:: INTEGR13 semantic presentation

REAL is non empty V51() V52() V53() V57() V62() set
NAT is V51() V52() V53() V54() V55() V56() V57() Element of K19(REAL)
K19(REAL) is set
COMPLEX is non empty V51() V57() V62() set
K20(NAT,REAL) is V1() V34() V35() V36() set
K19(K20(NAT,REAL)) is set
K20(NAT,COMPLEX) is V1() V34() set
K19(K20(NAT,COMPLEX)) is set
K20(COMPLEX,COMPLEX) is V1() V34() set
K19(K20(COMPLEX,COMPLEX)) is set
K20(REAL,REAL) is V1() V34() V35() V36() set
K19(K20(REAL,REAL)) is set
PFuncs (REAL,REAL) is set
K20(NAT,(PFuncs (REAL,REAL))) is V1() set
K19(K20(NAT,(PFuncs (REAL,REAL)))) is set
ExtREAL is non empty V52() set
RAT is non empty V51() V52() V53() V54() V57() V62() set
INT is non empty V51() V52() V53() V54() V55() V57() V62() set
NAT is V51() V52() V53() V54() V55() V56() V57() set
K19(NAT) is set
K19(NAT) is set
K20(COMPLEX,REAL) is V1() V34() V35() V36() set
K19(K20(COMPLEX,REAL)) is set
{} is set
the V1() V2() V3() V5( RAT ) empty V34() V35() V36() V37() V51() V52() V53() V54() V55() V56() V57() set is V1() V2() V3() V5( RAT ) empty V34() V35() V36() V37() V51() V52() V53() V54() V55() V56() V57() set
1 is V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() Element of NAT
{{},1} is set
0 is V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() Element of NAT
{0} is V51() V52() V53() V54() V55() V56() set
exp_R is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
ln is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R " is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ln is V51() V52() V53() Element of K19(REAL)
right_open_halfline 0 is V51() V52() V53() Element of K19(REAL)
+infty is set
K200(0,+infty) is set
rng ln is V51() V52() V53() Element of K19(REAL)
2 is V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() Element of NAT
sin is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
cos is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
sin + cos is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (sin + cos) is V51() V52() V53() Element of K19(REAL)
sin - cos is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- cos is V1() V6() V34() set
K98(1) is V28() V29() V30() set
K98(1) (#) cos is V1() V6() set
sin + (- cos) is V1() V6() set
dom (sin - cos) is V51() V52() V53() Element of K19(REAL)
exp_R (#) (sin - cos) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) (sin - cos)) is V51() V52() V53() Element of K19(REAL)
exp_R (#) (sin + cos) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) (sin + cos)) is V51() V52() V53() Element of K19(REAL)
(sin + cos) / exp_R is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin + cos) / exp_R) is V51() V52() V53() Element of K19(REAL)
(sin - cos) / exp_R is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin - cos) / exp_R) is V51() V52() V53() Element of K19(REAL)
exp_R (#) sin is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) sin) is V51() V52() V53() Element of K19(REAL)
exp_R (#) cos is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) cos) is V51() V52() V53() Element of K19(REAL)
sin / cos is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
cos / sin is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
tan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom tan is V51() V52() V53() Element of K19(REAL)
cot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom cot is V51() V52() V53() Element of K19(REAL)
tan * exp_R is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (tan * exp_R) is V51() V52() V53() Element of K19(REAL)
cot * exp_R is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (cot * exp_R) is V51() V52() V53() Element of K19(REAL)
tan * ln is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (tan * ln) is V51() V52() V53() Element of K19(REAL)
cot * ln is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (cot * ln) is V51() V52() V53() Element of K19(REAL)
exp_R * tan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R * tan) is V51() V52() V53() Element of K19(REAL)
exp_R * cot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R * cot) is V51() V52() V53() Element of K19(REAL)
ln * tan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * tan) is V51() V52() V53() Element of K19(REAL)
ln * cot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * cot) is V51() V52() V53() Element of K19(REAL)
ln (#) tan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln (#) tan) is V51() V52() V53() Element of K19(REAL)
ln (#) cot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln (#) cot) is V51() V52() V53() Element of K19(REAL)
- 1 is V28() V29() V30() ext-real Element of REAL
arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
].(- 1),1.[ is V51() V52() V53() open Element of K19(REAL)
arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
ln * arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * arctan) is V51() V52() V53() Element of K19(REAL)
#Z 2 is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z 2) * arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
1 / 2 is V28() V29() ext-real Element of REAL
K99(2) is V28() set
K97(1,K99(2)) is set
(1 / 2) (#) ((#Z 2) * arctan) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((1 / 2) (#) ((#Z 2) * arctan)) is V51() V52() V53() Element of K19(REAL)
(#Z 2) * arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(1 / 2) (#) ((#Z 2) * arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((1 / 2) (#) ((#Z 2) * arccot)) is V51() V52() V53() Element of K19(REAL)
exp_R * arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R * arctan) is V51() V52() V53() Element of K19(REAL)
exp_R * arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R * arccot) is V51() V52() V53() Element of K19(REAL)
ln (#) arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln (#) arctan) is V51() V52() V53() Element of K19(REAL)
ln (#) arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln (#) arccot) is V51() V52() V53() Element of K19(REAL)
{0} is V51() V52() V53() V54() V55() V56() Element of K19(REAL)
A is V51() V52() V53() open Element of K19(REAL)
id A is V1() V4( REAL ) V4(A) V5( REAL ) V5(A) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id A) " {0} is V51() V52() V53() Element of K19(REAL)
f1 is set
dom (id A) is V51() V52() V53() Element of K19(A)
K19(A) is set
(id A) . f1 is V28() V29() ext-real Element of REAL
f1 is set
(id A) . f1 is V28() V29() ext-real Element of REAL
dom (id A) is V51() V52() V53() Element of K19(A)
K19(A) is set
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : not b₁ <= 0 } is set
- (1 / 2) is V28() V29() ext-real Element of REAL
sin (#) cos is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin (#) cos) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln * tan) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln * tan) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln * tan) . (upper_bound A)) - ((ln * tan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln * tan) . (lower_bound A))) is V28() set
K96(((ln * tan) . (upper_bound A)),K98(((ln * tan) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(sin . Z) * (cos . Z) is V28() V29() ext-real Element of REAL
1 / ((sin . Z) * (cos . Z)) is V28() V29() ext-real Element of REAL
K99(((sin . Z) * (cos . Z))) is V28() set
K97(1,K99(((sin . Z) * (cos . Z)))) is set
((sin (#) cos) ^) . Z is V28() V29() ext-real Element of REAL
