL
D1 . D1 is V22() real ext-real Element of REAL
(A,g,D1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f | (A,g,D1) is Relation-like A -defined (A,g,D1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (f | (A,g,D1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (f | (A,g,D1))) is V22() real ext-real Element of REAL
((A,g,D1)) is V22() real ext-real Element of REAL
upper_bound (A,g,D1) is V22() real ext-real Element of REAL
sup (A,g,D1) is V22() real ext-real set
lower_bound (A,g,D1) is V22() real ext-real Element of REAL
inf (A,g,D1) is V22() real ext-real set
(upper_bound (A,g,D1)) - (lower_bound (A,g,D1)) is V22() real ext-real Element of REAL
(lower_bound (rng (f | (A,g,D1)))) * ((A,g,D1)) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,g) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,g) is V22() real ext-real Element of REAL
K295() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like non empty total V30([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K173(REAL,(A,f,g),K295()) is V22() real ext-real Element of REAL
(A,f,g) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,g) is V22() real ext-real Element of REAL
K173(REAL,(A,f,g),K295()) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
(A) is non empty set
[:(A),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A),REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
a1 is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 . D1 is V22() real ext-real Element of REAL
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
a1 . D1 is V22() real ext-real Element of REAL
a2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,a2) is V22() real ext-real Element of REAL
(A,f,a2) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,a2) is V22() real ext-real Element of REAL
K173(REAL,(A,f,a2),K295()) is V22() real ext-real Element of REAL
a1 is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
D1 is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
D1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
a1 . D1 is V22() real ext-real set
D1 . D1 is V22() real ext-real set
a1 . D1 is V22() real ext-real Element of REAL
a2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,a2) is V22() real ext-real Element of REAL
(A,f,a2) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,a2) is V22() real ext-real Element of REAL
K173(REAL,(A,f,a2),K295()) is V22() real ext-real Element of REAL
D1 . D1 is V22() real ext-real Element of REAL
a1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
a1 is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 . D1 is V22() real ext-real Element of REAL
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
a1 . D1 is V22() real ext-real Element of REAL
a2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,a2) is V22() real ext-real Element of REAL
(A,f,a2) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,a2) is V22() real ext-real Element of REAL
K173(REAL,(A,f,a2),K295()) is V22() real ext-real Element of REAL
a1 is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
D1 is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
D1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
a1 . D1 is V22() real ext-real set
D1 . D1 is V22() real ext-real set
a1 . D1 is V22() real ext-real Element of REAL
a2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,a2) is V22() real ext-real Element of REAL
(A,f,a2) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,a2) is V22() real ext-real Element of REAL
K173(REAL,(A,f,a2),K295()) is V22() real ext-real Element of REAL
D1 . D1 is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,f) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
(A) is non empty set
[:(A),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A),REAL:] is non empty non trivial V40() set
rng (A,f) is non empty V64() V65() V66() Element of bool REAL
lower_bound (rng (A,f)) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,f) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
(A) is non empty set
[:(A),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A),REAL:] is non empty non trivial V40() set
rng (A,f) is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng (A,f)) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,f) is V22() real ext-real Element of REAL
(A,f) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
(A) is non empty set
[:(A),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A),REAL:] is non empty non trivial V40() set
rng (A,f) is non empty V64() V65() V66() Element of bool REAL
lower_bound (rng (A,f)) is V22() real ext-real Element of REAL
A is non empty set
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
f + g is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (f + g) is V64() V65() V66() Element of bool REAL
rng f is V64() V65() V66() Element of bool REAL
rng g is V64() V65() V66() Element of bool REAL
(rng f) ++ (rng g) is V64() V65() V66() set
(rng f) ++ (rng g) is V65() set
a1 is set
dom (f + g) is Element of bool A
bool A is non empty set
D1 is set
(f + g) . D1 is V22() real ext-real Element of REAL
dom f is Element of bool A
dom g is Element of bool A
(dom f) /\ (dom g) is Element of bool A
f . D1 is V22() real ext-real Element of REAL
g . D1 is V22() real ext-real Element of REAL
(f . D1) + (g . D1) is V22() real ext-real Element of REAL
A is non empty set
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
f | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng f is V64() V65() V66() Element of bool REAL
dom f is Element of bool A
bool A is non empty set
A /\ (dom f) is Element of bool A
g is V22() real ext-real set
a1 is ext-real set
D1 is set
f . D1 is V22() real ext-real Element of REAL
A is non empty set
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng f is V64() V65() V66() Element of bool REAL
f | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g is V22() real ext-real set
dom f is Element of bool A
bool A is non empty set
A /\ (dom f) is Element of bool A
a1 is set
f . a1 is V22() real ext-real Element of REAL
A is non empty set
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
f | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng f is V64() V65() V66() Element of bool REAL
dom f is Element of bool A
bool A is non empty set
A /\ (dom f) is Element of bool A
g is V22() real ext-real set
a1 is ext-real set
D1 is set
f . D1 is V22() real ext-real Element of REAL
A is non empty set
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng f is V64() V65() V66() Element of bool REAL
f | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g is V22() real ext-real set
dom f is Element of bool A
bool A is non empty set
A /\ (dom f) is Element of bool A
a1 is set
f . a1 is V22() real ext-real Element of REAL
A is non empty set
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
f | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng f is V64() V65() V66() Element of bool REAL
A is non empty set
chi (A,A) is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
(chi (A,A)) | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
dom (chi (A,A)) is Element of bool A
bool A is non empty set
A /\ (dom (chi (A,A))) is Element of bool A
f is Element of A
(chi (A,A)) /. f is V22() real ext-real Element of REAL
(chi (A,A)) . f is V22() real ext-real Element of REAL
{1} is non empty trivial V40() V44() 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
A is non empty set
bool A is non empty set
f is non empty Element of bool A
chi (f,f) is Relation-like f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
rng (chi (f,f)) is V64() V65() V66() Element of bool REAL
(chi (f,f)) | f is Relation-like f -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
dom (chi (f,f)) is Element of bool f
bool f is non empty set
f /\ (dom (chi (f,f))) is Element of bool f
g is Element of A
(chi (f,f)) . g is V22() real ext-real Element of REAL
g is Element of A
(chi (f,f)) . g is V22() real ext-real Element of REAL
rng ((chi (f,f)) | f) is V64() V65() V66() Element of bool REAL
g is V22() real ext-real Element of REAL
{g} is non empty trivial V40() 1 -element V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
A is non empty set
bool A is non empty set
f is non empty Element of bool A
chi (f,f) is Relation-like f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
dom (chi (f,f)) is Element of bool f
bool f is non empty set
g is set
(chi (f,f)) | g is Relation-like f -defined g -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
rng ((chi (f,f)) | g) is V64() V65() V66() Element of bool REAL
dom ((chi (f,f)) | g) is Element of bool f
g /\ (dom (chi (f,f))) is Element of bool f
rng (chi (f,f)) is V64() V65() V66() Element of bool REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
chi (f,f) is Relation-like f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(f,g,A) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((f,g,A)) is V22() real ext-real Element of REAL
upper_bound (f,g,A) is V22() real ext-real Element of REAL
sup (f,g,A) is V22() real ext-real set
lower_bound (f,g,A) is V22() real ext-real Element of REAL
inf (f,g,A) is V22() real ext-real set
(upper_bound (f,g,A)) - (lower_bound (f,g,A)) is V22() real ext-real Element of REAL
(f,(chi (f,f)),g) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,(chi (f,f)),g) . A is V22() real ext-real Element of REAL
dom (chi (f,f)) is V64() V65() V66() Element of bool f
bool f is non empty set
(chi (f,f)) | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
rng ((chi (f,f)) | (f,g,A)) is V64() V65() V66() Element of bool REAL
lower_bound (rng ((chi (f,f)) | (f,g,A))) is V22() real ext-real Element of REAL
(lower_bound (rng ((chi (f,f)) | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
(f,g,A) /\ (dom (chi (f,f))) is V64() V65() V66() Element of bool f
rng (chi (f,f)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (chi (f,f))) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
chi (f,f) is Relation-like f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(f,g,A) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((f,g,A)) is V22() real ext-real Element of REAL
upper_bound (f,g,A) is V22() real ext-real Element of REAL
sup (f,g,A) is V22() real ext-real set
lower_bound (f,g,A) is V22() real ext-real Element of REAL
inf (f,g,A) is V22() real ext-real set
(upper_bound (f,g,A)) - (lower_bound (f,g,A)) is V22() real ext-real Element of REAL
(f,(chi (f,f)),g) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,(chi (f,f)),g) . A is V22() real ext-real Element of REAL
dom (chi (f,f)) is V64() V65() V66() Element of bool f
bool f is non empty set
(chi (f,f)) | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
rng ((chi (f,f)) | (f,g,A)) is V64() V65() V66() Element of bool REAL
upper_bound (rng ((chi (f,f)) | (f,g,A))) is V22() real ext-real Element of REAL
(upper_bound (rng ((chi (f,f)) | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
(f,g,A) /\ (dom (chi (f,f))) is V64() V65() V66() Element of bool f
rng (chi (f,f)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (chi (f,f))) is V22() real ext-real Element of REAL
A is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom A is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Sum A is V22() real ext-real Element of REAL
K173(REAL,A,K295()) is V22() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Sum f is V22() real ext-real Element of REAL
K173(REAL,f,K295()) is V22() real ext-real Element of REAL
(Sum A) + (Sum f) is V22() real ext-real Element of REAL
g is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Sum g is V22() real ext-real Element of REAL
K173(REAL,g,K295()) is V22() real ext-real Element of REAL
(len A) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
A + f is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K178(K295(),A,f) is set
len (A + f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom g is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len (A + f)) is V40() len (A + f) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g . a1 is V22() real ext-real Element of REAL
A . a1 is V22() real ext-real Element of REAL
f . a1 is V22() real ext-real Element of REAL
(A . a1) + (f . a1) is V22() real ext-real Element of REAL
Seg (len f) is V40() len f -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom f is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
f /. a1 is V22() real ext-real Element of REAL
A /. a1 is V22() real ext-real Element of REAL
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
g . a1 is V22() real ext-real Element of REAL
(A + f) . a1 is V22() real ext-real Element of REAL
A . a1 is V22() real ext-real Element of REAL
f . a1 is V22() real ext-real Element of REAL
(A . a1) + (f . a1) is V22() real ext-real Element of REAL
dom (A + f) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Sum (A + f) is V22() real ext-real Element of REAL
K173(REAL,(A + f),K295()) is V22() real ext-real Element of REAL
A is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom A is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Sum A is V22() real ext-real Element of REAL
K173(REAL,A,K295()) is V22() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Sum f is V22() real ext-real Element of REAL
K173(REAL,f,K295()) is V22() real ext-real Element of REAL
(Sum A) - (Sum f) is V22() real ext-real Element of REAL
g is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Sum g is V22() real ext-real Element of REAL
K173(REAL,g,K295()) is V22() real ext-real Element of REAL
(len A) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
A - f is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K296() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like non empty total V30([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K178(K296(),A,f) is set
len (A - f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom g is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len (A - f)) is V40() len (A - f) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g . a1 is V22() real ext-real Element of REAL
A . a1 is V22() real ext-real Element of REAL
f . a1 is V22() real ext-real Element of REAL
(A . a1) - (f . a1) is V22() real ext-real Element of REAL
Seg (len f) is V40() len f -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom f is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
f /. a1 is V22() real ext-real Element of REAL
A /. a1 is V22() real ext-real Element of REAL
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
g . a1 is V22() real ext-real Element of REAL
(A - f) . a1 is V22() real ext-real Element of REAL
Seg (len A) is V40() len A -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
A . a1 is V22() real ext-real Element of REAL
f . a1 is V22() real ext-real Element of REAL
(A . a1) - (f . a1) is V22() real ext-real Element of REAL
dom (A - f) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Sum (A - f) is V22() real ext-real Element of REAL
K173(REAL,(A - f),K295()) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
chi (A,A) is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
(A) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,(chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(chi (A,A)),f) is V22() real ext-real Element of REAL
K173(REAL,(A,(chi (A,A)),f),K295()) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom g is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g . a1 is V22() real ext-real Element of REAL
(A,f,a1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound (A,f,a1) is V22() real ext-real Element of REAL
sup (A,f,a1) is V22() real ext-real set
lower_bound (A,f,a1) is V22() real ext-real Element of REAL
inf (A,f,a1) is V22() real ext-real set
(upper_bound (A,f,a1)) - (lower_bound (A,f,a1)) is V22() real ext-real Element of REAL
((A,f,a1)) is V22() real ext-real Element of REAL
(len f) - 1 is V22() real ext-real Element of REAL
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + a1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom D1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom D1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg a1 is V40() a1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
<*(upper_bound A)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial V40() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
D1 ^ <*(upper_bound A)*> is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (D1 ^ <*(upper_bound A)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
<*(lower_bound A)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial V40() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
<*(lower_bound A)*> ^ D1 is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (<*(lower_bound A)*> ^ D1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom (D1 ^ <*(upper_bound A)*>) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom <*(upper_bound A)*> is non empty trivial V40() 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len <*(upper_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
a1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom <*(lower_bound A)*> is non empty trivial V40() 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len <*(lower_bound A)*> is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
a2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g . a2 is V22() real ext-real Element of REAL
(D1 ^ <*(upper_bound A)*>) /. a2 is V22() real ext-real Element of REAL
(<*(lower_bound A)*> ^ D1) /. a2 is V22() real ext-real Element of REAL
((D1 ^ <*(upper_bound A)*>) /. a2) - ((<*(lower_bound A)*> ^ D1) /. a2) is V22() real ext-real Element of REAL
(D1 ^ <*(upper_bound A)*>) . a2 is V22() real ext-real Element of REAL
Seg (len (D1 ^ <*(upper_bound A)*>)) is non empty V40() len (D1 ^ <*(upper_bound A)*>) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom (<*(lower_bound A)*> ^ D1) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(<*(lower_bound A)*> ^ D1) . a2 is V22() real ext-real Element of REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
f . a2 is V22() real ext-real Element of REAL
(A,f,a2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound (A,f,a2) is V22() real ext-real Element of REAL
sup (A,f,a2) is V22() real ext-real set
lower_bound (A,f,a2) is V22() real ext-real Element of REAL
inf (A,f,a2) is V22() real ext-real set
(upper_bound (A,f,a2)) - (lower_bound (A,f,a2)) is V22() real ext-real Element of REAL
2 - 1 is V22() real ext-real Element of REAL
<*(lower_bound A)*> . a2 is V22() real ext-real Element of REAL
D1 . a2 is V22() real ext-real Element of REAL
(A,f,a2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound (A,f,a2) is V22() real ext-real Element of REAL
sup (A,f,a2) is V22() real ext-real set
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
lower_bound (A,f,a2) is V22() real ext-real Element of REAL
inf (A,f,a2) is V22() real ext-real set
(upper_bound (A,f,a2)) - (lower_bound (A,f,a2)) is V22() real ext-real Element of REAL
a2 - (len D1) is V22() real ext-real Element of REAL
(a2 - (len D1)) + (len D1) is V22() real ext-real Element of REAL
<*(upper_bound A)*> . (a2 - (len D1)) is V22() real ext-real Element of REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,f,a2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound (A,f,a2) is V22() real ext-real Element of REAL
sup (A,f,a2) is V22() real ext-real set
lower_bound (A,f,a2) is V22() real ext-real Element of REAL
inf (A,f,a2) is V22() real ext-real set
(upper_bound (A,f,a2)) - (lower_bound (A,f,a2)) is V22() real ext-real Element of REAL
a2 - 1 is V22() real ext-real Element of REAL
f . (a2 - 1) is V22() real ext-real Element of REAL
(upper_bound (A,f,a2)) - (f . (a2 - 1)) is V22() real ext-real Element of REAL
f . a2 is V22() real ext-real Element of REAL
(f . a2) - (f . (a2 - 1)) is V22() real ext-real Element of REAL
a2 - (len <*(lower_bound A)*>) is V22() real ext-real Element of REAL
(len <*(lower_bound A)*>) + (a2 - (len <*(lower_bound A)*>)) is V22() real ext-real Element of REAL
D1 . (a2 - (len <*(lower_bound A)*>)) is V22() real ext-real Element of REAL
(len D1) + (len <*(upper_bound A)*>) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
a2 - 1 is V22() real ext-real Element of REAL
(a2 - 1) + 1 is V22() real ext-real Element of REAL
a2 - (len <*(lower_bound A)*>) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len D1) is V40() len D1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