(sin (#) cos) . Z is V28() V29() ext-real Element of REAL
1 / ((sin (#) cos) . Z) is V28() V29() ext-real Element of REAL
K99(((sin (#) cos) . Z)) is V28() set
K97(1,K99(((sin (#) cos) . Z))) is set
(ln * tan) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln * tan) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((ln * tan) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(sin . Z) * (cos . Z) is V28() V29() ext-real Element of REAL
1 / ((sin . Z) * (cos . Z)) is V28() V29() ext-real Element of REAL
K99(((sin . Z) * (cos . Z))) is V28() set
K97(1,K99(((sin . Z) * (cos . Z)))) is set
- ((sin (#) cos) ^) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((sin (#) cos) ^) is V1() V6() set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln * cot) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln * cot) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln * cot) . (upper_bound A)) - ((ln * cot) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln * cot) . (lower_bound A))) is V28() set
K96(((ln * cot) . (upper_bound A)),K98(((ln * cot) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom ((sin (#) cos) ^) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(sin . Z) * (cos . Z) is V28() V29() ext-real Element of REAL
1 / ((sin . Z) * (cos . Z)) is V28() V29() ext-real Element of REAL
K99(((sin . Z) * (cos . Z))) is V28() set
K97(1,K99(((sin . Z) * (cos . Z)))) is set
- (1 / ((sin . Z) * (cos . Z))) is V28() V29() ext-real Element of REAL
(- ((sin (#) cos) ^)) . Z is V28() V29() ext-real Element of REAL
((sin (#) cos) ^) . Z is V28() V29() ext-real Element of REAL
- (((sin (#) cos) ^) . Z) is V28() V29() ext-real Element of REAL
(sin (#) cos) . Z is V28() V29() ext-real Element of REAL
1 / ((sin (#) cos) . Z) is V28() V29() ext-real Element of REAL
K99(((sin (#) cos) . Z)) is V28() set
K97(1,K99(((sin (#) cos) . Z))) is set
- (1 / ((sin (#) cos) . Z)) is V28() V29() ext-real Element of REAL
(ln * cot) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln * cot) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((ln * cot) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(sin . Z) * (cos . Z) is V28() V29() ext-real Element of REAL
1 / ((sin . Z) * (cos . Z)) is V28() V29() ext-real Element of REAL
K99(((sin . Z) * (cos . Z))) is V28() set
K97(1,K99(((sin . Z) * (cos . Z)))) is set
- (1 / ((sin . Z) * (cos . Z))) is V28() V29() ext-real Element of REAL
2 (#) (exp_R (#) sin) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(exp_R (#) (sin - cos)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(exp_R (#) (sin - cos)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((exp_R (#) (sin - cos)) . (upper_bound A)) - ((exp_R (#) (sin - cos)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((exp_R (#) (sin - cos)) . (lower_bound A))) is V28() set
K96(((exp_R (#) (sin - cos)) . (upper_bound A)),K98(((exp_R (#) (sin - cos)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
2 * (exp_R . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(2 * (exp_R . Z)) * (sin . Z) is V28() V29() ext-real Element of REAL
(2 (#) (exp_R (#) sin)) . Z is V28() V29() ext-real Element of REAL
(exp_R (#) sin) . Z is V28() V29() ext-real Element of REAL
2 * ((exp_R (#) sin) . Z) is V28() V29() ext-real Element of REAL
(exp_R . Z) * (sin . Z) is V28() V29() ext-real Element of REAL
2 * ((exp_R . Z) * (sin . Z)) is V28() V29() ext-real Element of REAL
(exp_R (#) (sin - cos)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R (#) (sin - cos)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((exp_R (#) (sin - cos)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
2 * (exp_R . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(2 * (exp_R . Z)) * (sin . Z) is V28() V29() ext-real Element of REAL
2 (#) (exp_R (#) cos) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(exp_R (#) (sin + cos)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(exp_R (#) (sin + cos)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((exp_R (#) (sin + cos)) . (upper_bound A)) - ((exp_R (#) (sin + cos)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((exp_R (#) (sin + cos)) . (lower_bound A))) is V28() set
K96(((exp_R (#) (sin + cos)) . (upper_bound A)),K98(((exp_R (#) (sin + cos)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
2 * (exp_R . Z) is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(2 * (exp_R . Z)) * (cos . Z) is V28() V29() ext-real Element of REAL
(2 (#) (exp_R (#) cos)) . Z is V28() V29() ext-real Element of REAL
(exp_R (#) cos) . Z is V28() V29() ext-real Element of REAL
2 * ((exp_R (#) cos) . Z) is V28() V29() ext-real Element of REAL
(exp_R . Z) * (cos . Z) is V28() V29() ext-real Element of REAL
2 * ((exp_R . Z) * (cos . Z)) is V28() V29() ext-real Element of REAL
(exp_R (#) (sin + cos)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R (#) (sin + cos)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((exp_R (#) (sin + cos)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
2 * (exp_R . Z) is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(2 * (exp_R . Z)) * (cos . Z) is V28() V29() ext-real Element of REAL
cos - sin is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- sin is V1() V6() V34() set
K98(1) (#) sin is V1() V6() set
cos + (- sin) is V1() V6() set
dom (cos - sin) is V51() V52() V53() Element of K19(REAL)
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
(cos - sin) | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral ((cos - sin),A) is V28() V29() ext-real Element of REAL
upper_bound A is V28() V29() ext-real Element of REAL
(sin + cos) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(sin + cos) . (lower_bound A) is V28() V29() ext-real Element of REAL
((sin + cos) . (upper_bound A)) - ((sin + cos) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((sin + cos) . (lower_bound A))) is V28() set
K96(((sin + cos) . (upper_bound A)),K98(((sin + cos) . (lower_bound A)))) is set
f1 is V51() V52() V53() open Element of K19(REAL)
dom cos is V51() V52() V53() Element of K19(REAL)
dom sin is V51() V52() V53() Element of K19(REAL)
(dom cos) /\ (dom sin) is V51() V52() V53() Element of K19(REAL)
(sin + cos) `| f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin + cos) `| f1) is V51() V52() V53() Element of K19(REAL)
f is V28() V29() ext-real Element of REAL
((sin + cos) `| f1) . f is V28() V29() ext-real Element of REAL
(cos - sin) . f is V28() V29() ext-real Element of REAL
cos . f is V28() V29() ext-real Element of REAL
sin . f is V28() V29() ext-real Element of REAL
(cos . f) - (sin . f) is V28() V29() ext-real Element of REAL
K98((sin . f)) is V28() set
K96((cos . f),K98((sin . f))) is set
cos + sin is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (cos + sin) is V51() V52() V53() Element of K19(REAL)
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
(cos + sin) | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral ((cos + sin),A) is V28() V29() ext-real Element of REAL
upper_bound A is V28() V29() ext-real Element of REAL
(sin - cos) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(sin - cos) . (lower_bound A) is V28() V29() ext-real Element of REAL
((sin - cos) . (upper_bound A)) - ((sin - cos) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((sin - cos) . (lower_bound A))) is V28() set
K96(((sin - cos) . (upper_bound A)),K98(((sin - cos) . (lower_bound A)))) is set
f1 is V51() V52() V53() open Element of K19(REAL)
dom cos is V51() V52() V53() Element of K19(REAL)
dom sin is V51() V52() V53() Element of K19(REAL)
(dom cos) /\ (dom sin) is V51() V52() V53() Element of K19(REAL)
(sin - cos) `| f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin - cos) `| f1) is V51() V52() V53() Element of K19(REAL)
f is V28() V29() ext-real Element of REAL
((sin - cos) `| f1) . f is V28() V29() ext-real Element of REAL
(cos + sin) . f is V28() V29() ext-real Element of REAL
cos . f is V28() V29() ext-real Element of REAL
sin . f is V28() V29() ext-real Element of REAL
(cos . f) + (sin . f) is V28() V29() ext-real Element of REAL
(- (1 / 2)) (#) ((sin + cos) / exp_R) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- (1 / 2)) (#) ((sin + cos) / exp_R)) is V51() V52() V53() Element of K19(REAL)
A is V51() V52() V53() open Element of K19(REAL)
((- (1 / 2)) (#) ((sin + cos) / exp_R)) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom exp_R is V51() V52() V53() Element of K19(REAL)
exp_R " {0} is V51() V52() V53() Element of K19(REAL)
(dom exp_R) \ (exp_R " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (sin + cos)) /\ ((dom exp_R) \ (exp_R " {0})) is V51() V52() V53() Element of K19(REAL)
(sin + cos) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
((sin + cos) `| A) . f1 is V28() V29() ext-real Element of REAL
cos . f1 is V28() V29() ext-real Element of REAL
sin . f1 is V28() V29() ext-real Element of REAL
(cos . f1) - (sin . f1) is V28() V29() ext-real Element of REAL
K98((sin . f1)) is V28() set
K96((cos . f1),K98((sin . f1))) is set
f1 is V28() V29() ext-real Element of REAL
(((- (1 / 2)) (#) ((sin + cos) / exp_R)) `| A) . f1 is V28() V29() ext-real Element of REAL
sin . f1 is V28() V29() ext-real Element of REAL
exp_R . f1 is V28() V29() ext-real Element of REAL
(sin . f1) / (exp_R . f1) is V28() V29() ext-real Element of REAL
K99((exp_R . f1)) is V28() set
K97((sin . f1),K99((exp_R . f1))) is set
(sin + cos) . f1 is V28() V29() ext-real Element of REAL
cos . f1 is V28() V29() ext-real Element of REAL
(sin . f1) + (cos . f1) is V28() V29() ext-real Element of REAL
diff (((sin + cos) / exp_R),f1) is V28() V29() ext-real Element of REAL
(- (1 / 2)) * (diff (((sin + cos) / exp_R),f1)) is V28() V29() ext-real Element of REAL
diff ((sin + cos),f1) is V28() V29() ext-real Element of REAL
(diff ((sin + cos),f1)) * (exp_R . f1) is V28() V29() ext-real Element of REAL
diff (exp_R,f1) is V28() V29() ext-real Element of REAL
(diff (exp_R,f1)) * ((sin + cos) . f1) is V28() V29() ext-real Element of REAL
((diff ((sin + cos),f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1)) is V28() V29() ext-real Element of REAL
K98(((diff (exp_R,f1)) * ((sin + cos) . f1))) is V28() set
K96(((diff ((sin + cos),f1)) * (exp_R . f1)),K98(((diff (exp_R,f1)) * ((sin + cos) . f1)))) is set
(exp_R . f1) ^2 is V28() V29() ext-real Element of REAL
K97((exp_R . f1),(exp_R . f1)) is set
(((diff ((sin + cos),f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))) / ((exp_R . f1) ^2) is V28() V29() ext-real Element of REAL
K99(((exp_R . f1) ^2)) is V28() set
K97((((diff ((sin + cos),f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))),K99(((exp_R . f1) ^2))) is set
(- (1 / 2)) * ((((diff ((sin + cos),f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))) / ((exp_R . f1) ^2)) is V28() V29() ext-real Element of REAL
((sin + cos) `| A) . f1 is V28() V29() ext-real Element of REAL
(((sin + cos) `| A) . f1) * (exp_R . f1) is V28() V29() ext-real Element of REAL
((((sin + cos) `| A) . f1) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1)) is V28() V29() ext-real Element of REAL
K96(((((sin + cos) `| A) . f1) * (exp_R . f1)),K98(((diff (exp_R,f1)) * ((sin + cos) . f1)))) is set
(((((sin + cos) `| A) . f1) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))) / ((exp_R . f1) ^2) is V28() V29() ext-real Element of REAL
K97((((((sin + cos) `| A) . f1) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))),K99(((exp_R . f1) ^2))) is set
(- (1 / 2)) * ((((((sin + cos) `| A) . f1) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))) / ((exp_R . f1) ^2)) is V28() V29() ext-real Element of REAL
(cos . f1) - (sin . f1) is V28() V29() ext-real Element of REAL
K98((sin . f1)) is V28() set
K96((cos . f1),K98((sin . f1))) is set
((cos . f1) - (sin . f1)) * (exp_R . f1) is V28() V29() ext-real Element of REAL
(((cos . f1) - (sin . f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1)) is V28() V29() ext-real Element of REAL
K96((((cos . f1) - (sin . f1)) * (exp_R . f1)),K98(((diff (exp_R,f1)) * ((sin + cos) . f1)))) is set
((((cos . f1) - (sin . f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))) / ((exp_R . f1) ^2) is V28() V29() ext-real Element of REAL
K97(((((cos . f1) - (sin . f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))),K99(((exp_R . f1) ^2))) is set
(- (1 / 2)) * (((((cos . f1) - (sin . f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin + cos) . f1))) / ((exp_R . f1) ^2)) is V28() V29() ext-real Element of REAL
(exp_R . f1) * ((sin . f1) + (cos . f1)) is V28() V29() ext-real Element of REAL
(((cos . f1) - (sin . f1)) * (exp_R . f1)) - ((exp_R . f1) * ((sin . f1) + (cos . f1))) is V28() V29() ext-real Element of REAL
K98(((exp_R . f1) * ((sin . f1) + (cos . f1)))) is V28() set
K96((((cos . f1) - (sin . f1)) * (exp_R . f1)),K98(((exp_R . f1) * ((sin . f1) + (cos . f1))))) is set
((((cos . f1) - (sin . f1)) * (exp_R . f1)) - ((exp_R . f1) * ((sin . f1) + (cos . f1)))) / ((exp_R . f1) ^2) is V28() V29() ext-real Element of REAL
K97(((((cos . f1) - (sin . f1)) * (exp_R . f1)) - ((exp_R . f1) * ((sin . f1) + (cos . f1)))),K99(((exp_R . f1) ^2))) is set
(- (1 / 2)) * (((((cos . f1) - (sin . f1)) * (exp_R . f1)) - ((exp_R . f1) * ((sin . f1) + (cos . f1)))) / ((exp_R . f1) ^2)) is V28() V29() ext-real Element of REAL
2 * (sin . f1) is V28() V29() ext-real Element of REAL
- (2 * (sin . f1)) is V28() V29() ext-real Element of REAL
(exp_R . f1) * (exp_R . f1) is V28() V29() ext-real Element of REAL
(exp_R . f1) / ((exp_R . f1) * (exp_R . f1)) is V28() V29() ext-real Element of REAL
K99(((exp_R . f1) * (exp_R . f1))) is V28() set
K97((exp_R . f1),K99(((exp_R . f1) * (exp_R . f1)))) is set
(- (2 * (sin . f1))) * ((exp_R . f1) / ((exp_R . f1) * (exp_R . f1))) is V28() V29() ext-real Element of REAL
(- (1 / 2)) * ((- (2 * (sin . f1))) * ((exp_R . f1) / ((exp_R . f1) * (exp_R . f1)))) is V28() V29() ext-real Element of REAL
(exp_R . f1) / (exp_R . f1) is V28() V29() ext-real Element of REAL
K97((exp_R . f1),K99((exp_R . f1))) is set
((exp_R . f1) / (exp_R . f1)) / (exp_R . f1) is V28() V29() ext-real Element of REAL
K97(((exp_R . f1) / (exp_R . f1)),K99((exp_R . f1))) is set
(- (2 * (sin . f1))) * (((exp_R . f1) / (exp_R . f1)) / (exp_R . f1)) is V28() V29() ext-real Element of REAL
(- (1 / 2)) * ((- (2 * (sin . f1))) * (((exp_R . f1) / (exp_R . f1)) / (exp_R . f1))) is V28() V29() ext-real Element of REAL
1 / (exp_R . f1) is V28() V29() ext-real Element of REAL
K97(1,K99((exp_R . f1))) is set
(- (2 * (sin . f1))) * (1 / (exp_R . f1)) is V28() V29() ext-real Element of REAL
(- (1 / 2)) * ((- (2 * (sin . f1))) * (1 / (exp_R . f1))) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
(((- (1 / 2)) (#) ((sin + cos) / exp_R)) `| A) . f1 is V28() V29() ext-real Element of REAL
sin . f1 is V28() V29() ext-real Element of REAL