a1 - 1 is V22() real ext-real Element of REAL
(a2 - 1) + 0 is V22() real ext-real Element of REAL
(a1 - 1) + 1 is V22() real ext-real Element of REAL
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + D2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len D1) is V40() len D1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len D1) is V40() len D1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(len <*(lower_bound A)*>) + (len D1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len <*(lower_bound A)*>) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D1 . (a2 - (len <*(lower_bound A)*>)) is V22() real ext-real Element of REAL
D1 . (a2 - 1) is V22() real ext-real Element of REAL
(A,f,a2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,f,a2) is V22() real ext-real Element of REAL
inf (A,f,a2) is V22() real ext-real set
D1 . a2 is V22() real ext-real Element of REAL
upper_bound (A,f,a2) is V22() real ext-real Element of REAL
sup (A,f,a2) is V22() real ext-real set
Sum g is V22() real ext-real Element of REAL
K173(REAL,g,K295()) is V22() real ext-real Element of REAL
Sum (D1 ^ <*(upper_bound A)*>) is V22() real ext-real Element of REAL
K173(REAL,(D1 ^ <*(upper_bound A)*>),K295()) is V22() real ext-real Element of REAL
Sum (<*(lower_bound A)*> ^ D1) is V22() real ext-real Element of REAL
K173(REAL,(<*(lower_bound A)*> ^ D1),K295()) is V22() real ext-real Element of REAL
(Sum (D1 ^ <*(upper_bound A)*>)) - (Sum (<*(lower_bound A)*> ^ D1)) is V22() real ext-real Element of REAL
Sum D1 is V22() real ext-real Element of REAL
K173(REAL,D1,K295()) is V22() real ext-real Element of REAL
(Sum D1) + (upper_bound A) is V22() real ext-real Element of REAL
((Sum D1) + (upper_bound A)) - (Sum (<*(lower_bound A)*> ^ D1)) is V22() real ext-real Element of REAL
Sum D1 is V22() real ext-real Element of REAL
K173(REAL,D1,K295()) is V22() real ext-real Element of REAL
(lower_bound A) + (Sum D1) is V22() real ext-real Element of REAL
((Sum D1) + (upper_bound A)) - ((lower_bound A) + (Sum D1)) is V22() real ext-real Element of REAL
a2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,f,a2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound (A,f,a2) is V22() real ext-real Element of REAL
sup (A,f,a2) is V22() real ext-real set
a2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,(a2 + 1)) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,f,(a2 + 1)) is V22() real ext-real Element of REAL
inf (A,f,(a2 + 1)) is V22() real ext-real set
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
a1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(a2 + 1) - 1 is V22() real ext-real Element of REAL
f . a2 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f . a2 is V22() real ext-real Element of REAL
a2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
D1 . a2 is V22() real ext-real Element of REAL
D1 . a2 is V22() real ext-real Element of REAL
(A,f,a2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound (A,f,a2) is V22() real ext-real Element of REAL
sup (A,f,a2) is V22() real ext-real set
a2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,(a2 + 1)) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,f,(a2 + 1)) is V22() real ext-real Element of REAL
inf (A,f,(a2 + 1)) is V22() real ext-real set
len (A,(chi (A,A)),f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom (A,(chi (A,A)),f) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
a2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(A,(chi (A,A)),f) . a2 is V22() real ext-real Element of REAL
g . a2 is V22() real ext-real Element of REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,f,a2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((A,f,a2)) is V22() real ext-real Element of REAL
upper_bound (A,f,a2) is V22() real ext-real Element of REAL
sup (A,f,a2) is V22() real ext-real set
lower_bound (A,f,a2) is V22() real ext-real Element of REAL
inf (A,f,a2) is V22() real ext-real set
(upper_bound (A,f,a2)) - (lower_bound (A,f,a2)) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
chi (A,A) is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
(A) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,(chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(chi (A,A)),f) is V22() real ext-real Element of REAL
K173(REAL,(A,(chi (A,A)),f),K295()) is V22() real ext-real Element of REAL
(A,(chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (A,(chi (A,A)),f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(A,(chi (A,A)),f) . g is V22() real ext-real Element of REAL
(A,(chi (A,A)),f) . g 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,f,g) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((A,f,g)) is V22() real ext-real Element of REAL
upper_bound (A,f,g) is V22() real ext-real Element of REAL
sup (A,f,g) is V22() real ext-real set
lower_bound (A,f,g) is V22() real ext-real Element of REAL
inf (A,f,g) is V22() real ext-real set
(upper_bound (A,f,g)) - (lower_bound (A,f,g)) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (A,(chi (A,A)),f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,g) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (A,f,g) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (A,f,g) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
(A) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng g is non empty V64() V65() V66() Element of bool REAL
lower_bound (rng g) is V22() real ext-real Element of REAL
(lower_bound (rng g)) * (A) is V22() real ext-real Element of REAL
(A,g,f) is V22() real ext-real Element of REAL
(A,g,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,f) is V22() real ext-real Element of REAL
K173(REAL,(A,g,f),K295()) is V22() real ext-real Element of REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,f,a1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((A,f,a1)) is V22() real ext-real Element of REAL
upper_bound (A,f,a1) is V22() real ext-real Element of REAL
sup (A,f,a1) is V22() real ext-real set
lower_bound (A,f,a1) is V22() real ext-real Element of REAL
inf (A,f,a1) is V22() real ext-real set
(upper_bound (A,f,a1)) - (lower_bound (A,f,a1)) is V22() real ext-real Element of REAL
(lower_bound (rng g)) * ((A,f,a1)) is V22() real ext-real Element of REAL
g | (A,f,a1) is Relation-like A -defined (A,f,a1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (g | (A,f,a1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (g | (A,f,a1))) is V22() real ext-real Element of REAL
(lower_bound (rng (g | (A,f,a1)))) * ((A,f,a1)) is V22() real ext-real Element of REAL
dom g is non empty V64() V65() V66() Element of bool A
bool A is non empty set
dom (g | (A,f,a1)) is V64() V65() V66() Element of bool A
chi (A,A) is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,(chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,(chi (A,A)),f) . a1 is V22() real ext-real Element of REAL
(lower_bound (rng g)) * ((A,(chi (A,A)),f) . a1) is V22() real ext-real Element of REAL
(A,f,a1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g | (A,f,a1) is Relation-like A -defined (A,f,a1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (g | (A,f,a1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (g | (A,f,a1))) is V22() real ext-real Element of REAL
((A,f,a1)) is V22() real ext-real Element of REAL
upper_bound (A,f,a1) is V22() real ext-real Element of REAL
sup (A,f,a1) is V22() real ext-real set
lower_bound (A,f,a1) is V22() real ext-real Element of REAL
inf (A,f,a1) is V22() real ext-real set
(upper_bound (A,f,a1)) - (lower_bound (A,f,a1)) is V22() real ext-real Element of REAL
(lower_bound (rng (g | (A,f,a1)))) * ((A,f,a1)) is V22() real ext-real Element of REAL
(lower_bound (rng g)) * ((A,f,a1)) is V22() real ext-real Element of REAL
(lower_bound (rng g)) * (A,(chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K527((lower_bound (rng g))) is Relation-like REAL -defined REAL -valued Function-like non empty total V30( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K297() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like non empty total V30([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
id REAL is Relation-like REAL -defined REAL -valued non empty total V30( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K180(K297(),(lower_bound (rng g)),(id REAL)) is set
(A,(chi (A,A)),f) * K527((lower_bound (rng g))) is Relation-like NAT -defined REAL -valued V40() complex-valued ext-real-valued real-valued set
Sum ((lower_bound (rng g)) * (A,(chi (A,A)),f)) is V22() real ext-real Element of REAL
K173(REAL,((lower_bound (rng g)) * (A,(chi (A,A)),f)),K295()) is V22() real ext-real Element of REAL
len (A,(chi (A,A)),f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((lower_bound (rng g)) * (A,(chi (A,A)),f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom a1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
a1 . D1 is V22() real ext-real Element of REAL
((lower_bound (rng g)) * (A,(chi (A,A)),f)) . D1 is V22() real ext-real Element of REAL
(A,(chi (A,A)),f) . D1 is V22() real ext-real Element of REAL
(lower_bound (rng g)) * ((A,(chi (A,A)),f) . D1) is V22() real ext-real Element of REAL
(len f) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom D1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
D1 is Relation-like NAT -defined REAL -valued Function-like V40() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
D1 . D2 is V22() real ext-real Element of REAL
(A,(chi (A,A)),f) . D2 is V22() real ext-real Element of REAL
(lower_bound (rng g)) * ((A,(chi (A,A)),f) . D2) is V22() real ext-real Element of REAL
a2 is Relation-like NAT -defined REAL -valued Function-like V40() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
a2 . D2 is V22() real ext-real Element of REAL
(A,f,D2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g | (A,f,D2) is Relation-like A -defined (A,f,D2) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (g | (A,f,D2)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (g | (A,f,D2))) is V22() real ext-real Element of REAL
((A,f,D2)) is V22() real ext-real Element of REAL
upper_bound (A,f,D2) is V22() real ext-real Element of REAL
sup (A,f,D2) is V22() real ext-real set
lower_bound (A,f,D2) is V22() real ext-real Element of REAL
inf (A,f,D2) is V22() real ext-real set
(upper_bound (A,f,D2)) - (lower_bound (A,f,D2)) is V22() real ext-real Element of REAL
(lower_bound (rng (g | (A,f,D2)))) * ((A,f,D2)) is V22() real ext-real Element of REAL
Sum (A,(chi (A,A)),f) is V22() real ext-real Element of REAL
K173(REAL,(A,(chi (A,A)),f),K295()) is V22() real ext-real Element of REAL
(lower_bound (rng g)) * (Sum (A,(chi (A,A)),f)) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(f,g,A) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((f,g,A)) is V22() real ext-real Element of REAL
upper_bound (f,g,A) is V22() real ext-real Element of REAL
sup (f,g,A) is V22() real ext-real set
lower_bound (f,g,A) is V22() real ext-real Element of REAL
inf (f,g,A) is V22() real ext-real set
(upper_bound (f,g,A)) - (lower_bound (f,g,A)) is V22() real ext-real Element of REAL
a1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
a1 | f is Relation-like f -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
a1 | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
rng (a1 | (f,g,A)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (a1 | (f,g,A))) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
rng a1 is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng a1) is V22() real ext-real Element of REAL
(upper_bound (rng a1)) * ((f,g,A)) is V22() real ext-real Element of REAL
dom a1 is non empty V64() V65() V66() Element of bool f
bool f is non empty set
dom (a1 | (f,g,A)) is V64() V65() V66() Element of bool f
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
(A) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,g,f) is V22() real ext-real Element of REAL
(A,g,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,f) is V22() real ext-real Element of REAL
K173(REAL,(A,g,f),K295()) is V22() real ext-real Element of REAL
rng g is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng g) is V22() real ext-real Element of REAL
(upper_bound (rng g)) * (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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,f,a1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g | (A,f,a1) is Relation-like A -defined (A,f,a1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (g | (A,f,a1)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (g | (A,f,a1))) is V22() real ext-real Element of REAL
((A,f,a1)) is V22() real ext-real Element of REAL
upper_bound (A,f,a1) is V22() real ext-real Element of REAL
sup (A,f,a1) is V22() real ext-real set
lower_bound (A,f,a1) is V22() real ext-real Element of REAL
inf (A,f,a1) is V22() real ext-real set
(upper_bound (A,f,a1)) - (lower_bound (A,f,a1)) is V22() real ext-real Element of REAL
(upper_bound (rng (g | (A,f,a1)))) * ((A,f,a1)) is V22() real ext-real Element of REAL
(upper_bound (rng g)) * ((A,f,a1)) is V22() real ext-real Element of REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom g is non empty V64() V65() V66() Element of bool A
bool A is non empty set
dom (g | (A,f,a1)) is V64() V65() V66() Element of bool A
chi (A,A) is Relation-like A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,(chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,f,a1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g | (A,f,a1) is Relation-like A -defined (A,f,a1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (g | (A,f,a1)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (g | (A,f,a1))) is V22() real ext-real Element of REAL
((A,f,a1)) is V22() real ext-real Element of REAL
upper_bound (A,f,a1) is V22() real ext-real Element of REAL
sup (A,f,a1) is V22() real ext-real set
lower_bound (A,f,a1) is V22() real ext-real Element of REAL
inf (A,f,a1) is V22() real ext-real set
(upper_bound (A,f,a1)) - (lower_bound (A,f,a1)) is V22() real ext-real Element of REAL
(upper_bound (rng (g | (A,f,a1)))) * ((A,f,a1)) is V22() real ext-real Element of REAL
(A,(chi (A,A)),f) . a1 is V22() real ext-real Element of REAL
(upper_bound (rng g)) * ((A,(chi (A,A)),f) . a1) is V22() real ext-real Element of REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(upper_bound (rng g)) * ((A,f,a1)) is V22() real ext-real Element of REAL
(upper_bound (rng g)) * (A,(chi (A,A)),f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K527((upper_bound (rng g))) is Relation-like REAL -defined REAL -valued Function-like non empty total V30( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K297() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like non empty total V30([:REAL,REAL:], REAL ) complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
id REAL is Relation-like REAL -defined REAL -valued non empty total V30( REAL , REAL ) complex-valued ext-real-valued real-valued Element of bool [:REAL,REAL:]
K180(K297(),(upper_bound (rng g)),(id REAL)) is set
(A,(chi (A,A)),f) * K527((upper_bound (rng g))) is Relation-like NAT -defined REAL -valued V40() complex-valued ext-real-valued real-valued set
Sum ((upper_bound (rng g)) * (A,(chi (A,A)),f)) is V22() real ext-real Element of REAL
K173(REAL,((upper_bound (rng g)) * (A,(chi (A,A)),f)),K295()) is V22() real ext-real Element of REAL
len (A,(chi (A,A)),f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((upper_bound (rng g)) * (A,(chi (A,A)),f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom a1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
a1 . D1 is V22() real ext-real Element of REAL
((upper_bound (rng g)) * (A,(chi (A,A)),f)) . D1 is V22() real ext-real Element of REAL
(A,(chi (A,A)),f) . D1 is V22() real ext-real Element of REAL
(upper_bound (rng g)) * ((A,(chi (A,A)),f) . D1) is V22() real ext-real Element of REAL
(len f) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom D1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
a2 is Relation-like NAT -defined REAL -valued Function-like V40() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
a2 . D2 is V22() real ext-real Element of REAL
(A,f,D2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g | (A,f,D2) is Relation-like A -defined (A,f,D2) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (g | (A,f,D2)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (g | (A,f,D2))) is V22() real ext-real Element of REAL
((A,f,D2)) is V22() real ext-real Element of REAL
upper_bound (A,f,D2) is V22() real ext-real Element of REAL
sup (A,f,D2) is V22() real ext-real set
lower_bound (A,f,D2) is V22() real ext-real Element of REAL
inf (A,f,D2) is V22() real ext-real set
(upper_bound (A,f,D2)) - (lower_bound (A,f,D2)) is V22() real ext-real Element of REAL
(upper_bound (rng (g | (A,f,D2)))) * ((A,f,D2)) is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like V40() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
D1 . D2 is V22() real ext-real Element of REAL
(A,(chi (A,A)),f) . D2 is V22() real ext-real Element of REAL
(upper_bound (rng g)) * ((A,(chi (A,A)),f) . D2) is V22() real ext-real Element of REAL
Sum (A,(chi (A,A)),f) is V22() real ext-real Element of REAL
K173(REAL,(A,(chi (A,A)),f),K295()) is V22() real ext-real Element of REAL
(upper_bound (rng g)) * (Sum (A,(chi (A,A)),f)) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,g,f) is V22() real ext-real Element of REAL
(A,g,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,f) is V22() real ext-real Element of REAL
K173(REAL,(A,g,f),K295()) is V22() real ext-real Element of REAL
(A,g,f) is V22() real ext-real Element of REAL
(A,g,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,f) is V22() real ext-real Element of REAL
K173(REAL,(A,g,f),K295()) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom a1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
rng g is non empty V64() V65() V66() Element of bool REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(len f) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom D1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
D1 is Relation-like NAT -defined REAL -valued Function-like V40() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
a2 is Relation-like NAT -defined REAL -valued Function-like V40() len f -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (len f) -tuples_on REAL
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
D1 . D2 is V22() real ext-real Element of REAL
a2 . D2 is V22() real ext-real Element of REAL
dom g is non empty V64() V65() V66() Element of bool A
bool A is non empty set
(A,f,D2) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g | (A,f,D2) is Relation-like A -defined (A,f,D2) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
dom (g | (A,f,D2)) is V64() V65() V66() Element of bool A
rng (g | (A,f,D2)) is V64() V65() V66() Element of bool REAL
((A,f,D2)) is V22() real ext-real Element of REAL
upper_bound (A,f,D2) is V22() real ext-real Element of REAL
sup (A,f,D2) is V22() real ext-real set
lower_bound (A,f,D2) is V22() real ext-real Element of REAL
inf (A,f,D2) is V22() real ext-real set
(upper_bound (A,f,D2)) - (lower_bound (A,f,D2)) is V22() real ext-real Element of REAL
lower_bound (rng (g | (A,f,D2))) is V22() real ext-real Element of REAL
(lower_bound (rng (g | (A,f,D2)))) * ((A,f,D2)) is V22() real ext-real Element of REAL
upper_bound (rng (g | (A,f,D2))) is V22() real ext-real Element of REAL
(upper_bound (rng (g | (A,f,D2)))) * ((A,f,D2)) is V22() real ext-real Element of REAL
Sum D1 is V22() real ext-real Element of REAL
K173(REAL,D1,K295()) is V22() real ext-real Element of REAL
Sum a2 is V22() real ext-real Element of REAL
K173(REAL,a2,K295()) is V22() real ext-real Element of REAL
g is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
len g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
rng a1 is V40() set
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g . (len g) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
f . 1 is V22() real ext-real Element of REAL
<*(upper_bound A)*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial V40() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
rng f is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
{(upper_bound A)} is non empty trivial V40() 1 -element V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
Seg (len g) is non empty V40() len g -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
rng f is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded set
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded set
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
bool (rng g) is non empty V40() V44() set
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,a1,g) is V22() real ext-real Element of REAL
(A,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,g) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,g),K295()) is V22() real ext-real Element of REAL
(A,a1,f) is V22() real ext-real Element of REAL
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,f) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,f),K295()) is V22() real ext-real Element of REAL
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,f,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,f,1) is V22() real ext-real Element of REAL
inf (A,f,1) is V22() real ext-real set
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
upper_bound (A,f,1) is V22() real ext-real Element of REAL
sup (A,f,1) is V22() real ext-real set
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
f . 1 is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
(A,a1,f) . 1 is V22() real ext-real Element of REAL
a1 | (A,f,1) is Relation-like A -defined (A,f,1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (a1 | (A,f,1)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (a1 | (A,f,1))) is V22() real ext-real Element of REAL
((A,f,1)) is V22() real ext-real Element of REAL
(upper_bound (A,f,1)) - (lower_bound (A,f,1)) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | (A,f,1)))) * ((A,f,1)) is V22() real ext-real Element of REAL
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
<*((upper_bound (rng (a1 | (A,f,1)))) * ((A,f,1)))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial V40() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
Sum <*((upper_bound (rng (a1 | (A,f,1)))) * ((A,f,1)))*> is V22() real ext-real Element of REAL
K173(REAL,<*((upper_bound (rng (a1 | (A,f,1)))) * ((A,f,1)))*>,K295()) is V22() real ext-real Element of REAL
rng (a1 | A) is V64() V65() V66() Element of bool REAL
upper_bound (rng (a1 | A)) is V22() real ext-real Element of REAL
(A) is V22() real ext-real Element of REAL
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | A))) * (A) is V22() real ext-real Element of REAL
rng a1 is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng a1) is V22() real ext-real Element of REAL
(upper_bound (rng a1)) * (A) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,a1,f) is V22() real ext-real Element of REAL
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,f) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,f),K295()) is V22() real ext-real Element of REAL
(A,a1,g) is V22() real ext-real Element of REAL
(A,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,g) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,g),K295()) is V22() real ext-real Element of REAL
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,f,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,f,1) is V22() real ext-real Element of REAL
inf (A,f,1) is V22() real ext-real set
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
upper_bound (A,f,1) is V22() real ext-real Element of REAL
sup (A,f,1) is V22() real ext-real set
f . 1 is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
[.(lower_bound A),(upper_bound A).] is V64() V65() V66() interval Element of bool REAL
(A,a1,f) . 1 is V22() real ext-real Element of REAL
a1 | (A,f,1) is Relation-like A -defined (A,f,1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (a1 | (A,f,1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (a1 | (A,f,1))) is V22() real ext-real Element of REAL
((A,f,1)) is V22() real ext-real Element of REAL
(upper_bound (A,f,1)) - (lower_bound (A,f,1)) is V22() real ext-real Element of REAL
(lower_bound (rng (a1 | (A,f,1)))) * ((A,f,1)) is V22() real ext-real Element of REAL
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
<*((lower_bound (rng (a1 | (A,f,1)))) * ((A,f,1)))*> is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial V40() 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing FinSequence of REAL
Sum <*((lower_bound (rng (a1 | (A,f,1)))) * ((A,f,1)))*> is V22() real ext-real Element of REAL
K173(REAL,<*((lower_bound (rng (a1 | (A,f,1)))) * ((A,f,1)))*>,K295()) is V22() real ext-real Element of REAL
rng (a1 | A) is V64() V65() V66() Element of bool REAL
lower_bound (rng (a1 | A)) is V22() real ext-real Element of REAL
(A) is V22() real ext-real Element of REAL
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
(lower_bound (rng (a1 | A))) * (A) is V22() real ext-real Element of REAL
rng a1 is non empty V64() V65() V66() Element of bool REAL
lower_bound (rng a1) is V22() real ext-real Element of REAL
(lower_bound (rng a1)) * (A) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound f is V22() real ext-real Element of REAL
inf f is V22() real ext-real set
upper_bound f is V22() real ext-real Element of REAL
sup f is V22() real ext-real set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g . A is V22() real ext-real Element of REAL
[.(lower_bound f),(g . A).] is V64() V65() V66() interval Element of bool REAL
[.(g . A),(upper_bound f).] is V64() V65() V66() interval Element of bool REAL
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
[.(lower_bound f),(upper_bound f).] is V64() V65() V66() interval Element of bool REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound f <= b₁ & b₁ <= upper_bound f ) } is set
D1 is V22() real ext-real Element of REAL
D1 is V22() real ext-real Element of REAL
a1 is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
D1 is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
a1 \/ D1 is V64() V65() V66() Element of bool REAL
D1 is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g . A is V22() real ext-real Element of REAL
a1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom a1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
rng a1 is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
D1 is set
a1 . D1 is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a1 . D1 is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
f . a1 is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g . D1 is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g . D1 is V22() real ext-real Element of REAL
A is Relation-like NAT -defined REAL -valued Function-like one-to-one V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
A /^ f is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom (A /^ f) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A /^ f) . g is V22() real ext-real Element of REAL
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A /^ f) . a1 is V22() real ext-real Element of REAL
a1 + f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g + f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len (A /^ f) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Seg (len (A /^ f)) is V40() len (A /^ f) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
1 + f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len A) - f is V22() real ext-real Element of REAL
Seg (len A) is V40() len A -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom A is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
A . (a1 + f) is V22() real ext-real Element of REAL
A . (g + f) is V22() real ext-real Element of REAL
A is Relation-like NAT -defined REAL -valued Function-like one-to-one V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
dom A is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
mid (A,f,g) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Seg (len A) is V40() len A -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
f -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
A /^ (f -' 1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
g -' f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(g -' f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg ((g -' f) + 1) is non empty V40() (g -' f) + 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A /^ (f -' 1)) | (Seg ((g -' f) + 1)) is Relation-like NAT -defined Seg ((g -' f) + 1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
(A /^ (f -' 1)) | ((g -' f) + 1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f - A is V22() real ext-real Element of REAL
(f - A) + 1 is V22() real ext-real Element of REAL
g is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
dom g is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
mid (g,A,f) is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
len (mid (g,A,f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Seg (len g) is V40() len g -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
A -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g /^ (A -' 1) is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
len (g /^ (A -' 1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(len g) - (A -' 1) is V22() real ext-real Element of REAL
f -' A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(f -' A) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f - (A -' 1) is V22() real ext-real Element of REAL
A - 1 is V22() real ext-real Element of REAL
f - (A - 1) is V22() real ext-real Element of REAL
(g /^ (A -' 1)) | ((f -' A) + 1) is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
a1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Seg D1 is V40() D1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
a1 | (Seg D1) is Relation-like Seg D1 -defined NAT -defined V40() FinSubsequence-like set
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
a1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (g)
dom a1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
mid (a1,A,f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(mid (a1,A,f)) . 1 is V22() real ext-real Element of REAL
len (mid (a1,A,f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(mid (a1,A,f)) . (len (mid (a1,A,f))) is V22() real ext-real Element of REAL
len a1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len a1) is non empty V40() len a1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
A -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a1 /^ (A -' 1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (a1 /^ (A -' 1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(len a1) - (A -' 1) is V22() real ext-real Element of REAL
f -' A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(f -' A) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f - (A -' 1) is V22() real ext-real Element of REAL
A - 1 is V22() real ext-real Element of REAL
f - (A - 1) is V22() real ext-real Element of REAL
f - A is V22() real ext-real Element of REAL
(f - A) + 1 is V22() real ext-real Element of REAL
(a1 /^ (A -' 1)) | ((f -' A) + 1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Seg D1 is V40() D1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D1 | (Seg D1) is Relation-like NAT -defined Seg D1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
a2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
len a2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len a2) is non empty V40() len a2 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom a2 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
a2 . 1 is V22() real ext-real Element of REAL
a2 . (len a2) is V22() real ext-real Element of REAL
[.(a2 . 1),(a2 . (len a2)).] is V64() V65() V66() interval Element of bool REAL
D2 is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound D2 is V22() real ext-real Element of REAL
inf D2 is V22() real ext-real set
upper_bound D2 is V22() real ext-real Element of REAL
sup D2 is V22() real ext-real set
[.(lower_bound D2),(upper_bound D2).] is V64() V65() V66() interval Element of bool REAL
rng a2 is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
D2 is V22() real ext-real Element of REAL
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a2 . D is V22() real ext-real Element of REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( a2 . 1 <= b₁ & b₁ <= a2 . (len a2) ) } is set
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
a1 is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound a1 is V22() real ext-real Element of REAL
inf a1 is V22() real ext-real set
upper_bound a1 is V22() real ext-real Element of REAL
sup a1 is V22() real ext-real set
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (g)
dom D1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
D1 . A is V22() real ext-real Element of REAL
D1 . f is V22() real ext-real Element of REAL
mid (D1,A,f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f - A is V22() real ext-real Element of REAL
(f - A) + 1 is V22() real ext-real Element of REAL
((f - A) + 1) + A is V22() real ext-real Element of REAL
(((f - A) + 1) + A) - 1 is V22() real ext-real Element of REAL
len D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len D1) is non empty V40() len D1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(mid (D1,A,f)) . 1 is V22() real ext-real Element of REAL
len (mid (D1,A,f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(mid (D1,A,f)) . (len (mid (D1,A,f))) is V22() real ext-real Element of REAL
D1 is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound D1 is V22() real ext-real Element of REAL
inf D1 is V22() real ext-real set
upper_bound D1 is V22() real ext-real Element of REAL
sup D1 is V22() real ext-real set
1 + A is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + A) - 1 is V22() real ext-real Element of REAL
a2 is V22() real ext-real Element of REAL
[.(lower_bound D1),(upper_bound D1).] is V64() V65() V66() interval Element of bool REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound D1 <= b₁ & b₁ <= upper_bound D1 ) } is set
D2 is V22() real ext-real Element of REAL
D2 is V22() real ext-real Element of REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound a1 <= b₁ & b₁ <= upper_bound a1 ) } is set
[.(lower_bound a1),(upper_bound a1).] is V64() V65() V66() interval Element of bool REAL
rng (mid (D1,A,f)) is V40() V64() V65() V66() bounded_below bounded_above real-bounded Element of bool REAL
A is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom A is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
f is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom f is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
f . g is V22() real ext-real Element of REAL
A | g is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A | g) is V22() real ext-real Element of REAL
K173(REAL,(A | g),K295()) is V22() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len g is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
f . a1 is V22() real ext-real Element of REAL
g . a1 is V22() real ext-real Element of REAL
A | a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A | a1) is V22() real ext-real Element of REAL
K173(REAL,(A | a1),K295()) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,f,g,1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (A,f,g,1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,g) | (A,f,g,1)) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,g) | (A,f,g,1)),K295()) is V22() real ext-real Element of REAL
(A,a1,f) | 1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,f) | 1) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,f) | 1),K295()) is V22() real ext-real Element of REAL
(A,f,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:(A,f,1),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A,f,1),REAL:] is non empty non trivial V40() set
a1 | (A,f,1) is Relation-like A -defined (A,f,1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
mid (f,1,1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
D1 is Relation-like (A,f,1) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,1),REAL:]
dom D1 is V64() V65() V66() Element of bool (A,f,1)
bool (A,f,1) is non empty set
dom a1 is non empty V64() V65() V66() Element of bool A
bool A is non empty set
(dom a1) /\ (A,f,1) is V64() V65() V66() Element of bool REAL
f . 1 is V22() real ext-real Element of REAL
upper_bound (A,f,1) is V22() real ext-real Element of REAL
sup (A,f,1) is V22() real ext-real set
g . (A,f,g,1) is V22() real ext-real Element of REAL
lower_bound (A,f,1) is V22() real ext-real Element of REAL
inf (A,f,1) is V22() real ext-real set
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len (mid (f,1,1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(1 -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing ((A,f,1))
((A,f,1),D1,D2) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((A,f,1),D1,D2) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len D2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f | 1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (f | 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(mid (f,1,1)) . D2 is V22() real ext-real Element of REAL
(f | 1) . D2 is V22() real ext-real Element of REAL
Seg (len (f | 1)) is V40() len (f | 1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (f | 1) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
f | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (f | (Seg 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(f | (Seg 1)) . D2 is V22() real ext-real Element of REAL
f . D2 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + 1) - 1 is V22() real ext-real Element of REAL
f . ((1 + 1) - 1) is V22() real ext-real Element of REAL
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
((A,f,1),D1,D2) . D2 is V22() real ext-real Element of REAL
((A,a1,f) | 1) . D2 is V22() real ext-real Element of REAL
f | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (f | (Seg 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(dom f) /\ (Seg 1) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (A,a1,f) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (A,a1,f)) is non empty V40() len (A,a1,f) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,f) | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
dom ((A,a1,f) | (Seg 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(Seg (len f)) /\ (Seg 1) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom D2 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,f) | (Seg 1)) . D2 is V22() real ext-real Element of REAL
((A,a1,f) | (Seg 1)) . 1 is V22() real ext-real Element of REAL
(A,a1,f) . 1 is V22() real ext-real Element of REAL
rng (a1 | (A,f,1)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (a1 | (A,f,1))) is V22() real ext-real Element of REAL
((A,f,1)) is V22() real ext-real Element of REAL
(upper_bound (A,f,1)) - (lower_bound (A,f,1)) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | (A,f,1)))) * ((A,f,1)) is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
[.(lower_bound A),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound A),(f . 1).] is V64() V65() V66() interval Element of bool REAL
((A,f,1),D1,D2) . 1 is V22() real ext-real Element of REAL
((A,f,1),D2,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
D1 | ((A,f,1),D2,1) is Relation-like (A,f,1) -defined ((A,f,1),D2,1) -defined (A,f,1) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,1),REAL:]
rng (D1 | ((A,f,1),D2,1)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (D1 | ((A,f,1),D2,1))) is V22() real ext-real Element of REAL
(((A,f,1),D2,1)) is V22() real ext-real Element of REAL
upper_bound ((A,f,1),D2,1) is V22() real ext-real Element of REAL
sup ((A,f,1),D2,1) is V22() real ext-real set
lower_bound ((A,f,1),D2,1) is V22() real ext-real Element of REAL
inf ((A,f,1),D2,1) is V22() real ext-real set
(upper_bound ((A,f,1),D2,1)) - (lower_bound ((A,f,1),D2,1)) is V22() real ext-real Element of REAL
(upper_bound (rng (D1 | ((A,f,1),D2,1)))) * (((A,f,1),D2,1)) is V22() real ext-real Element of REAL
[.(lower_bound ((A,f,1),D2,1)),(upper_bound ((A,f,1),D2,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,1)),(upper_bound ((A,f,1),D2,1)).] is V64() V65() V66() interval Element of bool REAL
D2 . 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),(D2 . 1).] is V64() V65() V66() interval Element of bool REAL
(f | 1) . 1 is V22() real ext-real Element of REAL
[.(lower_bound A),((f | 1) . 1).] is V64() V65() V66() interval Element of bool REAL
(f | (Seg 1)) . 1 is V22() real ext-real Element of REAL
[.(lower_bound A),((f | (Seg 1)) . 1).] is V64() V65() V66() interval Element of bool REAL