exp_R . f1 is V28() V29() ext-real Element of REAL
(sin . f1) / (exp_R . f1) is V28() V29() ext-real Element of REAL
K99((exp_R . f1)) is V28() set
K97((sin . f1),K99((exp_R . f1))) is set
sin / exp_R is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (upper_bound A)) - (((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (lower_bound A))) is V28() set
K96((((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (upper_bound A)),K98((((- (1 / 2)) (#) ((sin + cos) / exp_R)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
(sin . Z) / (exp_R . Z) is V28() V29() ext-real Element of REAL
K99((exp_R . Z)) is V28() set
K97((sin . Z),K99((exp_R . Z))) is set
((- (1 / 2)) (#) ((sin + cos) / exp_R)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((- (1 / 2)) (#) ((sin + cos) / exp_R)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
(((- (1 / 2)) (#) ((sin + cos) / exp_R)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
(sin . Z) / (exp_R . Z) is V28() V29() ext-real Element of REAL
K99((exp_R . Z)) is V28() set
K97((sin . Z),K99((exp_R . Z))) is set
(1 / 2) (#) ((sin - cos) / exp_R) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((1 / 2) (#) ((sin - cos) / exp_R)) is V51() V52() V53() Element of K19(REAL)
A is V51() V52() V53() open Element of K19(REAL)
((1 / 2) (#) ((sin - cos) / exp_R)) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom exp_R is V51() V52() V53() Element of K19(REAL)
exp_R " {0} is V51() V52() V53() Element of K19(REAL)
(dom exp_R) \ (exp_R " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (sin - cos)) /\ ((dom exp_R) \ (exp_R " {0})) is V51() V52() V53() Element of K19(REAL)
(sin - cos) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
((sin - cos) `| A) . f1 is V28() V29() ext-real Element of REAL
cos . f1 is V28() V29() ext-real Element of REAL
sin . f1 is V28() V29() ext-real Element of REAL
(cos . f1) + (sin . f1) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
(((1 / 2) (#) ((sin - cos) / exp_R)) `| A) . f1 is V28() V29() ext-real Element of REAL
cos . f1 is V28() V29() ext-real Element of REAL
exp_R . f1 is V28() V29() ext-real Element of REAL
(cos . f1) / (exp_R . f1) is V28() V29() ext-real Element of REAL
K99((exp_R . f1)) is V28() set
K97((cos . f1),K99((exp_R . f1))) is set
(sin - cos) . f1 is V28() V29() ext-real Element of REAL
sin . f1 is V28() V29() ext-real Element of REAL
(sin . f1) - (cos . f1) is V28() V29() ext-real Element of REAL
K98((cos . f1)) is V28() set
K96((sin . f1),K98((cos . f1))) is set
diff (((sin - cos) / exp_R),f1) is V28() V29() ext-real Element of REAL
(1 / 2) * (diff (((sin - cos) / exp_R),f1)) is V28() V29() ext-real Element of REAL
diff ((sin - cos),f1) is V28() V29() ext-real Element of REAL
(diff ((sin - cos),f1)) * (exp_R . f1) is V28() V29() ext-real Element of REAL
diff (exp_R,f1) is V28() V29() ext-real Element of REAL
(diff (exp_R,f1)) * ((sin - cos) . f1) is V28() V29() ext-real Element of REAL
((diff ((sin - cos),f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1)) is V28() V29() ext-real Element of REAL
K98(((diff (exp_R,f1)) * ((sin - cos) . f1))) is V28() set
K96(((diff ((sin - cos),f1)) * (exp_R . f1)),K98(((diff (exp_R,f1)) * ((sin - cos) . f1)))) is set
(exp_R . f1) ^2 is V28() V29() ext-real Element of REAL
K97((exp_R . f1),(exp_R . f1)) is set
(((diff ((sin - cos),f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))) / ((exp_R . f1) ^2) is V28() V29() ext-real Element of REAL
K99(((exp_R . f1) ^2)) is V28() set
K97((((diff ((sin - cos),f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))),K99(((exp_R . f1) ^2))) is set
(1 / 2) * ((((diff ((sin - cos),f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))) / ((exp_R . f1) ^2)) is V28() V29() ext-real Element of REAL
((sin - cos) `| A) . f1 is V28() V29() ext-real Element of REAL
(((sin - cos) `| A) . f1) * (exp_R . f1) is V28() V29() ext-real Element of REAL
((((sin - cos) `| A) . f1) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1)) is V28() V29() ext-real Element of REAL
K96(((((sin - cos) `| A) . f1) * (exp_R . f1)),K98(((diff (exp_R,f1)) * ((sin - cos) . f1)))) is set
(((((sin - cos) `| A) . f1) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))) / ((exp_R . f1) ^2) is V28() V29() ext-real Element of REAL
K97((((((sin - cos) `| A) . f1) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))),K99(((exp_R . f1) ^2))) is set
(1 / 2) * ((((((sin - cos) `| A) . f1) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))) / ((exp_R . f1) ^2)) is V28() V29() ext-real Element of REAL
(cos . f1) + (sin . f1) is V28() V29() ext-real Element of REAL
((cos . f1) + (sin . f1)) * (exp_R . f1) is V28() V29() ext-real Element of REAL
(((cos . f1) + (sin . f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1)) is V28() V29() ext-real Element of REAL
K96((((cos . f1) + (sin . f1)) * (exp_R . f1)),K98(((diff (exp_R,f1)) * ((sin - cos) . f1)))) is set
((((cos . f1) + (sin . f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))) / ((exp_R . f1) ^2) is V28() V29() ext-real Element of REAL
K97(((((cos . f1) + (sin . f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))),K99(((exp_R . f1) ^2))) is set
(1 / 2) * (((((cos . f1) + (sin . f1)) * (exp_R . f1)) - ((diff (exp_R,f1)) * ((sin - cos) . f1))) / ((exp_R . f1) ^2)) is V28() V29() ext-real Element of REAL
(exp_R . f1) * ((sin . f1) - (cos . f1)) is V28() V29() ext-real Element of REAL
(((cos . f1) + (sin . f1)) * (exp_R . f1)) - ((exp_R . f1) * ((sin . f1) - (cos . f1))) is V28() V29() ext-real Element of REAL
K98(((exp_R . f1) * ((sin . f1) - (cos . f1)))) is V28() set
K96((((cos . f1) + (sin . f1)) * (exp_R . f1)),K98(((exp_R . f1) * ((sin . f1) - (cos . f1))))) is set
((((cos . f1) + (sin . f1)) * (exp_R . f1)) - ((exp_R . f1) * ((sin . f1) - (cos . f1)))) / ((exp_R . f1) ^2) is V28() V29() ext-real Element of REAL
K97(((((cos . f1) + (sin . f1)) * (exp_R . f1)) - ((exp_R . f1) * ((sin . f1) - (cos . f1)))),K99(((exp_R . f1) ^2))) is set
(1 / 2) * (((((cos . f1) + (sin . f1)) * (exp_R . f1)) - ((exp_R . f1) * ((sin . f1) - (cos . f1)))) / ((exp_R . f1) ^2)) is V28() V29() ext-real Element of REAL
2 * (cos . f1) is V28() V29() ext-real Element of REAL
(exp_R . f1) * (exp_R . f1) is V28() V29() ext-real Element of REAL
(exp_R . f1) / ((exp_R . f1) * (exp_R . f1)) is V28() V29() ext-real Element of REAL
K99(((exp_R . f1) * (exp_R . f1))) is V28() set
K97((exp_R . f1),K99(((exp_R . f1) * (exp_R . f1)))) is set
(2 * (cos . f1)) * ((exp_R . f1) / ((exp_R . f1) * (exp_R . f1))) is V28() V29() ext-real Element of REAL
(1 / 2) * ((2 * (cos . f1)) * ((exp_R . f1) / ((exp_R . f1) * (exp_R . f1)))) is V28() V29() ext-real Element of REAL
(exp_R . f1) / (exp_R . f1) is V28() V29() ext-real Element of REAL
K97((exp_R . f1),K99((exp_R . f1))) is set
((exp_R . f1) / (exp_R . f1)) / (exp_R . f1) is V28() V29() ext-real Element of REAL
K97(((exp_R . f1) / (exp_R . f1)),K99((exp_R . f1))) is set
(2 * (cos . f1)) * (((exp_R . f1) / (exp_R . f1)) / (exp_R . f1)) is V28() V29() ext-real Element of REAL
(1 / 2) * ((2 * (cos . f1)) * (((exp_R . f1) / (exp_R . f1)) / (exp_R . f1))) is V28() V29() ext-real Element of REAL
1 / (exp_R . f1) is V28() V29() ext-real Element of REAL
K97(1,K99((exp_R . f1))) is set
(2 * (cos . f1)) * (1 / (exp_R . f1)) is V28() V29() ext-real Element of REAL
(1 / 2) * ((2 * (cos . f1)) * (1 / (exp_R . f1))) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
(((1 / 2) (#) ((sin - cos) / exp_R)) `| A) . f1 is V28() V29() ext-real Element of REAL
cos . f1 is V28() V29() ext-real Element of REAL
exp_R . f1 is V28() V29() ext-real Element of REAL
(cos . f1) / (exp_R . f1) is V28() V29() ext-real Element of REAL
K99((exp_R . f1)) is V28() set
K97((cos . f1),K99((exp_R . f1))) is set
cos / exp_R is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
((1 / 2) (#) ((sin - cos) / exp_R)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
((1 / 2) (#) ((sin - cos) / exp_R)) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((1 / 2) (#) ((sin - cos) / exp_R)) . (upper_bound A)) - (((1 / 2) (#) ((sin - cos) / exp_R)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((1 / 2) (#) ((sin - cos) / exp_R)) . (lower_bound A))) is V28() set
K96((((1 / 2) (#) ((sin - cos) / exp_R)) . (upper_bound A)),K98((((1 / 2) (#) ((sin - cos) / exp_R)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
(cos . Z) / (exp_R . Z) is V28() V29() ext-real Element of REAL
K99((exp_R . Z)) is V28() set
K97((cos . Z),K99((exp_R . Z))) is set
((1 / 2) (#) ((sin - cos) / exp_R)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((1 / 2) (#) ((sin - cos) / exp_R)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
(((1 / 2) (#) ((sin - cos) / exp_R)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
(cos . Z) / (exp_R . Z) is V28() V29() ext-real Element of REAL
K99((exp_R . Z)) is V28() set
K97((cos . Z),K99((exp_R . Z))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(exp_R (#) sin) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(exp_R (#) sin) . (lower_bound A) is V28() V29() ext-real Element of REAL
((exp_R (#) sin) . (upper_bound A)) - ((exp_R (#) sin) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((exp_R (#) sin) . (lower_bound A))) is V28() set
K96(((exp_R (#) sin) . (upper_bound A)),K98(((exp_R (#) sin) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom exp_R is V51() V52() V53() Element of K19(REAL)
(dom exp_R) /\ (dom (sin + cos)) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(sin . Z) + (cos . Z) is V28() V29() ext-real Element of REAL
(exp_R . Z) * ((sin . Z) + (cos . Z)) is V28() V29() ext-real Element of REAL
(exp_R (#) (sin + cos)) . Z is V28() V29() ext-real Element of REAL
(sin + cos) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) * ((sin + cos) . Z) is V28() V29() ext-real Element of REAL
(exp_R (#) sin) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R (#) sin) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((exp_R (#) sin) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(sin . Z) + (cos . Z) is V28() V29() ext-real Element of REAL
(exp_R . Z) * ((sin . Z) + (cos . Z)) is V28() V29() ext-real Element of REAL
exp_R (#) (cos - sin) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(exp_R (#) cos) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(exp_R (#) cos) . (lower_bound A) is V28() V29() ext-real Element of REAL
((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((exp_R (#) cos) . (lower_bound A))) is V28() set
K96(((exp_R (#) cos) . (upper_bound A)),K98(((exp_R (#) cos) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom exp_R is V51() V52() V53() Element of K19(REAL)
(dom exp_R) /\ (dom (cos - sin)) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(cos . Z) - (sin . Z) is V28() V29() ext-real Element of REAL
K98((sin . Z)) is V28() set
K96((cos . Z),K98((sin . Z))) is set
(exp_R . Z) * ((cos . Z) - (sin . Z)) is V28() V29() ext-real Element of REAL
(exp_R (#) (cos - sin)) . Z is V28() V29() ext-real Element of REAL
(cos - sin) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) * ((cos - sin) . Z) is V28() V29() ext-real Element of REAL
(exp_R (#) cos) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R (#) cos) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((exp_R (#) cos) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(cos . Z) - (sin . Z) is V28() V29() ext-real Element of REAL
K98((sin . Z)) is V28() set
K96((cos . Z),K98((sin . Z))) is set
(exp_R . Z) * ((cos . Z) - (sin . Z)) is V28() V29() ext-real Element of REAL
cos ^2 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
cos (#) cos is V1() V6() set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin / cos) / f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((sin / cos) / f1) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((sin / cos) / f1) is V1() V6() set
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) ^) / (cos ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(- ((sin / cos) / f1)) + (((id Z) ^) / (cos ^2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) ^) (#) tan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id Z) ^) (#) tan) is V51() V52() V53() Element of K19(REAL)
(((id Z) ^) (#) tan) . (upper_bound A) is V28() V29() ext-real Element of REAL
(((id Z) ^) (#) tan) . (lower_bound A) is V28() V29() ext-real Element of REAL
((((id Z) ^) (#) tan) . (upper_bound A)) - ((((id Z) ^) (#) tan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((((id Z) ^) (#) tan) . (lower_bound A))) is V28() set
K96(((((id Z) ^) (#) tan) . (upper_bound A)),K98(((((id Z) ^) (#) tan) . (lower_bound A)))) is set
dom ((id Z) ^) is V51() V52() V53() Element of K19(REAL)
(dom ((id Z) ^)) /\ (dom tan) is V51() V52() V53() Element of K19(REAL)
dom (id Z) is V51() V52() V53() Element of K19(Z)
K19(Z) is set
(id Z) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id Z)) \ ((id Z) " {0}) is V51() V52() V53() Element of K19(Z)
(dom (id Z)) \ {0} is V51() V52() V53() Element of K19(Z)
dom (- ((sin / cos) / f1)) is V51() V52() V53() Element of K19(REAL)
dom (((id Z) ^) / (cos ^2)) is V51() V52() V53() Element of K19(REAL)
(dom (- ((sin / cos) / f1))) /\ (dom (((id Z) ^) / (cos ^2))) is V51() V52() V53() Element of K19(REAL)
dom ((sin / cos) / f1) is V51() V52() V53() Element of K19(REAL)
dom (sin / cos) is V51() V52() V53() Element of K19(REAL)
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 " {0} is V51() V52() V53() Element of K19(REAL)
(dom f1) \ (f1 " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (sin / cos)) /\ ((dom f1) \ (f1 " {0})) is V51() V52() V53() Element of K19(REAL)
dom (cos ^2) is V51() V52() V53() Element of K19(REAL)
(cos ^2) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (cos ^2)) \ ((cos ^2) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom ((id Z) ^)) /\ ((dom (cos ^2)) \ ((cos ^2) " {0})) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
f . h is V28() V29() ext-real Element of REAL
sin . h is V28() V29() ext-real Element of REAL
cos . h is V28() V29() ext-real Element of REAL
(sin . h) / (cos . h) is V28() V29() ext-real Element of REAL
K99((cos . h)) is V28() set
K97((sin . h),K99((cos . h))) is set
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
((sin . h) / (cos . h)) / (h ^2) is V28() V29() ext-real Element of REAL
K99((h ^2)) is V28() set
K97(((sin . h) / (cos . h)),K99((h ^2))) is set
- (((sin . h) / (cos . h)) / (h ^2)) is V28() V29() ext-real Element of REAL
1 / h is V28() V29() ext-real Element of REAL
K99(h) is V28() set
K97(1,K99(h)) is set
(cos . h) ^2 is V28() V29() ext-real Element of REAL
K97((cos . h),(cos . h)) is set
(1 / h) / ((cos . h) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . h) ^2)) is V28() set