D1 | (A,f,1) is Relation-like (A,f,1) -defined (A,f,1) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,1),REAL:]
A /\ (dom a1) is V64() V65() V66() Element of bool A
D2 is V22() real ext-real set
(A,f,1) /\ (dom D1) is V64() V65() V66() Element of bool (A,f,1)
D is set
D1 . D is V22() real ext-real Element of REAL
n is V22() real ext-real Element of A
a1 . n is V22() real ext-real Element of REAL
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len g) is non empty V40() len g -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g . 1 is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
[.(lower_bound A),(upper_bound A).] is V64() V65() V66() interval Element of bool REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound A <= b₁ & b₁ <= upper_bound A ) } is set
D2 is V22() real ext-real Element of REAL
mid (g,1,(A,f,g,1)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (mid (g,1,(A,f,g,1))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,f,g,1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
((A,f,g,1) -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g,1) - 1 is V22() real ext-real Element of REAL
((A,f,g,1) - 1) + 1 is V22() real ext-real Element of REAL
g | (A,f,g,1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (g | (A,f,g,1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
D is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(mid (g,1,(A,f,g,1))) . D is V22() real ext-real Element of REAL
(g | (A,f,g,1)) . D is V22() real ext-real Element of REAL
Seg (len (g | (A,f,g,1))) is V40() len (g | (A,f,g,1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (g | (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (A,f,g,1) is V40() (A,f,g,1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (g | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(g | (Seg (A,f,g,1))) . D is V22() real ext-real Element of REAL
g . 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(D + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g . ((D + 1) -' 1) is V22() real ext-real Element of REAL
(D + 1) - 1 is V22() real ext-real Element of REAL
g . ((D + 1) - 1) is V22() real ext-real Element of REAL
D2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing ((A,f,1))
((A,f,1),D1,D2) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((A,f,1),D1,D2) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() 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 V40() cardinal set
((A,f,1),D1,D2) . D is V22() real ext-real Element of REAL
((A,a1,g) | (A,f,g,1)) . D is V22() real ext-real Element of REAL
len D2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len D2) is non empty V40() len D2 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom D2 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,f,1),D2,D) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
(A,g,D) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
Seg (A,f,g,1) is V40() (A,f,g,1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (g | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(dom g) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,g,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,g,1) is V22() real ext-real Element of REAL
inf (A,g,1) is V22() real ext-real set
upper_bound (A,g,1) is V22() real ext-real Element of REAL
sup (A,g,1) is V22() real ext-real set
[.(lower_bound (A,g,1)),(upper_bound (A,g,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound A),(upper_bound (A,g,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound A),(g . 1).] is V64() V65() V66() interval Element of bool REAL
((A,f,1),D2,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound ((A,f,1),D2,1) is V22() real ext-real Element of REAL
inf ((A,f,1),D2,1) is V22() real ext-real set
upper_bound ((A,f,1),D2,1) is V22() real ext-real Element of REAL
sup ((A,f,1),D2,1) is V22() real ext-real set
[.(lower_bound ((A,f,1),D2,1)),(upper_bound ((A,f,1),D2,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,1)),(upper_bound ((A,f,1),D2,1)).] is V64() V65() V66() interval Element of bool REAL
D2 . 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),(D2 . 1).] is V64() V65() V66() interval Element of bool REAL
(g | (A,f,g,1)) . 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),((g | (A,f,g,1)) . 1).] is V64() V65() V66() interval Element of bool REAL
(g | (Seg (A,f,g,1))) . 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),((g | (Seg (A,f,g,1))) . 1).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,1)),(g . 1).] is V64() V65() V66() interval Element of bool REAL
D - 1 is V22() real ext-real Element of REAL
g | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (g | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g,1) - 0 is V22() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + n is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + n is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(dom g) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D2 . (D - 1) is V22() real ext-real Element of REAL
(g | (A,f,g,1)) . (D - 1) is V22() real ext-real Element of REAL
(g | (Seg (A,f,g,1))) . (D - 1) is V22() real ext-real Element of REAL
g . (D - 1) is V22() real ext-real Element of REAL
(dom g) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
D2 . D is V22() real ext-real Element of REAL
(g | (A,f,g,1)) . D is V22() real ext-real Element of REAL
(g | (Seg (A,f,g,1))) . D is V22() real ext-real Element of REAL
g . D is V22() real ext-real Element of REAL
lower_bound (A,g,D) is V22() real ext-real Element of REAL
inf (A,g,D) is V22() real ext-real set
upper_bound (A,g,D) is V22() real ext-real Element of REAL
sup (A,g,D) is V22() real ext-real set
[.(lower_bound (A,g,D)),(upper_bound (A,g,D)).] is V64() V65() V66() interval Element of bool REAL
[.(g . (D - 1)),(upper_bound (A,g,D)).] is V64() V65() V66() interval Element of bool REAL
[.(g . (D - 1)),(g . D).] is V64() V65() V66() interval Element of bool REAL
lower_bound ((A,f,1),D2,D) is V22() real ext-real Element of REAL
inf ((A,f,1),D2,D) is V22() real ext-real set
upper_bound ((A,f,1),D2,D) is V22() real ext-real Element of REAL
sup ((A,f,1),D2,D) is V22() real ext-real set
[.(lower_bound ((A,f,1),D2,D)),(upper_bound ((A,f,1),D2,D)).] is V64() V65() V66() interval Element of bool REAL
[.(D2 . (D - 1)),(upper_bound ((A,f,1),D2,D)).] is V64() V65() V66() interval Element of bool REAL
D1 | (A,g,D) is Relation-like (A,f,1) -defined (A,g,D) -defined (A,f,1) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,1),REAL:]
rng (D1 | (A,g,D)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (D1 | (A,g,D))) is V22() real ext-real Element of REAL
((A,g,D)) is V22() real ext-real Element of REAL
upper_bound (A,g,D) is V22() real ext-real Element of REAL
sup (A,g,D) is V22() real ext-real set
lower_bound (A,g,D) is V22() real ext-real Element of REAL
inf (A,g,D) is V22() real ext-real set
(upper_bound (A,g,D)) - (lower_bound (A,g,D)) is V22() real ext-real Element of REAL
(upper_bound (rng (D1 | (A,g,D)))) * ((A,g,D)) is V22() real ext-real Element of REAL
Seg (A,f,g,1) is V40() (A,f,g,1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (g | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(dom g) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound (A,f,1) <= b₁ & b₁ <= upper_bound (A,f,1) ) } is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D - 1 is V22() real ext-real Element of REAL
g . (D - 1) is V22() real ext-real Element of REAL
(A,f,g,1) - 0 is V22() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + n is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound (A,f,1) <= b₁ & b₁ <= upper_bound (A,f,1) ) } is set
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
g . D is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,g,D)),(upper_bound (A,g,D)).] is V64() V65() V66() interval Element of bool REAL
(A,a1,g) | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,g) | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (A,a1,g) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(dom (A,a1,g)) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
len (A,a1,g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (A,a1,g)) is non empty V40() len (A,a1,g) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(Seg (len (A,a1,g))) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(Seg (len g)) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,g) | (Seg (A,f,g,1))) . D is V22() real ext-real Element of REAL
(A,a1,g) . D is V22() real ext-real Element of REAL
a1 | (A,g,D) is Relation-like A -defined (A,g,D) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (a1 | (A,g,D)) is V64() V65() V66() Element of bool REAL
upper_bound (rng (a1 | (A,g,D))) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | (A,g,D)))) * ((A,g,D)) is V22() real ext-real Element of REAL
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((A,a1,f) | 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len (A,a1,g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((A,a1,g) | (A,f,g,1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
A /\ (A,f,1) is V64() V65() V66() interval Element of bool REAL
((A,f,1),D1,D2) is V22() real ext-real Element of REAL
Sum ((A,f,1),D1,D2) is V22() real ext-real Element of REAL
K173(REAL,((A,f,1),D1,D2),K295()) is V22() real ext-real Element of REAL
((A,f,1),D1,D2) is V22() real ext-real Element of REAL
Sum ((A,f,1),D1,D2) is V22() real ext-real Element of REAL
K173(REAL,((A,f,1),D1,D2),K295()) is V22() real ext-real Element of REAL
D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal set
(A,f,g,D1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (A,f,g,D1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,g) | (A,f,g,D1)) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,g) | (A,f,g,D1)),K295()) is V22() real ext-real Element of REAL
(A,a1,f) | D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,f) | D1) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,f) | D1),K295()) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g,(D1 + 1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (A,f,g,(D1 + 1)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,g) | (A,f,g,(D1 + 1))) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,g) | (A,f,g,(D1 + 1))),K295()) is V22() real ext-real Element of REAL
(A,a1,f) | (D1 + 1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,f) | (D1 + 1)) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,f) | (D1 + 1)),K295()) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((A,a1,f) | (D1 + 1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
p1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len p1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
q1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len q1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
p1 ^ q1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((A,a1,f) | D1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
ID is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
p1 . ID is V22() real ext-real Element of REAL
((A,a1,f) | D1) . ID is V22() real ext-real Element of REAL
Seg (len p1) is V40() len p1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,f) | D1) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg D1 is non empty V40() D1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,f) | (Seg D1) is Relation-like NAT -defined Seg D1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,f) | (Seg D1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (D1 + 1) is non empty V40() D1 + 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,f) | (D1 + 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,f) | (D1 + 1))) is V40() len ((A,a1,f) | (D1 + 1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,f) | D1)) is V40() len ((A,a1,f) | D1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,f) | (Seg (D1 + 1)) is Relation-like NAT -defined Seg (D1 + 1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
dom ((A,a1,f) | (Seg (D1 + 1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,f) | (D1 + 1)) . ID is V22() real ext-real Element of REAL
((A,a1,f) | (Seg (D1 + 1))) . ID is V22() real ext-real Element of REAL
(A,a1,f) . ID is V22() real ext-real Element of REAL
((A,a1,f) | (Seg D1)) . ID is V22() real ext-real Element of REAL
dom p1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len g) is non empty V40() len g -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + 1) - 1 is V22() real ext-real Element of REAL
f . (D1 + 1) is V22() real ext-real Element of REAL
f . D1 is V22() real ext-real Element of REAL
g . (A,f,g,(D1 + 1)) is V22() real ext-real Element of REAL
g . (A,f,g,D1) is V22() real ext-real Element of REAL
g . (A,f,g,D1) is V22() real ext-real Element of REAL
g . (A,f,g,(D1 + 1)) is V22() real ext-real Element of REAL
f . (D1 + 1) is V22() real ext-real Element of REAL
f . D1 is V22() real ext-real Element of REAL
ID is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(A,f,g,D1) + ID is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len (A,a1,g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((A,a1,g) | (A,f,g,(D1 + 1))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,f,g,(D1 + 1)) - (A,f,g,D1) is V22() real ext-real Element of REAL
(A,f,g,D1) + ((A,f,g,(D1 + 1)) - (A,f,g,D1)) is V22() real ext-real Element of REAL
p2 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len p2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
q2 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len q2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
p2 ^ q2 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (len (A,a1,g)) is non empty V40() len (A,a1,g) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
Sum p2 is V22() real ext-real Element of REAL
K173(REAL,p2,K295()) is V22() real ext-real Element of REAL
Sum q2 is V22() real ext-real Element of REAL
K173(REAL,q2,K295()) is V22() real ext-real Element of REAL
(Sum p2) + (Sum q2) is V22() real ext-real Element of REAL
len ((A,a1,g) | (A,f,g,D1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
p2 . i is V22() real ext-real Element of REAL
((A,a1,g) | (A,f,g,D1)) . i is V22() real ext-real Element of REAL
Seg (len p2) is V40() len p2 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,g) | (A,f,g,D1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (A,f,g,D1) is V40() (A,f,g,D1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,g) | (Seg (A,f,g,D1)) is Relation-like NAT -defined Seg (A,f,g,D1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,g) | (Seg (A,f,g,D1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,g) | (A,f,g,(D1 + 1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,g) | (A,f,g,(D1 + 1)))) is V40() len ((A,a1,g) | (A,f,g,(D1 + 1))) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (A,f,g,(D1 + 1)) is V40() (A,f,g,(D1 + 1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,g) | (A,f,g,D1))) is V40() len ((A,a1,g) | (A,f,g,D1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,g) | (Seg (A,f,g,(D1 + 1))) is Relation-like NAT -defined Seg (A,f,g,(D1 + 1)) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
dom ((A,a1,g) | (Seg (A,f,g,(D1 + 1)))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,g) | (A,f,g,(D1 + 1))) . i is V22() real ext-real Element of REAL
((A,a1,g) | (Seg (A,f,g,(D1 + 1)))) . i is V22() real ext-real Element of REAL
(A,a1,g) . i is V22() real ext-real Element of REAL
((A,a1,g) | (Seg (A,f,g,D1))) . i is V22() real ext-real Element of REAL
dom p2 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
ID is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Sum q1 is V22() real ext-real Element of REAL
K173(REAL,q1,K295()) is V22() real ext-real Element of REAL
(A,f,g,D1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1))) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mid (f,(D1 + 1),(D1 + 1)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,f,(D1 + 1)) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
a1 | (A,f,(D1 + 1)) is Relation-like A -defined (A,f,(D1 + 1)) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
[:(A,f,(D1 + 1)),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A,f,(D1 + 1)),REAL:] is non empty non trivial V40() set
(D1 + 1) - 1 is V22() real ext-real Element of REAL
lower_bound (A,f,(D1 + 1)) is V22() real ext-real Element of REAL
inf (A,f,(D1 + 1)) is V22() real ext-real set
f . D1 is V22() real ext-real Element of REAL
g . (A,f,g,(D1 + 1)) is V22() real ext-real Element of REAL
f . (D1 + 1) is V22() real ext-real Element of REAL
upper_bound (A,f,(D1 + 1)) is V22() real ext-real Element of REAL
sup (A,f,(D1 + 1)) is V22() real ext-real set
g . (A,f,g,D1) is V22() real ext-real Element of REAL
g . ((A,f,g,D1) + 1) is V22() real ext-real Element of REAL
(A,f,g,(D1 + 1)) -' ((A,f,g,D1) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
((A,f,g,(D1 + 1)) -' ((A,f,g,D1) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g,(D1 + 1)) - ((A,f,g,D1) + 1) is V22() real ext-real Element of REAL
((A,f,g,(D1 + 1)) - ((A,f,g,D1) + 1)) + 1 is V22() real ext-real Element of REAL
MD2 is Relation-like (A,f,(D1 + 1)) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,(D1 + 1)),REAL:]
MD1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing ((A,f,(D1 + 1)))
((A,f,(D1 + 1)),MD2,MD1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom ((A,a1,g) | (A,f,g,(D1 + 1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,g) | (A,f,g,(D1 + 1)))) is V40() len ((A,a1,g) | (A,f,g,(D1 + 1))) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (A,f,g,(D1 + 1)) is V40() (A,f,g,(D1 + 1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
len (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
q2 . n is V22() real ext-real Element of REAL
((A,f,(D1 + 1)),MD2,MD1) . n is V22() real ext-real Element of REAL
Seg (len q2) is V40() len q2 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom q2 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,f,g,D1) + n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (Seg (A,f,g,(D1 + 1))) is Relation-like NAT -defined Seg (A,f,g,(D1 + 1)) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,g) | (Seg (A,f,g,(D1 + 1)))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,g) | (A,f,g,(D1 + 1))) . ((A,f,g,D1) + n) is V22() real ext-real Element of REAL
((A,a1,g) | (Seg (A,f,g,(D1 + 1)))) . ((A,f,g,D1) + n) is V22() real ext-real Element of REAL
(A,a1,g) . ((A,f,g,D1) + n) is V22() real ext-real Element of REAL
(A,g,((A,f,g,D1) + n)) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
a1 | (A,g,((A,f,g,D1) + n)) is Relation-like A -defined (A,g,((A,f,g,D1) + n)) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (a1 | (A,g,((A,f,g,D1) + n))) is V64() V65() V66() Element of bool REAL
upper_bound (rng (a1 | (A,g,((A,f,g,D1) + n)))) is V22() real ext-real Element of REAL
((A,g,((A,f,g,D1) + n))) is V22() real ext-real Element of REAL
upper_bound (A,g,((A,f,g,D1) + n)) is V22() real ext-real Element of REAL
sup (A,g,((A,f,g,D1) + n)) is V22() real ext-real set
lower_bound (A,g,((A,f,g,D1) + n)) is V22() real ext-real Element of REAL
inf (A,g,((A,f,g,D1) + n)) is V22() real ext-real set
(upper_bound (A,g,((A,f,g,D1) + n))) - (lower_bound (A,g,((A,f,g,D1) + n))) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | (A,g,((A,f,g,D1) + n))))) * ((A,g,((A,f,g,D1) + n))) is V22() real ext-real Element of REAL
Seg ID is V40() ID -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom MD1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
Seg (len (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1))))) is V40() len (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,f,(D1 + 1)),MD1,n) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((A,f,(D1 + 1)),MD1,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound ((A,f,(D1 + 1)),MD1,1) is V22() real ext-real Element of REAL
sup ((A,f,(D1 + 1)),MD1,1) is V22() real ext-real set
(mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) . 1 is V22() real ext-real Element of REAL
1 + (A,f,g,D1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g . (1 + (A,f,g,D1)) is V22() real ext-real Element of REAL
lower_bound ((A,f,(D1 + 1)),MD1,1) is V22() real ext-real Element of REAL
inf ((A,f,(D1 + 1)),MD1,1) is V22() real ext-real set
(A,g,((A,f,g,D1) + 1)) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,g,((A,f,g,D1) + 1)) is V22() real ext-real Element of REAL