K97((1 / h),K99(((cos . h) ^2))) is set
(- (((sin . h) / (cos . h)) / (h ^2))) + ((1 / h) / ((cos . h) ^2)) is V28() V29() ext-real Element of REAL
((- ((sin / cos) / f1)) + (((id Z) ^) / (cos ^2))) . h is V28() V29() ext-real Element of REAL
(- ((sin / cos) / f1)) . h is V28() V29() ext-real Element of REAL
(((id Z) ^) / (cos ^2)) . h is V28() V29() ext-real Element of REAL
((- ((sin / cos) / f1)) . h) + ((((id Z) ^) / (cos ^2)) . h) is V28() V29() ext-real Element of REAL
((sin / cos) / f1) . h is V28() V29() ext-real Element of REAL
- (((sin / cos) / f1) . h) is V28() V29() ext-real Element of REAL
(- (((sin / cos) / f1) . h)) + ((((id Z) ^) / (cos ^2)) . h) is V28() V29() ext-real Element of REAL
(sin / cos) . h is V28() V29() ext-real Element of REAL
f1 . h is V28() V29() ext-real Element of REAL
((sin / cos) . h) / (f1 . h) is V28() V29() ext-real Element of REAL
K99((f1 . h)) is V28() set
K97(((sin / cos) . h),K99((f1 . h))) is set
- (((sin / cos) . h) / (f1 . h)) is V28() V29() ext-real Element of REAL
(- (((sin / cos) . h) / (f1 . h))) + ((((id Z) ^) / (cos ^2)) . h) is V28() V29() ext-real Element of REAL
(cos . h) " is V28() V29() ext-real Element of REAL
(sin . h) * ((cos . h) ") is V28() V29() ext-real Element of REAL
((sin . h) * ((cos . h) ")) / (f1 . h) is V28() V29() ext-real Element of REAL
K97(((sin . h) * ((cos . h) ")),K99((f1 . h))) is set
- (((sin . h) * ((cos . h) ")) / (f1 . h)) is V28() V29() ext-real Element of REAL
(- (((sin . h) * ((cos . h) ")) / (f1 . h))) + ((((id Z) ^) / (cos ^2)) . h) is V28() V29() ext-real Element of REAL
((sin . h) / (cos . h)) / (f1 . h) is V28() V29() ext-real Element of REAL
K97(((sin . h) / (cos . h)),K99((f1 . h))) is set
- (((sin . h) / (cos . h)) / (f1 . h)) is V28() V29() ext-real Element of REAL
((id Z) ^) . h is V28() V29() ext-real Element of REAL
(cos ^2) . h is V28() V29() ext-real Element of REAL
(((id Z) ^) . h) / ((cos ^2) . h) is V28() V29() ext-real Element of REAL
K99(((cos ^2) . h)) is V28() set
K97((((id Z) ^) . h),K99(((cos ^2) . h))) is set
(- (((sin . h) / (cos . h)) / (f1 . h))) + ((((id Z) ^) . h) / ((cos ^2) . h)) is V28() V29() ext-real Element of REAL
(id Z) . h is V28() V29() ext-real Element of REAL
((id Z) . h) " is V28() V29() ext-real Element of REAL
(((id Z) . h) ") / ((cos ^2) . h) is V28() V29() ext-real Element of REAL
K97((((id Z) . h) "),K99(((cos ^2) . h))) is set
(- (((sin . h) / (cos . h)) / (f1 . h))) + ((((id Z) . h) ") / ((cos ^2) . h)) is V28() V29() ext-real Element of REAL
(1 / h) / ((cos ^2) . h) is V28() V29() ext-real Element of REAL
K97((1 / h),K99(((cos ^2) . h))) is set
(- (((sin . h) / (cos . h)) / (f1 . h))) + ((1 / h) / ((cos ^2) . h)) is V28() V29() ext-real Element of REAL
(- (((sin . h) / (cos . h)) / (f1 . h))) + ((1 / h) / ((cos . h) ^2)) is V28() V29() ext-real Element of REAL
h #Z 2 is V28() V29() ext-real Element of REAL
((sin . h) / (cos . h)) / (h #Z 2) is V28() V29() ext-real Element of REAL
K99((h #Z 2)) is V28() set
K97(((sin . h) / (cos . h)),K99((h #Z 2))) is set
- (((sin . h) / (cos . h)) / (h #Z 2)) is V28() V29() ext-real Element of REAL
(- (((sin . h) / (cos . h)) / (h #Z 2))) + ((1 / h) / ((cos . h) ^2)) is V28() V29() ext-real Element of REAL
(((id Z) ^) (#) tan) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((((id Z) ^) (#) tan) `| Z) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
((((id Z) ^) (#) tan) `| Z) . h is V28() V29() ext-real Element of REAL
f . h is V28() V29() ext-real Element of REAL
sin . h is V28() V29() ext-real Element of REAL
cos . h is V28() V29() ext-real Element of REAL
(sin . h) / (cos . h) is V28() V29() ext-real Element of REAL
K99((cos . h)) is V28() set
K97((sin . h),K99((cos . h))) is set
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
((sin . h) / (cos . h)) / (h ^2) is V28() V29() ext-real Element of REAL
K99((h ^2)) is V28() set
K97(((sin . h) / (cos . h)),K99((h ^2))) is set
- (((sin . h) / (cos . h)) / (h ^2)) is V28() V29() ext-real Element of REAL
1 / h is V28() V29() ext-real Element of REAL
K99(h) is V28() set
K97(1,K99(h)) is set
(cos . h) ^2 is V28() V29() ext-real Element of REAL
K97((cos . h),(cos . h)) is set
(1 / h) / ((cos . h) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . h) ^2)) is V28() set
K97((1 / h),K99(((cos . h) ^2))) is set
(- (((sin . h) / (cos . h)) / (h ^2))) + ((1 / h) / ((cos . h) ^2)) is V28() V29() ext-real Element of REAL
sin ^2 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
sin (#) sin is V1() V6() set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos / sin) / f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((cos / sin) / f) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((cos / sin) / f) is V1() V6() set
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) ^) / (sin ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(- ((cos / sin) / f)) - (((id Z) ^) / (sin ^2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (((id Z) ^) / (sin ^2)) is V1() V6() V34() set
K98(1) (#) (((id Z) ^) / (sin ^2)) is V1() V6() set
(- ((cos / sin) / f)) + (- (((id Z) ^) / (sin ^2))) is V1() V6() set
((id Z) ^) (#) cot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id Z) ^) (#) cot) is V51() V52() V53() Element of K19(REAL)
(((id Z) ^) (#) cot) . (upper_bound A) is V28() V29() ext-real Element of REAL
(((id Z) ^) (#) cot) . (lower_bound A) is V28() V29() ext-real Element of REAL
((((id Z) ^) (#) cot) . (upper_bound A)) - ((((id Z) ^) (#) cot) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((((id Z) ^) (#) cot) . (lower_bound A))) is V28() set
K96(((((id Z) ^) (#) cot) . (upper_bound A)),K98(((((id Z) ^) (#) cot) . (lower_bound A)))) is set
dom ((id Z) ^) is V51() V52() V53() Element of K19(REAL)
(dom ((id Z) ^)) /\ (dom cot) is V51() V52() V53() Element of K19(REAL)
dom (id Z) is V51() V52() V53() Element of K19(Z)
K19(Z) is set
(id Z) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id Z)) \ ((id Z) " {0}) is V51() V52() V53() Element of K19(Z)
(dom (id Z)) \ {0} is V51() V52() V53() Element of K19(Z)
dom (- ((cos / sin) / f)) is V51() V52() V53() Element of K19(REAL)
dom (((id Z) ^) / (sin ^2)) is V51() V52() V53() Element of K19(REAL)
(dom (- ((cos / sin) / f))) /\ (dom (((id Z) ^) / (sin ^2))) is V51() V52() V53() Element of K19(REAL)
dom ((cos / sin) / f) is V51() V52() V53() Element of K19(REAL)
dom (cos / sin) is V51() V52() V53() Element of K19(REAL)
dom f is V51() V52() V53() Element of K19(REAL)
f " {0} is V51() V52() V53() Element of K19(REAL)
(dom f) \ (f " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (cos / sin)) /\ ((dom f) \ (f " {0})) is V51() V52() V53() Element of K19(REAL)
dom (sin ^2) is V51() V52() V53() Element of K19(REAL)
(sin ^2) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (sin ^2)) \ ((sin ^2) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom ((id Z) ^)) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
f1 . h is V28() V29() ext-real Element of REAL
cos . h is V28() V29() ext-real Element of REAL
sin . h is V28() V29() ext-real Element of REAL
(cos . h) / (sin . h) is V28() V29() ext-real Element of REAL
K99((sin . h)) is V28() set
K97((cos . h),K99((sin . h))) is set
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
((cos . h) / (sin . h)) / (h ^2) is V28() V29() ext-real Element of REAL
K99((h ^2)) is V28() set
K97(((cos . h) / (sin . h)),K99((h ^2))) is set