inf (A,g,((A,f,g,D1) + 1)) is V22() real ext-real set
upper_bound (A,g,((A,f,g,D1) + 1)) is V22() real ext-real Element of REAL
sup (A,g,((A,f,g,D1) + 1)) is V22() real ext-real set
[.(lower_bound (A,g,((A,f,g,D1) + 1))),(upper_bound (A,g,((A,f,g,D1) + 1))).] is V64() V65() V66() interval Element of bool REAL
((A,f,g,D1) + 1) - 1 is V22() real ext-real Element of REAL
g . (((A,f,g,D1) + 1) - 1) is V22() real ext-real Element of REAL
[.(g . (((A,f,g,D1) + 1) - 1)),(upper_bound (A,g,((A,f,g,D1) + 1))).] is V64() V65() V66() interval Element of bool REAL
[.(g . (A,f,g,D1)),(g . ((A,f,g,D1) + 1)).] is V64() V65() V66() interval Element of bool REAL
[.(f . D1),(g . ((A,f,g,D1) + 1)).] is V64() V65() V66() interval Element of bool REAL
1 - n is V22() real ext-real Element of REAL
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 - 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,g,((A,f,g,D1) + n))),(upper_bound (A,g,((A,f,g,D1) + n))).] is V64() V65() V66() interval Element of bool REAL
((A,f,g,D1) + n) - 1 is V22() real ext-real Element of REAL
g . (((A,f,g,D1) + n) - 1) is V22() real ext-real Element of REAL
[.(g . (((A,f,g,D1) + n) - 1)),(upper_bound (A,g,((A,f,g,D1) + n))).] is V64() V65() V66() interval Element of bool REAL
g . ((A,f,g,D1) + n) is V22() real ext-real Element of REAL
[.(g . (((A,f,g,D1) + n) - 1)),(g . ((A,f,g,D1) + n)).] is V64() V65() V66() interval Element of bool REAL
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
n - 1 is V22() real ext-real Element of REAL
len MD1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + x is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
x + ((A,f,g,D1) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(x + ((A,f,g,D1) + 1)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
n + ((A,f,g,D1) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(n + ((A,f,g,D1) + 1)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
n + (A,f,g,D1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(n + (A,f,g,D1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((n + (A,f,g,D1)) + 1) - 1 is V22() real ext-real Element of REAL
lower_bound ((A,f,(D1 + 1)),MD1,n) is V22() real ext-real Element of REAL
inf ((A,f,(D1 + 1)),MD1,n) is V22() real ext-real set
MD1 . (n - 1) is V22() real ext-real Element of REAL
upper_bound ((A,f,(D1 + 1)),MD1,n) is V22() real ext-real Element of REAL
sup ((A,f,(D1 + 1)),MD1,n) is V22() real ext-real set
MD1 . n is V22() real ext-real Element of REAL
MD2 | ((A,f,(D1 + 1)),MD1,n) is Relation-like (A,f,(D1 + 1)) -defined ((A,f,(D1 + 1)),MD1,n) -defined (A,f,(D1 + 1)) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,(D1 + 1)),REAL:]
MD2 | (A,f,(D1 + 1)) is Relation-like (A,f,(D1 + 1)) -defined (A,f,(D1 + 1)) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,(D1 + 1)),REAL:]
dom a1 is non empty V64() V65() V66() Element of bool A
bool A is non empty set
A /\ (dom a1) is V64() V65() V66() Element of bool A
x is V22() real ext-real set
dom MD2 is V64() V65() V66() Element of bool (A,f,(D1 + 1))
bool (A,f,(D1 + 1)) is non empty set
(A,f,(D1 + 1)) /\ (dom MD2) is V64() V65() V66() Element of bool (A,f,(D1 + 1))
x is set
MD2 . x is V22() real ext-real Element of REAL
x is V22() real ext-real Element of A
a1 . x is V22() real ext-real Element of REAL
n is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing ((A,f,(D1 + 1)))
len n is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(D1 + 1) -' (D1 + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
((D1 + 1) -' (D1 + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(D1 + 1) - (D1 + 1) is V22() real ext-real Element of REAL
((D1 + 1) - (D1 + 1)) + 1 is V22() real ext-real Element of REAL
dom q1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom n is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,f,(D1 + 1)),MD2,n) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (len ((A,a1,f) | (D1 + 1))) is V40() len ((A,a1,f) | (D1 + 1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,f) | (D1 + 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (D1 + 1) is non empty V40() D1 + 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,f) | (Seg (D1 + 1)) is Relation-like NAT -defined Seg (D1 + 1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,f) | (Seg (D1 + 1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
n . 1 is V22() real ext-real Element of REAL
Seg (len n) is non empty V40() len n -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,f,(D1 + 1)),n,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound ((A,f,(D1 + 1)),n,1) is V22() real ext-real Element of REAL
inf ((A,f,(D1 + 1)),n,1) is V22() real ext-real set
upper_bound ((A,f,(D1 + 1)),n,1) is V22() real ext-real Element of REAL
sup ((A,f,(D1 + 1)),n,1) is V22() real ext-real set
[.(lower_bound ((A,f,(D1 + 1)),n,1)),(upper_bound ((A,f,(D1 + 1)),n,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,(D1 + 1))),(upper_bound ((A,f,(D1 + 1)),n,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,(D1 + 1))),(f . (D1 + 1)).] is V64() V65() V66() interval Element of bool REAL
x is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
q1 . x is V22() real ext-real Element of REAL
((A,f,(D1 + 1)),MD2,n) . x is V22() real ext-real Element of REAL
Seg (len q1) is V40() len q1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
[.(f . D1),(f . (D1 + 1)).] is V64() V65() V66() interval Element of bool REAL
rng (MD2 | (A,f,(D1 + 1))) is V64() V65() V66() Element of bool REAL
upper_bound (rng (MD2 | (A,f,(D1 + 1)))) is V22() real ext-real Element of REAL
((A,f,(D1 + 1))) is V22() real ext-real Element of REAL
(upper_bound (A,f,(D1 + 1))) - (lower_bound (A,f,(D1 + 1))) is V22() real ext-real Element of REAL
(upper_bound (rng (MD2 | (A,f,(D1 + 1))))) * ((A,f,(D1 + 1))) is V22() real ext-real Element of REAL
((A,a1,f) | (D1 + 1)) . (D1 + 1) is V22() real ext-real Element of REAL
((A,a1,f) | (Seg (D1 + 1))) . (D1 + 1) is V22() real ext-real Element of REAL
(A,a1,f) . (D1 + 1) is V22() real ext-real Element of REAL
rng (a1 | (A,f,(D1 + 1))) is V64() V65() V66() Element of bool REAL
upper_bound (rng (a1 | (A,f,(D1 + 1)))) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | (A,f,(D1 + 1))))) * ((A,f,(D1 + 1))) is V22() real ext-real Element of REAL
len ((A,f,(D1 + 1)),MD2,n) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
dom MD2 is V64() V65() V66() Element of bool (A,f,(D1 + 1))
bool (A,f,(D1 + 1)) is non empty set
dom a1 is non empty V64() V65() V66() Element of bool A
bool A is non empty set
(dom a1) /\ (A,f,(D1 + 1)) is V64() V65() V66() Element of bool REAL
A /\ (A,f,(D1 + 1)) is V64() V65() V66() interval Element of bool REAL
((A,f,(D1 + 1)),MD2,MD1) is V22() real ext-real Element of REAL
Sum ((A,f,(D1 + 1)),MD2,MD1) is V22() real ext-real Element of REAL
K173(REAL,((A,f,(D1 + 1)),MD2,MD1),K295()) is V22() real ext-real Element of REAL
((A,f,(D1 + 1)),MD2,n) is V22() real ext-real Element of REAL
Sum ((A,f,(D1 + 1)),MD2,n) is V22() real ext-real Element of REAL
K173(REAL,((A,f,(D1 + 1)),MD2,n),K295()) is V22() real ext-real Element of REAL
len ((A,f,(D1 + 1)),MD2,MD1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Sum p1 is V22() real ext-real Element of REAL
K173(REAL,p1,K295()) is V22() real ext-real Element of REAL
(Sum p1) + (Sum q1) is V22() real ext-real Element of REAL
(Sum ((A,a1,f) | (D1 + 1))) - (Sum p1) is V22() real ext-real Element of REAL
(Sum q2) + (Sum p1) is V22() real ext-real Element of REAL
(Sum ((A,a1,f) | (D1 + 1))) - (Sum q2) is V22() real ext-real Element of REAL
D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g,D1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (A,f,g,D1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,g) | (A,f,g,D1)) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,g) | (A,f,g,D1)),K295()) is V22() real ext-real Element of REAL
(A,a1,f) | D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,f) | D1) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,f) | D1),K295()) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,a1,f) | 1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,f) | 1) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,f) | 1),K295()) is V22() real ext-real Element of REAL
(A,f,g,1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (A,f,g,1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,g) | (A,f,g,1)) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,g) | (A,f,g,1)),K295()) is V22() real ext-real Element of REAL
(A,f,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
a1 | (A,f,1) is Relation-like A -defined (A,f,1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
mid (g,1,(A,f,g,1)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mid (f,1,1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
[:(A,f,1),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A,f,1),REAL:] is non empty non trivial V40() set
D2 is Relation-like (A,f,1) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,1),REAL:]
dom D2 is V64() V65() V66() Element of bool (A,f,1)
bool (A,f,1) is non empty set
dom a1 is non empty V64() V65() V66() Element of bool A
bool A is non empty set
(dom a1) /\ (A,f,1) is V64() V65() V66() Element of bool REAL
f . 1 is V22() real ext-real Element of REAL
upper_bound (A,f,1) is V22() real ext-real Element of REAL
sup (A,f,1) is V22() real ext-real set
g . (A,f,g,1) is V22() real ext-real Element of REAL
lower_bound (A,f,1) is V22() real ext-real Element of REAL
inf (A,f,1) is V22() real ext-real set
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len (mid (f,1,1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(1 -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing ((A,f,1))
((A,f,1),D2,D) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((A,f,1),D2,D) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len D is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
f | 1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (f | 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(mid (f,1,1)) . n is V22() real ext-real Element of REAL
(f | 1) . n is V22() real ext-real Element of REAL
Seg (len (f | 1)) is V40() len (f | 1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (f | 1) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
f | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (f | (Seg 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(f | (Seg 1)) . n is V22() real ext-real Element of REAL
f . n 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + 1) - 1 is V22() real ext-real Element of REAL
f . ((1 + 1) - 1) is V22() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
((A,f,1),D2,D) . n is V22() real ext-real Element of REAL
((A,a1,f) | 1) . n is V22() real ext-real Element of REAL
f | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (f | (Seg 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(dom f) /\ (Seg 1) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (A,a1,f) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (A,a1,f)) is non empty V40() len (A,a1,f) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,f) | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
dom ((A,a1,f) | (Seg 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(Seg (len f)) /\ (Seg 1) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom D is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,f) | (Seg 1)) . n is V22() real ext-real Element of REAL
((A,a1,f) | (Seg 1)) . 1 is V22() real ext-real Element of REAL
(A,a1,f) . 1 is V22() real ext-real Element of REAL
rng (a1 | (A,f,1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (a1 | (A,f,1))) is V22() real ext-real Element of REAL
((A,f,1)) is V22() real ext-real Element of REAL
(upper_bound (A,f,1)) - (lower_bound (A,f,1)) is V22() real ext-real Element of REAL
(lower_bound (rng (a1 | (A,f,1)))) * ((A,f,1)) is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
[.(lower_bound A),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound A),(f . 1).] is V64() V65() V66() interval Element of bool REAL
((A,f,1),D2,D) . 1 is V22() real ext-real Element of REAL
((A,f,1),D,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
D2 | ((A,f,1),D,1) is Relation-like (A,f,1) -defined ((A,f,1),D,1) -defined (A,f,1) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,1),REAL:]
rng (D2 | ((A,f,1),D,1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (D2 | ((A,f,1),D,1))) is V22() real ext-real Element of REAL
(((A,f,1),D,1)) is V22() real ext-real Element of REAL
upper_bound ((A,f,1),D,1) is V22() real ext-real Element of REAL
sup ((A,f,1),D,1) is V22() real ext-real set
lower_bound ((A,f,1),D,1) is V22() real ext-real Element of REAL
inf ((A,f,1),D,1) is V22() real ext-real set
(upper_bound ((A,f,1),D,1)) - (lower_bound ((A,f,1),D,1)) is V22() real ext-real Element of REAL
(lower_bound (rng (D2 | ((A,f,1),D,1)))) * (((A,f,1),D,1)) is V22() real ext-real Element of REAL
[.(lower_bound ((A,f,1),D,1)),(upper_bound ((A,f,1),D,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,1)),(upper_bound ((A,f,1),D,1)).] is V64() V65() V66() interval Element of bool REAL
D . 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),(D . 1).] is V64() V65() V66() interval Element of bool REAL
(f | 1) . 1 is V22() real ext-real Element of REAL
[.(lower_bound A),((f | 1) . 1).] is V64() V65() V66() interval Element of bool REAL
(f | (Seg 1)) . 1 is V22() real ext-real Element of REAL
[.(lower_bound A),((f | (Seg 1)) . 1).] is V64() V65() V66() interval Element of bool REAL
D2 | (A,f,1) is Relation-like (A,f,1) -defined (A,f,1) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,1),REAL:]
A /\ (dom a1) is V64() V65() V66() Element of bool A
n is V22() real ext-real set
(A,f,1) /\ (dom D2) is V64() V65() V66() Element of bool (A,f,1)
p1 is set
D2 . p1 is V22() real ext-real Element of REAL
q1 is V22() real ext-real Element of A
a1 . q1 is V22() real ext-real Element of REAL
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len g) is non empty V40() len g -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g . 1 is V22() real ext-real Element of REAL
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
[.(lower_bound A),(upper_bound A).] is V64() V65() V66() interval Element of bool REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound A <= b₁ & b₁ <= upper_bound A ) } is set
n is V22() real ext-real Element of REAL
len (mid (g,1,(A,f,g,1))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,f,g,1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
((A,f,g,1) -' 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g,1) - 1 is V22() real ext-real Element of REAL
((A,f,g,1) - 1) + 1 is V22() real ext-real Element of REAL
g | (A,f,g,1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (g | (A,f,g,1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
p1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(mid (g,1,(A,f,g,1))) . p1 is V22() real ext-real Element of REAL
(g | (A,f,g,1)) . p1 is V22() real ext-real Element of REAL
Seg (len (g | (A,f,g,1))) is V40() len (g | (A,f,g,1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (g | (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (A,f,g,1) is V40() (A,f,g,1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (g | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(g | (Seg (A,f,g,1))) . p1 is V22() real ext-real Element of REAL
g . p1 is V22() real ext-real Element of REAL
p1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(p1 + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g . ((p1 + 1) -' 1) is V22() real ext-real Element of REAL
(p1 + 1) - 1 is V22() real ext-real Element of REAL
g . ((p1 + 1) - 1) is V22() real ext-real Element of REAL
n is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing ((A,f,1))
((A,f,1),D2,n) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((A,f,1),D2,n) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
p1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
((A,f,1),D2,n) . p1 is V22() real ext-real Element of REAL
((A,a1,g) | (A,f,g,1)) . p1 is V22() real ext-real Element of REAL
len n is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len n) is non empty V40() len n -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom n is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,f,1),n,p1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
(A,g,p1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
Seg (A,f,g,1) is V40() (A,f,g,1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (g | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(dom g) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,g,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,g,1) is V22() real ext-real Element of REAL
inf (A,g,1) is V22() real ext-real set
upper_bound (A,g,1) is V22() real ext-real Element of REAL
sup (A,g,1) is V22() real ext-real set
[.(lower_bound (A,g,1)),(upper_bound (A,g,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound A),(upper_bound (A,g,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound A),(g . 1).] is V64() V65() V66() interval Element of bool REAL
((A,f,1),n,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound ((A,f,1),n,1) is V22() real ext-real Element of REAL
inf ((A,f,1),n,1) is V22() real ext-real set
upper_bound ((A,f,1),n,1) is V22() real ext-real Element of REAL
sup ((A,f,1),n,1) is V22() real ext-real set
[.(lower_bound ((A,f,1),n,1)),(upper_bound ((A,f,1),n,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,1)),(upper_bound ((A,f,1),n,1)).] is V64() V65() V66() interval Element of bool REAL
n . 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),(n . 1).] is V64() V65() V66() interval Element of bool REAL
(g | (A,f,g,1)) . 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),((g | (A,f,g,1)) . 1).] is V64() V65() V66() interval Element of bool REAL
(g | (Seg (A,f,g,1))) . 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),((g | (Seg (A,f,g,1))) . 1).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,1)),(g . 1).] is V64() V65() V66() interval Element of bool REAL
p1 - 1 is V22() real ext-real Element of REAL
g | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (g | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g,1) - 0 is V22() real ext-real Element of REAL
q1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + q1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
q1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + q1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(dom g) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
n . (p1 - 1) is V22() real ext-real Element of REAL
(g | (A,f,g,1)) . (p1 - 1) is V22() real ext-real Element of REAL
(g | (Seg (A,f,g,1))) . (p1 - 1) is V22() real ext-real Element of REAL
g . (p1 - 1) is V22() real ext-real Element of REAL
(dom g) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
n . p1 is V22() real ext-real Element of REAL
(g | (A,f,g,1)) . p1 is V22() real ext-real Element of REAL
(g | (Seg (A,f,g,1))) . p1 is V22() real ext-real Element of REAL
g . p1 is V22() real ext-real Element of REAL
lower_bound (A,g,p1) is V22() real ext-real Element of REAL
inf (A,g,p1) is V22() real ext-real set
upper_bound (A,g,p1) is V22() real ext-real Element of REAL
sup (A,g,p1) is V22() real ext-real set
[.(lower_bound (A,g,p1)),(upper_bound (A,g,p1)).] is V64() V65() V66() interval Element of bool REAL
[.(g . (p1 - 1)),(upper_bound (A,g,p1)).] is V64() V65() V66() interval Element of bool REAL
[.(g . (p1 - 1)),(g . p1).] is V64() V65() V66() interval Element of bool REAL
lower_bound ((A,f,1),n,p1) is V22() real ext-real Element of REAL
inf ((A,f,1),n,p1) is V22() real ext-real set
upper_bound ((A,f,1),n,p1) is V22() real ext-real Element of REAL
sup ((A,f,1),n,p1) is V22() real ext-real set
[.(lower_bound ((A,f,1),n,p1)),(upper_bound ((A,f,1),n,p1)).] is V64() V65() V66() interval Element of bool REAL
[.(n . (p1 - 1)),(upper_bound ((A,f,1),n,p1)).] is V64() V65() V66() interval Element of bool REAL
D2 | (A,g,p1) is Relation-like (A,f,1) -defined (A,g,p1) -defined (A,f,1) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,1),REAL:]
rng (D2 | (A,g,p1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (D2 | (A,g,p1))) is V22() real ext-real Element of REAL
((A,g,p1)) is V22() real ext-real Element of REAL
upper_bound (A,g,p1) is V22() real ext-real Element of REAL
sup (A,g,p1) is V22() real ext-real set
lower_bound (A,g,p1) is V22() real ext-real Element of REAL
inf (A,g,p1) is V22() real ext-real set
(upper_bound (A,g,p1)) - (lower_bound (A,g,p1)) is V22() real ext-real Element of REAL
(lower_bound (rng (D2 | (A,g,p1)))) * ((A,g,p1)) is V22() real ext-real Element of REAL
Seg (A,f,g,1) is V40() (A,f,g,1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
g | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom (g | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(dom g) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound (A,f,1) <= b₁ & b₁ <= upper_bound (A,f,1) ) } is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
p1 - 1 is V22() real ext-real Element of REAL
g . (p1 - 1) is V22() real ext-real Element of REAL
(A,f,g,1) - 0 is V22() real ext-real Element of REAL
q1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + q1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound (A,f,1) <= b₁ & b₁ <= upper_bound (A,f,1) ) } is set
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
g . p1 is V22() real ext-real Element of REAL
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,f,1)),(upper_bound (A,f,1)).] is V64() V65() V66() interval Element of bool REAL
[.(lower_bound (A,g,p1)),(upper_bound (A,g,p1)).] is V64() V65() V66() interval Element of bool REAL
(A,a1,g) | (Seg (A,f,g,1)) is Relation-like NAT -defined Seg (A,f,g,1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,g) | (Seg (A,f,g,1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom (A,a1,g) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(dom (A,a1,g)) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
len (A,a1,g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (A,a1,g)) is non empty V40() len (A,a1,g) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(Seg (len (A,a1,g))) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(Seg (len g)) /\ (Seg (A,f,g,1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,g) | (Seg (A,f,g,1))) . p1 is V22() real ext-real Element of REAL
(A,a1,g) . p1 is V22() real ext-real Element of REAL
a1 | (A,g,p1) is Relation-like A -defined (A,g,p1) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (a1 | (A,g,p1)) is V64() V65() V66() Element of bool REAL
lower_bound (rng (a1 | (A,g,p1))) is V22() real ext-real Element of REAL
(lower_bound (rng (a1 | (A,g,p1)))) * ((A,g,p1)) is V22() real ext-real Element of REAL
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((A,a1,f) | 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len (A,a1,g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((A,a1,g) | (A,f,g,1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
A /\ (A,f,1) is V64() V65() V66() interval Element of bool REAL
((A,f,1),D2,D) is V22() real ext-real Element of REAL
Sum ((A,f,1),D2,D) is V22() real ext-real Element of REAL
K173(REAL,((A,f,1),D2,D),K295()) is V22() real ext-real Element of REAL
((A,f,1),D2,n) is V22() real ext-real Element of REAL
Sum ((A,f,1),D2,n) is V22() real ext-real Element of REAL
K173(REAL,((A,f,1),D2,n),K295()) is V22() real ext-real Element of REAL
D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal set
(A,a1,f) | D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,f) | D1) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,f) | D1),K295()) is V22() real ext-real Element of REAL
(A,f,g,D1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (A,f,g,D1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,g) | (A,f,g,D1)) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,g) | (A,f,g,D1)),K295()) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,a1,f) | (D1 + 1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,f) | (D1 + 1)) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,f) | (D1 + 1)),K295()) is V22() real ext-real Element of REAL
(A,f,g,(D1 + 1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (A,f,g,(D1 + 1)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,g) | (A,f,g,(D1 + 1))) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,g) | (A,f,g,(D1 + 1))),K295()) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((A,a1,f) | (D1 + 1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
p1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len p1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
q1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len q1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
p1 ^ q1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((A,a1,f) | D1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
ID is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
p1 . ID is V22() real ext-real Element of REAL
((A,a1,f) | D1) . ID is V22() real ext-real Element of REAL
Seg (len p1) is V40() len p1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,f) | D1) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg D1 is non empty V40() D1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,f) | (Seg D1) is Relation-like NAT -defined Seg D1 -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,f) | (Seg D1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (D1 + 1) is non empty V40() D1 + 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,f) | (D1 + 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,f) | (D1 + 1))) is V40() len ((A,a1,f) | (D1 + 1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,f) | D1)) is V40() len ((A,a1,f) | D1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,f) | (Seg (D1 + 1)) is Relation-like NAT -defined Seg (D1 + 1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
dom ((A,a1,f) | (Seg (D1 + 1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,f) | (D1 + 1)) . ID is V22() real ext-real Element of REAL
((A,a1,f) | (Seg (D1 + 1))) . ID is V22() real ext-real Element of REAL
(A,a1,f) . ID is V22() real ext-real Element of REAL
((A,a1,f) | (Seg D1)) . ID is V22() real ext-real Element of REAL
dom p1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len g) is non empty V40() len g -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(1 + 1) - 1 is V22() real ext-real Element of REAL
f . (D1 + 1) is V22() real ext-real Element of REAL
f . D1 is V22() real ext-real Element of REAL
g . (A,f,g,(D1 + 1)) is V22() real ext-real Element of REAL
g . (A,f,g,D1) is V22() real ext-real Element of REAL
g . (A,f,g,D1) is V22() real ext-real Element of REAL
g . (A,f,g,(D1 + 1)) is V22() real ext-real Element of REAL
f . (D1 + 1) is V22() real ext-real Element of REAL
f . D1 is V22() real ext-real Element of REAL
ID is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(A,f,g,D1) + ID is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len (A,a1,g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len ((A,a1,g) | (A,f,g,(D1 + 1))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,f,g,(D1 + 1)) - (A,f,g,D1) is V22() real ext-real Element of REAL
(A,f,g,D1) + ((A,f,g,(D1 + 1)) - (A,f,g,D1)) is V22() real ext-real Element of REAL
p2 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len p2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
q2 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len q2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
p2 ^ q2 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum q1 is V22() real ext-real Element of REAL
K173(REAL,q1,K295()) is V22() real ext-real Element of REAL
Sum q2 is V22() real ext-real Element of REAL
K173(REAL,q2,K295()) is V22() real ext-real Element of REAL
(A,f,g,D1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1))) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
mid (f,(D1 + 1),(D1 + 1)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,f,(D1 + 1)) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
a1 | (A,f,(D1 + 1)) is Relation-like A -defined (A,f,(D1 + 1)) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
[:(A,f,(D1 + 1)),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A,f,(D1 + 1)),REAL:] is non empty non trivial V40() set
(D1 + 1) - 1 is V22() real ext-real Element of REAL
lower_bound (A,f,(D1 + 1)) is V22() real ext-real Element of REAL
inf (A,f,(D1 + 1)) is V22() real ext-real set
f . D1 is V22() real ext-real Element of REAL
g is Relation-like (A,f,(D1 + 1)) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,(D1 + 1)),REAL:]
dom g is V64() V65() V66() Element of bool (A,f,(D1 + 1))
bool (A,f,(D1 + 1)) is non empty set
dom a1 is non empty V64() V65() V66() Element of bool A
bool A is non empty set
(dom a1) /\ (A,f,(D1 + 1)) is V64() V65() V66() Element of bool REAL
A /\ (A,f,(D1 + 1)) is V64() V65() V66() interval Element of bool REAL
upper_bound (A,f,(D1 + 1)) is V22() real ext-real Element of REAL
sup (A,f,(D1 + 1)) is V22() real ext-real set
f . (D1 + 1) is V22() real ext-real Element of REAL
g . (A,f,g,(D1 + 1)) is V22() real ext-real Element of REAL
g . (A,f,g,D1) is V22() real ext-real Element of REAL
g . ((A,f,g,D1) + 1) is V22() real ext-real Element of REAL
(A,f,g,(D1 + 1)) -' ((A,f,g,D1) + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
((A,f,g,(D1 + 1)) -' ((A,f,g,D1) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,f,g,(D1 + 1)) - ((A,f,g,D1) + 1) is V22() real ext-real Element of REAL
((A,f,g,(D1 + 1)) - ((A,f,g,D1) + 1)) + 1 is V22() real ext-real Element of REAL
MD2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing ((A,f,(D1 + 1)))
((A,f,(D1 + 1)),g,MD2) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
MD1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
q2 . MD1 is V22() real ext-real Element of REAL
((A,f,(D1 + 1)),g,MD2) . MD1 is V22() real ext-real Element of REAL
Seg (len q2) is V40() len q2 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom q2 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,f,g,D1) + MD1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
dom ((A,a1,g) | (A,f,g,(D1 + 1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (A,f,g,(D1 + 1)) is V40() (A,f,g,(D1 + 1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,g) | (Seg (A,f,g,(D1 + 1))) is Relation-like NAT -defined Seg (A,f,g,(D1 + 1)) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,g) | (Seg (A,f,g,(D1 + 1)))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
len (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
ID is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Seg (len ((A,a1,g) | (A,f,g,(D1 + 1)))) is V40() len ((A,a1,g) | (A,f,g,(D1 + 1))) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1))))) is V40() len (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom MD2 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,g) | (A,f,g,(D1 + 1))) . ((A,f,g,D1) + MD1) is V22() real ext-real Element of REAL
((A,a1,g) | (Seg (A,f,g,(D1 + 1)))) . ((A,f,g,D1) + MD1) is V22() real ext-real Element of REAL
(A,a1,g) . ((A,f,g,D1) + MD1) is V22() real ext-real Element of REAL
(A,g,((A,f,g,D1) + MD1)) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
a1 | (A,g,((A,f,g,D1) + MD1)) is Relation-like A -defined (A,g,((A,f,g,D1) + MD1)) -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
rng (a1 | (A,g,((A,f,g,D1) + MD1))) is V64() V65() V66() Element of bool REAL
lower_bound (rng (a1 | (A,g,((A,f,g,D1) + MD1)))) is V22() real ext-real Element of REAL
((A,g,((A,f,g,D1) + MD1))) is V22() real ext-real Element of REAL
upper_bound (A,g,((A,f,g,D1) + MD1)) is V22() real ext-real Element of REAL
sup (A,g,((A,f,g,D1) + MD1)) is V22() real ext-real set
lower_bound (A,g,((A,f,g,D1) + MD1)) is V22() real ext-real Element of REAL
inf (A,g,((A,f,g,D1) + MD1)) is V22() real ext-real set
(upper_bound (A,g,((A,f,g,D1) + MD1))) - (lower_bound (A,g,((A,f,g,D1) + MD1))) is V22() real ext-real Element of REAL
(lower_bound (rng (a1 | (A,g,((A,f,g,D1) + MD1))))) * ((A,g,((A,f,g,D1) + MD1))) is V22() real ext-real Element of REAL
((A,f,(D1 + 1)),MD2,MD1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((A,f,(D1 + 1)),MD2,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound ((A,f,(D1 + 1)),MD2,1) is V22() real ext-real Element of REAL
sup ((A,f,(D1 + 1)),MD2,1) is V22() real ext-real set
(mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) . 1 is V22() real ext-real Element of REAL
1 + (A,f,g,D1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g . (1 + (A,f,g,D1)) is V22() real ext-real Element of REAL
lower_bound ((A,f,(D1 + 1)),MD2,1) is V22() real ext-real Element of REAL
inf ((A,f,(D1 + 1)),MD2,1) is V22() real ext-real set
(A,g,((A,f,g,D1) + 1)) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
lower_bound (A,g,((A,f,g,D1) + 1)) is V22() real ext-real Element of REAL
inf (A,g,((A,f,g,D1) + 1)) is V22() real ext-real set
upper_bound (A,g,((A,f,g,D1) + 1)) is V22() real ext-real Element of REAL
sup (A,g,((A,f,g,D1) + 1)) is V22() real ext-real set
[.(lower_bound (A,g,((A,f,g,D1) + 1))),(upper_bound (A,g,((A,f,g,D1) + 1))).] is V64() V65() V66() interval Element of bool REAL
((A,f,g,D1) + 1) - 1 is V22() real ext-real Element of REAL
g . (((A,f,g,D1) + 1) - 1) is V22() real ext-real Element of REAL
[.(g . (((A,f,g,D1) + 1) - 1)),(upper_bound (A,g,((A,f,g,D1) + 1))).] is V64() V65() V66() interval Element of bool REAL
[.(g . (A,f,g,D1)),(g . ((A,f,g,D1) + 1)).] is V64() V65() V66() interval Element of bool REAL
[.(f . D1),(g . ((A,f,g,D1) + 1)).] is V64() V65() V66() interval Element of bool REAL
1 - MD1 is V22() real ext-real Element of REAL
MD1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
1 - 1 is V22() real ext-real Element of REAL
[.(lower_bound (A,g,((A,f,g,D1) + MD1))),(upper_bound (A,g,((A,f,g,D1) + MD1))).] is V64() V65() V66() interval Element of bool REAL
((A,f,g,D1) + MD1) - 1 is V22() real ext-real Element of REAL
g . (((A,f,g,D1) + MD1) - 1) is V22() real ext-real Element of REAL
[.(g . (((A,f,g,D1) + MD1) - 1)),(upper_bound (A,g,((A,f,g,D1) + MD1))).] is V64() V65() V66() interval Element of bool REAL
g . ((A,f,g,D1) + MD1) is V22() real ext-real Element of REAL
[.(g . (((A,f,g,D1) + MD1) - 1)),(g . ((A,f,g,D1) + MD1)).] is V64() V65() V66() interval Element of bool REAL
MD1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
MD1 - 1 is V22() real ext-real Element of REAL
len MD2 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
1 + n is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
x + ((A,f,g,D1) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(x + ((A,f,g,D1) + 1)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
MD1 + ((A,f,g,D1) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(MD1 + ((A,f,g,D1) + 1)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
MD1 + (A,f,g,D1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(MD1 + (A,f,g,D1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((MD1 + (A,f,g,D1)) + 1) - 1 is V22() real ext-real Element of REAL
lower_bound ((A,f,(D1 + 1)),MD2,MD1) is V22() real ext-real Element of REAL
inf ((A,f,(D1 + 1)),MD2,MD1) is V22() real ext-real set
MD2 . (MD1 - 1) is V22() real ext-real Element of REAL
upper_bound ((A,f,(D1 + 1)),MD2,MD1) is V22() real ext-real Element of REAL
sup ((A,f,(D1 + 1)),MD2,MD1) is V22() real ext-real set
MD2 . MD1 is V22() real ext-real Element of REAL
g | ((A,f,(D1 + 1)),MD2,MD1) is Relation-like (A,f,(D1 + 1)) -defined ((A,f,(D1 + 1)),MD2,MD1) -defined (A,f,(D1 + 1)) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,(D1 + 1)),REAL:]
g | (A,f,(D1 + 1)) is Relation-like (A,f,(D1 + 1)) -defined (A,f,(D1 + 1)) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A,f,(D1 + 1)),REAL:]
A /\ (dom a1) is V64() V65() V66() Element of bool A
n is V22() real ext-real set
(A,f,(D1 + 1)) /\ (dom g) is V64() V65() V66() Element of bool (A,f,(D1 + 1))
x is set
g . x is V22() real ext-real Element of REAL
x is V22() real ext-real Element of A
a1 . x is V22() real ext-real Element of REAL
MD1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing ((A,f,(D1 + 1)))
len MD1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(D1 + 1) -' (D1 + 1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
((D1 + 1) -' (D1 + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(D1 + 1) - (D1 + 1) is V22() real ext-real Element of REAL
((D1 + 1) - (D1 + 1)) + 1 is V22() real ext-real Element of REAL
((A,f,(D1 + 1)),g,MD1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (len ((A,a1,f) | (D1 + 1))) is V40() len ((A,a1,f) | (D1 + 1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,f) | (D1 + 1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (D1 + 1) is non empty V40() D1 + 1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,f) | (Seg (D1 + 1)) is Relation-like NAT -defined Seg (D1 + 1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,f) | (Seg (D1 + 1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,f) | (D1 + 1)) . (D1 + 1) is V22() real ext-real Element of REAL
((A,a1,f) | (Seg (D1 + 1))) . (D1 + 1) is V22() real ext-real Element of REAL
(A,a1,f) . (D1 + 1) is V22() real ext-real Element of REAL
MD1 . 1 is V22() real ext-real Element of REAL
dom MD1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,f,(D1 + 1)),MD1,1) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
upper_bound ((A,f,(D1 + 1)),MD1,1) is V22() real ext-real Element of REAL
sup ((A,f,(D1 + 1)),MD1,1) is V22() real ext-real set
n is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
q1 . n is V22() real ext-real Element of REAL
((A,f,(D1 + 1)),g,MD1) . n is V22() real ext-real Element of REAL
lower_bound ((A,f,(D1 + 1)),MD1,1) is V22() real ext-real Element of REAL
inf ((A,f,(D1 + 1)),MD1,1) is V22() real ext-real set
[.(lower_bound (A,f,(D1 + 1))),(f . (D1 + 1)).] is V64() V65() V66() interval Element of bool REAL
[.(f . D1),(f . (D1 + 1)).] is V64() V65() V66() interval Element of bool REAL
Seg (len q1) is V40() len q1 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
rng (g | (A,f,(D1 + 1))) is V64() V65() V66() Element of bool REAL
lower_bound (rng (g | (A,f,(D1 + 1)))) is V22() real ext-real Element of REAL
((A,f,(D1 + 1))) is V22() real ext-real Element of REAL
(upper_bound (A,f,(D1 + 1))) - (lower_bound (A,f,(D1 + 1))) is V22() real ext-real Element of REAL
(lower_bound (rng (g | (A,f,(D1 + 1))))) * ((A,f,(D1 + 1))) is V22() real ext-real Element of REAL
dom q1 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
rng (a1 | (A,f,(D1 + 1))) is V64() V65() V66() Element of bool REAL
lower_bound (rng (a1 | (A,f,(D1 + 1)))) is V22() real ext-real Element of REAL
(lower_bound (rng (a1 | (A,f,(D1 + 1))))) * ((A,f,(D1 + 1))) is V22() real ext-real Element of REAL
len ((A,f,(D1 + 1)),g,MD1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((A,f,(D1 + 1)),g,MD1) is V22() real ext-real Element of REAL
Sum ((A,f,(D1 + 1)),g,MD1) is V22() real ext-real Element of REAL
K173(REAL,((A,f,(D1 + 1)),g,MD1),K295()) is V22() real ext-real Element of REAL
((A,f,(D1 + 1)),g,MD2) is V22() real ext-real Element of REAL
Sum ((A,f,(D1 + 1)),g,MD2) is V22() real ext-real Element of REAL
K173(REAL,((A,f,(D1 + 1)),g,MD2),K295()) is V22() real ext-real Element of REAL
len ((A,f,(D1 + 1)),g,MD2) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len (mid (g,((A,f,g,D1) + 1),(A,f,g,(D1 + 1)))) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
Sum p2 is V22() real ext-real Element of REAL
K173(REAL,p2,K295()) is V22() real ext-real Element of REAL
(Sum p2) + (Sum q2) is V22() real ext-real Element of REAL
len ((A,a1,g) | (A,f,g,D1)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
p2 . i is V22() real ext-real Element of REAL
((A,a1,g) | (A,f,g,D1)) . i is V22() real ext-real Element of REAL
Seg (len p2) is V40() len p2 -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,g) | (A,f,g,D1)) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (A,f,g,D1) is V40() (A,f,g,D1) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,g) | (Seg (A,f,g,D1)) is Relation-like NAT -defined Seg (A,f,g,D1) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
dom ((A,a1,g) | (Seg (A,f,g,D1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
dom ((A,a1,g) | (A,f,g,(D1 + 1))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,g) | (A,f,g,(D1 + 1)))) is V40() len ((A,a1,g) | (A,f,g,(D1 + 1))) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (A,f,g,(D1 + 1)) is V40() (A,f,g,(D1 + 1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Seg (len ((A,a1,g) | (A,f,g,D1))) is V40() len ((A,a1,g) | (A,f,g,D1)) -element V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
(A,a1,g) | (Seg (A,f,g,(D1 + 1))) is Relation-like NAT -defined Seg (A,f,g,(D1 + 1)) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