- (((cos . h) / (sin . h)) / (h ^2)) is V28() V29() ext-real Element of REAL
1 / h is V28() V29() ext-real Element of REAL
K99(h) is V28() set
K97(1,K99(h)) is set
(sin . h) ^2 is V28() V29() ext-real Element of REAL
K97((sin . h),(sin . h)) is set
(1 / h) / ((sin . h) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . h) ^2)) is V28() set
K97((1 / h),K99(((sin . h) ^2))) is set
(- (((cos . h) / (sin . h)) / (h ^2))) - ((1 / h) / ((sin . h) ^2)) is V28() V29() ext-real Element of REAL
K98(((1 / h) / ((sin . h) ^2))) is V28() set
K96((- (((cos . h) / (sin . h)) / (h ^2))),K98(((1 / h) / ((sin . h) ^2)))) is set
((- ((cos / sin) / f)) - (((id Z) ^) / (sin ^2))) . h is V28() V29() ext-real Element of REAL
(- ((cos / sin) / f)) . h is V28() V29() ext-real Element of REAL
(((id Z) ^) / (sin ^2)) . h is V28() V29() ext-real Element of REAL
((- ((cos / sin) / f)) . h) - ((((id Z) ^) / (sin ^2)) . h) is V28() V29() ext-real Element of REAL
K98(((((id Z) ^) / (sin ^2)) . h)) is V28() set
K96(((- ((cos / sin) / f)) . h),K98(((((id Z) ^) / (sin ^2)) . h))) is set
((cos / sin) / f) . h is V28() V29() ext-real Element of REAL
- (((cos / sin) / f) . h) is V28() V29() ext-real Element of REAL
(- (((cos / sin) / f) . h)) - ((((id Z) ^) / (sin ^2)) . h) is V28() V29() ext-real Element of REAL
K96((- (((cos / sin) / f) . h)),K98(((((id Z) ^) / (sin ^2)) . h))) is set
(cos / sin) . h is V28() V29() ext-real Element of REAL
f . h is V28() V29() ext-real Element of REAL
((cos / sin) . h) / (f . h) is V28() V29() ext-real Element of REAL
K99((f . h)) is V28() set
K97(((cos / sin) . h),K99((f . h))) is set
- (((cos / sin) . h) / (f . h)) is V28() V29() ext-real Element of REAL
(- (((cos / sin) . h) / (f . h))) - ((((id Z) ^) / (sin ^2)) . h) is V28() V29() ext-real Element of REAL
K96((- (((cos / sin) . h) / (f . h))),K98(((((id Z) ^) / (sin ^2)) . h))) is set
((cos . h) / (sin . h)) / (f . h) is V28() V29() ext-real Element of REAL
K97(((cos . h) / (sin . h)),K99((f . h))) is set
- (((cos . h) / (sin . h)) / (f . h)) is V28() V29() ext-real Element of REAL
(- (((cos . h) / (sin . h)) / (f . h))) - ((((id Z) ^) / (sin ^2)) . h) is V28() V29() ext-real Element of REAL
K96((- (((cos . h) / (sin . h)) / (f . h))),K98(((((id Z) ^) / (sin ^2)) . h))) is set
((id Z) ^) . h is V28() V29() ext-real Element of REAL
(sin ^2) . h is V28() V29() ext-real Element of REAL
(((id Z) ^) . h) / ((sin ^2) . h) is V28() V29() ext-real Element of REAL
K99(((sin ^2) . h)) is V28() set
K97((((id Z) ^) . h),K99(((sin ^2) . h))) is set
(- (((cos . h) / (sin . h)) / (f . h))) - ((((id Z) ^) . h) / ((sin ^2) . h)) is V28() V29() ext-real Element of REAL
K98(((((id Z) ^) . h) / ((sin ^2) . h))) is V28() set
K96((- (((cos . h) / (sin . h)) / (f . h))),K98(((((id Z) ^) . h) / ((sin ^2) . h)))) is set
(id Z) . h is V28() V29() ext-real Element of REAL
((id Z) . h) " is V28() V29() ext-real Element of REAL
(((id Z) . h) ") / ((sin ^2) . h) is V28() V29() ext-real Element of REAL
K97((((id Z) . h) "),K99(((sin ^2) . h))) is set
(- (((cos . h) / (sin . h)) / (f . h))) - ((((id Z) . h) ") / ((sin ^2) . h)) is V28() V29() ext-real Element of REAL
K98(((((id Z) . h) ") / ((sin ^2) . h))) is V28() set
K96((- (((cos . h) / (sin . h)) / (f . h))),K98(((((id Z) . h) ") / ((sin ^2) . h)))) is set
(1 / h) / ((sin ^2) . h) is V28() V29() ext-real Element of REAL
K97((1 / h),K99(((sin ^2) . h))) is set
(- (((cos . h) / (sin . h)) / (f . h))) - ((1 / h) / ((sin ^2) . h)) is V28() V29() ext-real Element of REAL
K98(((1 / h) / ((sin ^2) . h))) is V28() set
K96((- (((cos . h) / (sin . h)) / (f . h))),K98(((1 / h) / ((sin ^2) . h)))) is set
(- (((cos . h) / (sin . h)) / (f . h))) - ((1 / h) / ((sin . h) ^2)) is V28() V29() ext-real Element of REAL
K96((- (((cos . h) / (sin . h)) / (f . h))),K98(((1 / h) / ((sin . h) ^2)))) is set
h #Z 2 is V28() V29() ext-real Element of REAL
((cos . h) / (sin . h)) / (h #Z 2) is V28() V29() ext-real Element of REAL
K99((h #Z 2)) is V28() set
K97(((cos . h) / (sin . h)),K99((h #Z 2))) is set
- (((cos . h) / (sin . h)) / (h #Z 2)) is V28() V29() ext-real Element of REAL
(- (((cos . h) / (sin . h)) / (h #Z 2))) - ((1 / h) / ((sin . h) ^2)) is V28() V29() ext-real Element of REAL
K96((- (((cos . h) / (sin . h)) / (h #Z 2))),K98(((1 / h) / ((sin . h) ^2)))) is set
(((id Z) ^) (#) cot) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((((id Z) ^) (#) cot) `| Z) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
((((id Z) ^) (#) cot) `| Z) . h is V28() V29() ext-real Element of REAL
f1 . h is V28() V29() ext-real Element of REAL
cos . h is V28() V29() ext-real Element of REAL
sin . h is V28() V29() ext-real Element of REAL
(cos . h) / (sin . h) is V28() V29() ext-real Element of REAL
K99((sin . h)) is V28() set
K97((cos . h),K99((sin . h))) is set
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
((cos . h) / (sin . h)) / (h ^2) is V28() V29() ext-real Element of REAL
K99((h ^2)) is V28() set
K97(((cos . h) / (sin . h)),K99((h ^2))) is set
- (((cos . h) / (sin . h)) / (h ^2)) is V28() V29() ext-real Element of REAL
1 / h is V28() V29() ext-real Element of REAL
K99(h) is V28() set
K97(1,K99(h)) is set
(sin . h) ^2 is V28() V29() ext-real Element of REAL
K97((sin . h),(sin . h)) is set
(1 / h) / ((sin . h) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . h) ^2)) is V28() set
K97((1 / h),K99(((sin . h) ^2))) is set
(- (((cos . h) / (sin . h)) / (h ^2))) - ((1 / h) / ((sin . h) ^2)) is V28() V29() ext-real Element of REAL
K98(((1 / h) / ((sin . h) ^2))) is V28() set
K96((- (((cos . h) / (sin . h)) / (h ^2))),K98(((1 / h) / ((sin . h) ^2)))) is set
ln / (cos ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln (#) tan) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln (#) tan) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln (#) tan) . (upper_bound A)) - ((ln (#) tan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln (#) tan) . (lower_bound A))) is V28() set
K96(((ln (#) tan) . (upper_bound A)),K98(((ln (#) tan) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin / cos) / (id f) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((sin / cos) / (id f)) + (ln / (cos ^2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin / cos) / (id f)) is V51() V52() V53() Element of K19(REAL)
dom (ln / (cos ^2)) is V51() V52() V53() Element of K19(REAL)
(dom ((sin / cos) / (id f))) /\ (dom (ln / (cos ^2))) is V51() V52() V53() Element of K19(REAL)
dom (sin / cos) is V51() V52() V53() Element of K19(REAL)
dom (id f) is V51() V52() V53() Element of K19(f)
K19(f) is set
(id f) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id f)) \ ((id f) " {0}) is V51() V52() V53() Element of K19(f)
(dom (sin / cos)) /\ ((dom (id f)) \ ((id f) " {0})) is V51() V52() V53() Element of K19(f)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(sin . Z) / (cos . Z) is V28() V29() ext-real Element of REAL
K99((cos . Z)) is V28() set
K97((sin . Z),K99((cos . Z))) is set
((sin . Z) / (cos . Z)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97(((sin . Z) / (cos . Z)),K99(Z)) is set
ln . Z is V28() V29() ext-real Element of REAL
(cos . Z) ^2 is V28() V29() ext-real Element of REAL
K97((cos . Z),(cos . Z)) is set
(ln . Z) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() set
K97((ln . Z),K99(((cos . Z) ^2))) is set
(((sin . Z) / (cos . Z)) / Z) + ((ln . Z) / ((cos . Z) ^2)) is V28() V29() ext-real Element of REAL