dom ((A,a1,g) | (Seg (A,f,g,(D1 + 1)))) is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,g) | (A,f,g,(D1 + 1))) . i is V22() real ext-real Element of REAL
((A,a1,g) | (Seg (A,f,g,(D1 + 1)))) . i is V22() real ext-real Element of REAL
(A,a1,g) . i is V22() real ext-real Element of REAL
((A,a1,g) | (Seg (A,f,g,D1))) . i is V22() real ext-real Element of REAL
dom p2 is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
Sum p1 is V22() real ext-real Element of REAL
K173(REAL,p1,K295()) is V22() real ext-real Element of REAL
(Sum p1) + (Sum q1) is V22() real ext-real Element of REAL
(Sum ((A,a1,f) | (D1 + 1))) - (Sum p1) is V22() real ext-real Element of REAL
(Sum q2) + (Sum p1) is V22() real ext-real Element of REAL
(Sum ((A,a1,f) | (D1 + 1))) - (Sum q2) is V22() real ext-real Element of REAL
D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(A,a1,f) | D1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,f) | D1) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,f) | D1),K295()) is V22() real ext-real Element of REAL
(A,f,g,D1) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
(A,a1,g) | (A,f,g,D1) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,a1,g) | (A,f,g,D1)) is V22() real ext-real Element of REAL
K173(REAL,((A,a1,g) | (A,f,g,D1)),K295()) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
a1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
(f,g,a1,A) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
D1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
D1 | f is Relation-like f -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(f,D1,a1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((f,D1,a1)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((f,D1,a1)) . (f,g,a1,A) is V22() real ext-real Element of REAL
(f,D1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((f,D1,g)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((f,D1,g)) . A is V22() real ext-real Element of REAL
len (f,D1,a1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len a1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len g) is non empty V40() len g -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len (f,D1,g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (f,D1,g)) is non empty V40() len (f,D1,g) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom (f,D1,g) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(f,D1,g) | A is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((f,D1,g) | A) is V22() real ext-real Element of REAL
K173(REAL,((f,D1,g) | A),K295()) is V22() real ext-real Element of REAL
dom a1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
Seg (len (f,D1,a1)) is non empty V40() len (f,D1,a1) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom (f,D1,a1) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(f,D1,a1) | (f,g,a1,A) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((f,D1,a1) | (f,g,a1,A)) is V22() real ext-real Element of REAL
K173(REAL,((f,D1,a1) | (f,g,a1,A)),K295()) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
a1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
(f,g,a1,A) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
D1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
D1 | f is Relation-like f -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(f,D1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((f,D1,g)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((f,D1,g)) . A is V22() real ext-real Element of REAL
(f,D1,a1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((f,D1,a1)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((f,D1,a1)) . (f,g,a1,A) is V22() real ext-real Element of REAL
len (f,D1,a1) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len a1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len g) is non empty V40() len g -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
len (f,D1,g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (f,D1,g)) is non empty V40() len (f,D1,g) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom (f,D1,g) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(f,D1,g) | A is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((f,D1,g) | A) is V22() real ext-real Element of REAL
K173(REAL,((f,D1,g) | A),K295()) is V22() real ext-real Element of REAL
dom a1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
Seg (len (f,D1,a1)) is non empty V40() len (f,D1,a1) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom (f,D1,a1) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(f,D1,a1) | (f,g,a1,A) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((f,D1,a1) | (f,g,a1,A)) is V22() real ext-real Element of REAL
K173(REAL,((f,D1,a1) | (f,g,a1,A)),K295()) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,g,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A,g,f)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A,g,f)) . (len f) is V22() real ext-real Element of REAL
(A,g,f) is V22() real ext-real Element of REAL
Sum (A,g,f) is V22() real ext-real Element of REAL
K173(REAL,(A,g,f),K295()) is V22() real ext-real Element of REAL
len (A,g,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (A,g,f)) is non empty V40() len (A,g,f) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom (A,g,f) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,g,f) | (len f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,g,f) | (len f)) is V22() real ext-real Element of REAL
K173(REAL,((A,g,f) | (len f)),K295()) is V22() real ext-real Element of REAL
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,g,f) | (Seg (len f)) is Relation-like NAT -defined Seg (len f) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,g,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A,g,f)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A,g,f)) . (len f) is V22() real ext-real Element of REAL
(A,g,f) is V22() real ext-real Element of REAL
Sum (A,g,f) is V22() real ext-real Element of REAL
K173(REAL,(A,g,f),K295()) is V22() real ext-real Element of REAL
len (A,g,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len (A,g,f)) is non empty V40() len (A,g,f) -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom (A,g,f) is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,g,f) | (len f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum ((A,g,f) | (len f)) is V22() real ext-real Element of REAL
K173(REAL,((A,g,f) | (len f)),K295()) is V22() real ext-real Element of REAL
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,g,f) | (Seg (len f)) is Relation-like NAT -defined Seg (len f) -defined NAT -defined REAL -valued Function-like V40() FinSubsequence-like complex-valued ext-real-valued real-valued Element of bool [:NAT,REAL:]
[:NAT,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V40() set
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len f is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,g,(len f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
f . (len f) is V22() real ext-real Element of REAL
g . (A,f,g,(len f)) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
Seg (len g) is non empty V40() len g -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
g . (len g) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,a1,g) is V22() real ext-real Element of REAL
(A,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,g) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,g),K295()) is V22() real ext-real Element of REAL
(A,a1,f) is V22() real ext-real Element of REAL
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,f) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,f),K295()) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,g)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,f,g,(len f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
((A,a1,g)) . (A,f,g,(len f)) is V22() real ext-real Element of REAL
((A,a1,f)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A,a1,f)) . (len f) is V22() real ext-real Element of REAL
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((A,a1,g)) . (len g) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,a1,f) is V22() real ext-real Element of REAL
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,f) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,f),K295()) is V22() real ext-real Element of REAL
(A,a1,g) is V22() real ext-real Element of REAL
(A,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,g) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,g),K295()) 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
((A,a1,f)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A,a1,f)) . (len f) is V22() real ext-real Element of REAL
((A,a1,g)) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A,f,g,(len f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
((A,a1,g)) . (A,f,g,(len f)) is V22() real ext-real Element of REAL
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
((A,a1,g)) . (len g) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
f ^ g is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng (f ^ g) is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
card (rng (f ^ g)) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of omega
a1 is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
rng a1 is V40() V64() V65() V66() bounded_below bounded_above real-bounded Element of bool REAL
len a1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
rng g is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
rng f is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
(rng f) \/ (rng g) is V40() V64() V65() V66() bounded_below bounded_above real-bounded Element of bool REAL
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing FinSequence of REAL
rng D1 is non empty V40() V64() V65() V66() left_end right_end bounded_below bounded_above real-bounded Element of bool REAL
len D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
D1 . (len D1) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
Seg (len D1) is non empty V40() len D1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom D1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
[.(lower_bound A),(upper_bound A).] is V64() V65() V66() interval Element of bool REAL
{ b₁ where b₁ is V22() real ext-real Element of REAL : ( lower_bound A <= b₁ & b₁ <= upper_bound A ) } is set
D1 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
Seg (len D1) is non empty V40() len D1 -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
dom D1 is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
f . (len f) is V22() real ext-real Element of REAL
D1 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
D1 . D1 is V22() real ext-real Element of REAL
len g is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
card (rng g) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of omega
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
len D1 is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
card (rng f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of omega
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,a1,f) is V22() real ext-real Element of REAL
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,f) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,f),K295()) is V22() real ext-real Element of REAL
(A,a1,g) is V22() real ext-real Element of REAL
(A,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,g) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,g),K295()) is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,a1,D1) is V22() real ext-real Element of REAL
(A,a1,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,D1),K295()) is V22() real ext-real Element of REAL
(A,a1,D1) is V22() real ext-real Element of REAL
(A,a1,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,D1),K295()) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
f | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,f) is V22() real ext-real Element of REAL
(A,f) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
(A) is non empty set
[:(A),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A),REAL:] is non empty non trivial V40() set
rng (A,f) is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng (A,f)) is V22() real ext-real Element of REAL
(A,f) is V22() real ext-real Element of REAL
(A,f) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
rng (A,f) is non empty V64() V65() V66() Element of bool REAL
lower_bound (rng (A,f)) is V22() real ext-real Element of REAL
g is V22() real ext-real set
dom (A,f) is non empty Element of bool (A)
bool (A) is non empty set
a1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
(A,f) . a1 is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
D1 is V22() real ext-real set
dom (A,f) is non empty Element of bool (A)
a2 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
(A,f) . a2 is V22() real ext-real Element of REAL
D2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,D2) is V22() real ext-real Element of REAL
(A,f,D2) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D2) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D2),K295()) is V22() real ext-real Element of REAL
A is V64() V65() V66() Element of bool REAL
-- A is V64() V65() V66() Element of bool REAL
-- A is V65() set
f is V64() V65() V66() Element of bool REAL
-- f is V64() V65() V66() Element of bool REAL
-- f is V65() set
(-- A) ++ (-- f) is V64() V65() V66() set
(-- A) ++ (-- f) is V65() set
A ++ f is V64() V65() V66() set
A ++ f is V65() set
-- (A ++ f) is V64() V65() V66() set
-- (A ++ f) is V65() set
g is set
a1 is V64() V65() V66() Element of bool REAL
-- a1 is V64() V65() V66() Element of bool REAL
-- a1 is V65() set
D1 is V22() real ext-real Element of REAL
- D1 is V22() real ext-real Element of REAL
D1 is V22() real ext-real Element of REAL
a2 is V22() real ext-real Element of REAL
D1 + a2 is V22() real ext-real Element of REAL
- D1 is V22() real ext-real Element of REAL
- a2 is V22() real ext-real Element of REAL
(- D1) + (- a2) is V22() real ext-real Element of REAL
g is set
a1 is V22() real ext-real Element of REAL
D1 is V22() real ext-real Element of REAL
a1 + D1 is V22() real ext-real Element of REAL
D1 is V22() real ext-real Element of REAL
- D1 is V22() real ext-real Element of REAL
a2 is V64() V65() V66() Element of bool REAL
D2 is V22() real ext-real Element of REAL
- D2 is V22() real ext-real Element of REAL
D2 + D1 is V22() real ext-real Element of REAL
a2 ++ f is V64() V65() V66() set
a2 ++ f is V65() set
- (D2 + D1) is V22() real ext-real Element of REAL
A is V64() V65() V66() Element of bool REAL
f is V64() V65() V66() Element of bool REAL
A ++ f is V64() V65() V66() set
A ++ f is V65() set
-- f is V64() V65() V66() Element of bool REAL
-- f is V65() set
-- A is V64() V65() V66() Element of bool REAL
-- A is V65() set
(-- A) ++ (-- f) is V64() V65() V66() set
(-- A) ++ (-- f) is V65() set
g is V64() V65() V66() Element of bool REAL
-- g is V64() V65() V66() Element of bool REAL
-- g is V65() set
A is non empty V64() V65() V66() Element of bool REAL
f is non empty V64() V65() V66() Element of bool REAL
A ++ f is non empty V64() V65() V66() set
A ++ f is non empty V65() set
upper_bound (A ++ f) is V22() real ext-real set
upper_bound A is V22() real ext-real Element of REAL
upper_bound f is V22() real ext-real Element of REAL
(upper_bound A) + (upper_bound f) is V22() real ext-real Element of REAL
-- f is non empty V64() V65() V66() Element of bool REAL
-- f is non empty V65() set
-- A is non empty V64() V65() V66() Element of bool REAL
-- A is non empty V65() set
(-- A) ++ (-- f) is non empty V64() V65() V66() set
(-- A) ++ (-- f) is non empty V65() set
lower_bound ((-- A) ++ (-- f)) is V22() real ext-real set
lower_bound (-- A) is V22() real ext-real Element of REAL
lower_bound (-- f) is V22() real ext-real Element of REAL
(lower_bound (-- A)) + (lower_bound (-- f)) is V22() real ext-real Element of REAL
-- (-- A) is non empty V64() V65() V66() Element of bool REAL
-- (-- A) is non empty V65() set
upper_bound (-- (-- A)) is V22() real ext-real Element of REAL
- (upper_bound (-- (-- A))) is V22() real ext-real Element of REAL
(- (upper_bound (-- (-- A)))) + (lower_bound (-- f)) is V22() real ext-real Element of REAL
- (upper_bound A) is V22() real ext-real Element of REAL
-- (-- f) is non empty V64() V65() V66() Element of bool REAL
-- (-- f) is non empty V65() set
upper_bound (-- (-- f)) is V22() real ext-real Element of REAL
- (upper_bound (-- (-- f))) is V22() real ext-real Element of REAL
(- (upper_bound A)) + (- (upper_bound (-- (-- f)))) is V22() real ext-real Element of REAL
- ((upper_bound A) + (upper_bound f)) is V22() real ext-real Element of REAL
-- (A ++ f) is non empty V64() V65() V66() set
-- (A ++ f) is non empty V65() set
g is V64() V65() V66() Element of bool REAL
-- g is V64() V65() V66() Element of bool REAL
-- g is V65() set
-- (-- g) is V64() V65() V66() Element of bool REAL
-- (-- g) is V65() set
upper_bound (-- (-- g)) is V22() real ext-real Element of REAL
- (upper_bound (-- (-- g))) is V22() real ext-real Element of REAL
upper_bound g is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
a1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
a1 | f is Relation-like f -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(f,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,a1,g) . A is V22() real ext-real Element of REAL
D1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
D1 | f is Relation-like f -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
a1 + D1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(f,(a1 + D1),g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,(a1 + D1),g) . A is V22() real ext-real Element of REAL
(f,D1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,D1,g) . A is V22() real ext-real Element of REAL
((f,a1,g) . A) + ((f,D1,g) . A) is V22() real ext-real Element of REAL
dom (a1 + D1) is non empty V64() V65() V66() Element of bool f
bool f is non empty set
f /\ f is V64() V65() V66() interval Element of bool REAL
(f,g,A) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
(a1 + D1) | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
dom ((a1 + D1) | (f,g,A)) is V64() V65() V66() Element of bool f
rng ((a1 + D1) | (f,g,A)) is V64() V65() V66() Element of bool REAL
a1 | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
D1 | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(a1 | (f,g,A)) + (D1 | (f,g,A)) is Relation-like f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
rng (a1 | (f,g,A)) is V64() V65() V66() Element of bool REAL
rng (D1 | (f,g,A)) is V64() V65() V66() Element of bool REAL
(rng (a1 | (f,g,A))) ++ (rng (D1 | (f,g,A))) is V64() V65() V66() set
(rng (a1 | (f,g,A))) ++ (rng (D1 | (f,g,A))) is V65() set
rng a1 is non empty V64() V65() V66() Element of bool REAL
dom D1 is non empty V64() V65() V66() Element of bool f
dom (D1 | (f,g,A)) is V64() V65() V66() Element of bool f
((f,g,A)) is V22() real ext-real Element of REAL
upper_bound (f,g,A) is V22() real ext-real Element of REAL
sup (f,g,A) is V22() real ext-real set
lower_bound (f,g,A) is V22() real ext-real Element of REAL
inf (f,g,A) is V22() real ext-real set
(upper_bound (f,g,A)) - (lower_bound (f,g,A)) is V22() real ext-real Element of REAL
rng D1 is non empty V64() V65() V66() Element of bool REAL
dom a1 is non empty V64() V65() V66() Element of bool f
dom (a1 | (f,g,A)) is V64() V65() V66() Element of bool f
upper_bound ((rng (a1 | (f,g,A))) ++ (rng (D1 | (f,g,A)))) is V22() real ext-real set
upper_bound (rng (a1 | (f,g,A))) is V22() real ext-real Element of REAL
upper_bound (rng (D1 | (f,g,A))) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | (f,g,A)))) + (upper_bound (rng (D1 | (f,g,A)))) is V22() real ext-real Element of REAL
upper_bound (rng ((a1 + D1) | (f,g,A))) is V22() real ext-real Element of REAL
(upper_bound (rng ((a1 + D1) | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
((upper_bound (rng (a1 | (f,g,A)))) + (upper_bound (rng (D1 | (f,g,A))))) * ((f,g,A)) is V22() real ext-real Element of REAL
(upper_bound (rng (a1 | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
(upper_bound (rng (D1 | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
((upper_bound (rng (a1 | (f,g,A)))) * ((f,g,A))) + ((upper_bound (rng (D1 | (f,g,A)))) * ((f,g,A))) is V22() real ext-real Element of REAL
((f,a1,g) . A) + ((upper_bound (rng (D1 | (f,g,A)))) * ((f,g,A))) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:f,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:f,REAL:] is non empty non trivial V40() set
g is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (f)
dom g is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
a1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
a1 | f is Relation-like f -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(f,a1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,a1,g) . A is V22() real ext-real Element of REAL
D1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
D1 | f is Relation-like f -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(f,D1,g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,D1,g) . A is V22() real ext-real Element of REAL
((f,a1,g) . A) + ((f,D1,g) . A) is V22() real ext-real Element of REAL
a1 + D1 is Relation-like f -defined REAL -valued Function-like non empty total V30(f, REAL ) complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(f,(a1 + D1),g) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(f,(a1 + D1),g) . A is V22() real ext-real Element of REAL
(f,g,A) is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
((f,g,A)) is V22() real ext-real Element of REAL
upper_bound (f,g,A) is V22() real ext-real Element of REAL
sup (f,g,A) is V22() real ext-real set
lower_bound (f,g,A) is V22() real ext-real Element of REAL
inf (f,g,A) is V22() real ext-real set
(upper_bound (f,g,A)) - (lower_bound (f,g,A)) is V22() real ext-real Element of REAL
dom (a1 + D1) is non empty V64() V65() V66() Element of bool f
bool f is non empty set
f /\ f is V64() V65() V66() interval Element of bool REAL
(a1 + D1) | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
dom ((a1 + D1) | (f,g,A)) is V64() V65() V66() Element of bool f
rng ((a1 + D1) | (f,g,A)) is V64() V65() V66() Element of bool REAL
rng D1 is non empty V64() V65() V66() Element of bool REAL
D1 | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
rng (D1 | (f,g,A)) is V64() V65() V66() Element of bool REAL
dom D1 is non empty V64() V65() V66() Element of bool f
dom (D1 | (f,g,A)) is V64() V65() V66() Element of bool f
a1 | (f,g,A) is Relation-like f -defined (f,g,A) -defined f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
(a1 | (f,g,A)) + (D1 | (f,g,A)) is Relation-like f -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:f,REAL:]
rng (a1 | (f,g,A)) is V64() V65() V66() Element of bool REAL
(rng (a1 | (f,g,A))) ++ (rng (D1 | (f,g,A))) is V64() V65() V66() set
(rng (a1 | (f,g,A))) ++ (rng (D1 | (f,g,A))) is V65() set
rng a1 is non empty V64() V65() V66() Element of bool REAL
dom a1 is non empty V64() V65() V66() Element of bool f
dom (a1 | (f,g,A)) is V64() V65() V66() Element of bool f
lower_bound ((rng (a1 | (f,g,A))) ++ (rng (D1 | (f,g,A)))) is V22() real ext-real set
lower_bound (rng (a1 | (f,g,A))) is V22() real ext-real Element of REAL
lower_bound (rng (D1 | (f,g,A))) is V22() real ext-real Element of REAL
(lower_bound (rng (a1 | (f,g,A)))) + (lower_bound (rng (D1 | (f,g,A)))) is V22() real ext-real Element of REAL
((lower_bound (rng (a1 | (f,g,A)))) + (lower_bound (rng (D1 | (f,g,A))))) * ((f,g,A)) is V22() real ext-real Element of REAL
lower_bound (rng ((a1 + D1) | (f,g,A))) is V22() real ext-real Element of REAL
(lower_bound (rng ((a1 + D1) | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
(lower_bound (rng (a1 | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
(lower_bound (rng (D1 | (f,g,A)))) * ((f,g,A)) is V22() real ext-real Element of REAL
((lower_bound (rng (a1 | (f,g,A)))) * ((f,g,A))) + ((lower_bound (rng (D1 | (f,g,A)))) * ((f,g,A))) is V22() real ext-real Element of REAL
((f,a1,g) . A) + ((lower_bound (rng (D1 | (f,g,A)))) * ((f,g,A))) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,g,f) is V22() real ext-real Element of REAL
(A,g,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,f) is V22() real ext-real Element of REAL
K173(REAL,(A,g,f),K295()) is V22() real ext-real Element of REAL
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g + a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,(g + a1),f) is V22() real ext-real Element of REAL
(A,(g + a1),f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(g + a1),f) is V22() real ext-real Element of REAL
K173(REAL,(A,(g + a1),f),K295()) is V22() real ext-real Element of REAL
(A,a1,f) is V22() real ext-real Element of REAL
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,f) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,f),K295()) is V22() real ext-real Element of REAL
(A,g,f) + (A,a1,f) is V22() real ext-real Element of REAL
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len f) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
len (A,g,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,g,f) + (A,a1,f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K178(K295(),(A,g,f),(A,a1,f)) is set
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
(A,(g + a1),f) . D2 is V22() real ext-real Element of REAL
((A,g,f) + (A,a1,f)) . D2 is V22() real ext-real Element of REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,g,f) . D2 is V22() real ext-real Element of REAL
(A,a1,f) . D2 is V22() real ext-real Element of REAL
((A,g,f) . D2) + ((A,a1,f) . D2) is V22() real ext-real Element of REAL
len (A,(g + a1),f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Sum ((A,g,f) + (A,a1,f)) is V22() real ext-real Element of REAL
K173(REAL,((A,g,f) + (A,a1,f)),K295()) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,g,f) is V22() real ext-real Element of REAL
(A,g,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,f) is V22() real ext-real Element of REAL
K173(REAL,(A,g,f),K295()) is V22() real ext-real Element of REAL
a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
a1 | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,a1,f) is V22() real ext-real Element of REAL
(A,a1,f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,a1,f) is V22() real ext-real Element of REAL
K173(REAL,(A,a1,f),K295()) is V22() real ext-real Element of REAL
(A,g,f) + (A,a1,f) is V22() real ext-real Element of REAL
g + a1 is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,(g + a1),f) is V22() real ext-real Element of REAL
(A,(g + a1),f) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(g + a1),f) is V22() real ext-real Element of REAL
K173(REAL,(A,(g + a1),f),K295()) is V22() real ext-real Element of REAL
len (A,a1,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() 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 V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
(len f) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
len (A,g,f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Seg (len f) is non empty V40() len f -element V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,g,f) + (A,a1,f) is Relation-like NAT -defined REAL -valued Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
K178(K295(),(A,g,f),(A,a1,f)) is set
D2 is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal set
((A,g,f) + (A,a1,f)) . D2 is V22() real ext-real Element of REAL
(A,(g + a1),f) . D2 is V22() real ext-real Element of REAL
dom f is non empty V40() V64() V65() V66() V67() V68() V69() left_end right_end bounded_below bounded_above real-bounded Element of bool NAT
(A,g,f) . D2 is V22() real ext-real Element of REAL
(A,a1,f) . D2 is V22() real ext-real Element of REAL
((A,g,f) . D2) + ((A,a1,f) . D2) is V22() real ext-real Element of REAL
len (A,(g + a1),f) is non empty epsilon-transitive epsilon-connected ordinal natural V22() real ext-real positive non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() left_end right_end bounded_below bounded_above real-bounded Element of NAT
Sum ((A,g,f) + (A,a1,f)) is V22() real ext-real Element of REAL
K173(REAL,((A,g,f) + (A,a1,f)),K295()) is V22() real ext-real Element of REAL
A is non empty compact V64() V65() V66() bounded_below bounded_above real-bounded interval closed_interval Element of bool REAL
[:A,REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:A,REAL:] is non empty non trivial V40() set
f is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
f | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,f) is V22() real ext-real Element of REAL
(A,f) is V22() real ext-real Element of REAL
(A,f) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
(A) is non empty set
[:(A),REAL:] is Relation-like non empty non trivial V40() complex-valued ext-real-valued real-valued set
bool [:(A),REAL:] is non empty non trivial V40() set
rng (A,f) is non empty V64() V65() V66() Element of bool REAL
lower_bound (rng (A,f)) is V22() real ext-real Element of REAL
g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
g | A is Relation-like A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
f + g is Relation-like A -defined REAL -valued Function-like non empty total V30(A, REAL ) complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,(f + g)) is V22() real ext-real Element of REAL
(A,(f + g)) is V22() real ext-real Element of REAL
(A,(f + g)) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
rng (A,(f + g)) is non empty V64() V65() V66() Element of bool REAL
lower_bound (rng (A,(f + g))) is V22() real ext-real Element of REAL
(A,g) is V22() real ext-real Element of REAL
(A,g) is V22() real ext-real Element of REAL
(A,g) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
rng (A,g) is non empty V64() V65() V66() Element of bool REAL
lower_bound (rng (A,g)) is V22() real ext-real Element of REAL
(A,f) + (A,g) is V22() real ext-real Element of REAL
(A,f) is V22() real ext-real Element of REAL
(A,f) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
rng (A,f) is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng (A,f)) is V22() real ext-real Element of REAL
(A,g) is V22() real ext-real Element of REAL
(A,g) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
rng (A,g) is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng (A,g)) is V22() real ext-real Element of REAL
(A,f) + (A,g) is V22() real ext-real Element of REAL
(A,f) + (A,g) is V22() real ext-real Element of REAL
A /\ A is V64() V65() V66() interval Element of bool REAL
(f + g) | (A /\ A) is Relation-like A -defined A /\ A -defined A -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:A,REAL:]
(A,(f + g)) is Relation-like (A) -defined REAL -valued Function-like non empty total V30((A), REAL ) complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
dom (A,(f + g)) is non empty Element of bool (A)
bool (A) is non empty set
(A) /\ (dom (A,(f + g))) is Element of bool (A)
rng f is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng f) is V22() real ext-real Element of REAL
(A) is V22() real ext-real Element of REAL
upper_bound A is V22() real ext-real Element of REAL
sup A is V22() real ext-real set
lower_bound A is V22() real ext-real Element of REAL
inf A is V22() real ext-real set
(upper_bound A) - (lower_bound A) is V22() real ext-real Element of REAL
(upper_bound (rng f)) * (A) is V22() real ext-real Element of REAL
rng g is non empty V64() V65() V66() Element of bool REAL
upper_bound (rng g) is V22() real ext-real Element of REAL
(upper_bound (rng g)) * (A) is V22() real ext-real Element of REAL
((upper_bound (rng f)) * (A)) + ((upper_bound (rng g)) * (A)) is V22() real ext-real Element of REAL
a1 is set
(A,(f + g)) . a1 is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,(f + g)) . D1 is V22() real ext-real Element of REAL
(A,(f + g),D1) is V22() real ext-real Element of REAL
(A,(f + g),D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(f + g),D1) is V22() real ext-real Element of REAL
K173(REAL,(A,(f + g),D1),K295()) is V22() real ext-real Element of REAL
(A,(f + g),D1) is V22() real ext-real Element of REAL
(A,(f + g),D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(f + g),D1) is V22() real ext-real Element of REAL
K173(REAL,(A,(f + g),D1),K295()) is V22() real ext-real Element of REAL
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
(A,g,D1) is V22() real ext-real Element of REAL
(A,g,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,g,D1),K295()) is V22() real ext-real Element of REAL
(A,f,D1) + (A,g,D1) is V22() real ext-real Element of REAL
((upper_bound (rng f)) * (A)) + (A,g,D1) is V22() real ext-real Element of REAL
(A,(f + g)) | (A) is Relation-like (A) -defined (A) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
rng (A,(f + g)) is non empty V64() V65() V66() Element of bool REAL
dom (A,(f + g)) is non empty Element of bool (A)
(A) /\ (dom (A,(f + g))) is Element of bool (A)
lower_bound (rng f) is V22() real ext-real Element of REAL
(lower_bound (rng f)) * (A) is V22() real ext-real Element of REAL
lower_bound (rng g) is V22() real ext-real Element of REAL
(lower_bound (rng g)) * (A) is V22() real ext-real Element of REAL
((lower_bound (rng f)) * (A)) + ((lower_bound (rng g)) * (A)) is V22() real ext-real Element of REAL
a1 is set
(A,(f + g)) . a1 is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,(f + g)) . D1 is V22() real ext-real Element of REAL
(A,(f + g),D1) is V22() real ext-real Element of REAL
(A,(f + g),D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(f + g),D1) is V22() real ext-real Element of REAL
K173(REAL,(A,(f + g),D1),K295()) is V22() real ext-real Element of REAL
(A,(f + g),D1) is V22() real ext-real Element of REAL
(A,(f + g),D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(f + g),D1) is V22() real ext-real Element of REAL
K173(REAL,(A,(f + g),D1),K295()) is V22() real ext-real Element of REAL
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
(A,g,D1) is V22() real ext-real Element of REAL
(A,g,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,g,D1),K295()) is V22() real ext-real Element of REAL
((lower_bound (rng f)) * (A)) + (A,g,D1) is V22() real ext-real Element of REAL
(A,f,D1) + (A,g,D1) is V22() real ext-real Element of REAL
(A,(f + g)) | (A) is Relation-like (A) -defined (A) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [:(A),REAL:]
a1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f) . a1 is V22() real ext-real Element of REAL
(A,g) . a1 is V22() real ext-real Element of REAL
((A,f) . a1) + ((A,g) . a1) is V22() real ext-real Element of REAL
(A,(f + g),a1) is V22() real ext-real Element of REAL
(A,(f + g),a1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(f + g),a1) is V22() real ext-real Element of REAL
K173(REAL,(A,(f + g),a1),K295()) is V22() real ext-real Element of REAL
(A,f,a1) is V22() real ext-real Element of REAL
(A,f,a1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,a1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,a1),K295()) is V22() real ext-real Element of REAL
(A,g,a1) is V22() real ext-real Element of REAL
(A,g,a1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,a1) is V22() real ext-real Element of REAL
K173(REAL,(A,g,a1),K295()) is V22() real ext-real Element of REAL
(A,f,a1) + (A,g,a1) is V22() real ext-real Element of REAL
((A,f) . a1) + (A,g,a1) is V22() real ext-real Element of REAL
(A,(f + g)) . a1 is V22() real ext-real Element of REAL
(A,(f + g)) - (A,g) is V22() real ext-real Element of REAL
a1 is V22() real ext-real set
dom (A,f) is non empty Element of bool (A)
D1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
(A,f) . D1 is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
(A,(f + g)) - a1 is V22() real ext-real Element of REAL
a2 is V22() real ext-real set
dom (A,g) is non empty Element of bool (A)
D2 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
(A,g) . D2 is V22() real ext-real Element of REAL
D2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,g) . D is V22() real ext-real Element of REAL
(A,g,D) is V22() real ext-real Element of REAL
(A,g,D) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,D) is V22() real ext-real Element of REAL
K173(REAL,(A,g,D),K295()) is V22() real ext-real Element of REAL
(A,g,D2) is V22() real ext-real Element of REAL
(A,g,D2) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,D2) is V22() real ext-real Element of REAL
K173(REAL,(A,g,D2),K295()) is V22() real ext-real Element of REAL
(A,f) . D is V22() real ext-real Element of REAL
(A,f,D) is V22() real ext-real Element of REAL
(A,f,D) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D),K295()) is V22() real ext-real Element of REAL
((A,f) . D) + ((A,g) . D) is V22() real ext-real Element of REAL
(A,f,D1) + (A,g,D2) is V22() real ext-real Element of REAL
a1 + (lower_bound (rng (A,g))) is V22() real ext-real Element of REAL
(A,f) + (A,g) is V22() real ext-real Element of REAL
(A,(f + g)) is V22() real ext-real Element of REAL
upper_bound (rng (A,(f + g))) is V22() real ext-real Element of REAL
a1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f) . a1 is V22() real ext-real Element of REAL
(A,g) . a1 is V22() real ext-real Element of REAL
((A,f) . a1) + ((A,g) . a1) is V22() real ext-real Element of REAL
(A,f,a1) is V22() real ext-real Element of REAL
(A,f,a1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,a1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,a1),K295()) is V22() real ext-real Element of REAL
(A,g,a1) is V22() real ext-real Element of REAL
(A,g,a1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,a1) is V22() real ext-real Element of REAL
K173(REAL,(A,g,a1),K295()) is V22() real ext-real Element of REAL
(A,f,a1) + (A,g,a1) is V22() real ext-real Element of REAL
(A,(f + g),a1) is V22() real ext-real Element of REAL
(A,(f + g),a1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,(f + g),a1) is V22() real ext-real Element of REAL
K173(REAL,(A,(f + g),a1),K295()) is V22() real ext-real Element of REAL
((A,f) . a1) + (A,g,a1) is V22() real ext-real Element of REAL
(A,(f + g)) . a1 is V22() real ext-real Element of REAL
(A,(f + g)) - (A,g) is V22() real ext-real Element of REAL
a1 is V22() real ext-real set
dom (A,f) is non empty Element of bool (A)
D1 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
(A,f) . D1 is V22() real ext-real Element of REAL
D1 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,f,D1) is V22() real ext-real Element of REAL
(A,f,D1) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D1) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D1),K295()) is V22() real ext-real Element of REAL
(A,(f + g)) - a1 is V22() real ext-real Element of REAL
a2 is V22() real ext-real set
dom (A,g) is non empty Element of bool (A)
D2 is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of (A)
(A,g) . D2 is V22() real ext-real Element of REAL
D2 is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
D is Relation-like NAT -defined REAL -valued Function-like one-to-one non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing non-decreasing (A)
(A,g) . D is V22() real ext-real Element of REAL
(A,g,D) is V22() real ext-real Element of REAL
(A,g,D) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,D) is V22() real ext-real Element of REAL
K173(REAL,(A,g,D),K295()) is V22() real ext-real Element of REAL
(A,g,D2) is V22() real ext-real Element of REAL
(A,g,D2) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,g,D2) is V22() real ext-real Element of REAL
K173(REAL,(A,g,D2),K295()) is V22() real ext-real Element of REAL
(A,f) . D is V22() real ext-real Element of REAL
(A,f,D) is V22() real ext-real Element of REAL
(A,f,D) is Relation-like NAT -defined REAL -valued Function-like non empty V40() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Sum (A,f,D) is V22() real ext-real Element of REAL
K173(REAL,(A,f,D),K295()) is V22() real ext-real Element of REAL
(A,f,D1) + (A,g,D2) is V22() real ext-real Element of REAL
((A,f) . D) + ((A,g) . D) is V22() real ext-real Element of REAL
a1 + (upper_bound (rng (A,g))) is V22() real ext-real Element of REAL
A is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
g is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
dom g is V40() V64() V65() V66() V67() V68() V69() bounded_below bounded_above real-bounded Element of bool NAT
f is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
mid (g,A,f) is Relation-like NAT -defined Function-like V40() FinSequence-like FinSubsequence-like set
len (mid (g,A,f)) is epsilon-transitive epsilon-connected ordinal natural V22() real ext-real non negative V40() cardinal V64() V65() V66() V67() V68() V69() V71() V72() bounded_below bounded_above real-bounded Element of NAT
f - A is V22() real ext-real Element of REAL
(f - A) + 1 is V22() real ext-real Element of REAL