(((sin / cos) / (id f)) + (ln / (cos ^2))) . Z is V28() V29() ext-real Element of REAL
((sin / cos) / (id f)) . Z is V28() V29() ext-real Element of REAL
(ln / (cos ^2)) . Z is V28() V29() ext-real Element of REAL
(((sin / cos) / (id f)) . Z) + ((ln / (cos ^2)) . Z) is V28() V29() ext-real Element of REAL
(sin / cos) . Z is V28() V29() ext-real Element of REAL
(id f) . Z is V28() V29() ext-real Element of REAL
((sin / cos) . Z) / ((id f) . Z) is V28() V29() ext-real Element of REAL
K99(((id f) . Z)) is V28() set
K97(((sin / cos) . Z),K99(((id f) . Z))) is set
(((sin / cos) . Z) / ((id f) . Z)) + ((ln / (cos ^2)) . Z) is V28() V29() ext-real Element of REAL
((sin . Z) / (cos . Z)) / ((id f) . Z) is V28() V29() ext-real Element of REAL
K97(((sin . Z) / (cos . Z)),K99(((id f) . Z))) is set
(((sin . Z) / (cos . Z)) / ((id f) . Z)) + ((ln / (cos ^2)) . Z) is V28() V29() ext-real Element of REAL
(((sin . Z) / (cos . Z)) / Z) + ((ln / (cos ^2)) . Z) is V28() V29() ext-real Element of REAL
(cos ^2) . Z is V28() V29() ext-real Element of REAL
(ln . Z) / ((cos ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((cos ^2) . Z)) is V28() set
K97((ln . Z),K99(((cos ^2) . Z))) is set
(((sin . Z) / (cos . Z)) / Z) + ((ln . Z) / ((cos ^2) . Z)) is V28() V29() ext-real Element of REAL
(ln (#) tan) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln (#) tan) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((ln (#) tan) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(sin . Z) / (cos . Z) is V28() V29() ext-real Element of REAL
K99((cos . Z)) is V28() set
K97((sin . Z),K99((cos . Z))) is set
((sin . Z) / (cos . Z)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97(((sin . Z) / (cos . Z)),K99(Z)) is set
ln . Z is V28() V29() ext-real Element of REAL
(cos . Z) ^2 is V28() V29() ext-real Element of REAL
K97((cos . Z),(cos . Z)) is set
(ln . Z) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() set
K97((ln . Z),K99(((cos . Z) ^2))) is set
(((sin . Z) / (cos . Z)) / Z) + ((ln . Z) / ((cos . Z) ^2)) is V28() V29() ext-real Element of REAL
ln / (sin ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln (#) cot) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln (#) cot) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln (#) cot) . (upper_bound A)) - ((ln (#) cot) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln (#) cot) . (lower_bound A))) is V28() set
K96(((ln (#) cot) . (upper_bound A)),K98(((ln (#) cot) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos / sin) / (id f) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((cos / sin) / (id f)) - (ln / (sin ^2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (ln / (sin ^2)) is V1() V6() V34() set
K98(1) (#) (ln / (sin ^2)) is V1() V6() set
((cos / sin) / (id f)) + (- (ln / (sin ^2))) is V1() V6() set
dom ((cos / sin) / (id f)) is V51() V52() V53() Element of K19(REAL)
dom (ln / (sin ^2)) is V51() V52() V53() Element of K19(REAL)
(dom ((cos / sin) / (id f))) /\ (dom (ln / (sin ^2))) is V51() V52() V53() Element of K19(REAL)
dom (cos / sin) is V51() V52() V53() Element of K19(REAL)
dom (id f) is V51() V52() V53() Element of K19(f)
K19(f) is set
(id f) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id f)) \ ((id f) " {0}) is V51() V52() V53() Element of K19(f)
(dom (cos / sin)) /\ ((dom (id f)) \ ((id f) " {0})) is V51() V52() V53() Element of K19(f)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(cos . Z) / (sin . Z) is V28() V29() ext-real Element of REAL
K99((sin . Z)) is V28() set
K97((cos . Z),K99((sin . Z))) is set
((cos . Z) / (sin . Z)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97(((cos . Z) / (sin . Z)),K99(Z)) is set
ln . Z is V28() V29() ext-real Element of REAL
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is set
(ln . Z) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() set
K97((ln . Z),K99(((sin . Z) ^2))) is set
(((cos . Z) / (sin . Z)) / Z) - ((ln . Z) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
K98(((ln . Z) / ((sin . Z) ^2))) is V28() set
K96((((cos . Z) / (sin . Z)) / Z),K98(((ln . Z) / ((sin . Z) ^2)))) is set
(((cos / sin) / (id f)) - (ln / (sin ^2))) . Z is V28() V29() ext-real Element of REAL
((cos / sin) / (id f)) . Z is V28() V29() ext-real Element of REAL
(ln / (sin ^2)) . Z is V28() V29() ext-real Element of REAL
(((cos / sin) / (id f)) . Z) - ((ln / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K98(((ln / (sin ^2)) . Z)) is V28() set
K96((((cos / sin) / (id f)) . Z),K98(((ln / (sin ^2)) . Z))) is set
(cos / sin) . Z is V28() V29() ext-real Element of REAL
(id f) . Z is V28() V29() ext-real Element of REAL
((cos / sin) . Z) / ((id f) . Z) is V28() V29() ext-real Element of REAL
K99(((id f) . Z)) is V28() set
K97(((cos / sin) . Z),K99(((id f) . Z))) is set
(((cos / sin) . Z) / ((id f) . Z)) - ((ln / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K96((((cos / sin) . Z) / ((id f) . Z)),K98(((ln / (sin ^2)) . Z))) is set
((cos . Z) / (sin . Z)) / ((id f) . Z) is V28() V29() ext-real Element of REAL
K97(((cos . Z) / (sin . Z)),K99(((id f) . Z))) is set
(((cos . Z) / (sin . Z)) / ((id f) . Z)) - ((ln / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K96((((cos . Z) / (sin . Z)) / ((id f) . Z)),K98(((ln / (sin ^2)) . Z))) is set
(((cos . Z) / (sin . Z)) / Z) - ((ln / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K96((((cos . Z) / (sin . Z)) / Z),K98(((ln / (sin ^2)) . Z))) is set
(sin ^2) . Z is V28() V29() ext-real Element of REAL
(ln . Z) / ((sin ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((sin ^2) . Z)) is V28() set
K97((ln . Z),K99(((sin ^2) . Z))) is set
(((cos . Z) / (sin . Z)) / Z) - ((ln . Z) / ((sin ^2) . Z)) is V28() V29() ext-real Element of REAL
K98(((ln . Z) / ((sin ^2) . Z))) is V28() set
K96((((cos . Z) / (sin . Z)) / Z),K98(((ln . Z) / ((sin ^2) . Z)))) is set
(ln (#) cot) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln (#) cot) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((ln (#) cot) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(cos . Z) / (sin . Z) is V28() V29() ext-real Element of REAL
K99((sin . Z)) is V28() set
K97((cos . Z),K99((sin . Z))) is set
((cos . Z) / (sin . Z)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97(((cos . Z) / (sin . Z)),K99(Z)) is set
ln . Z is V28() V29() ext-real Element of REAL
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is set
(ln . Z) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() set
K97((ln . Z),K99(((sin . Z) ^2))) is set
(((cos . Z) / (sin . Z)) / Z) - ((ln . Z) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
K98(((ln . Z) / ((sin . Z) ^2))) is V28() set
K96((((cos . Z) / (sin . Z)) / Z),K98(((ln . Z) / ((sin . Z) ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln (#) tan) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln (#) tan) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln (#) tan) . (upper_bound A)) - ((ln (#) tan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln (#) tan) . (lower_bound A))) is V28() set
K96(((ln (#) tan) . (upper_bound A)),K98(((ln (#) tan) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
tan / (id f) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(tan / (id f)) + (ln / (cos ^2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (tan / (id f)) is V51() V52() V53() Element of K19(REAL)
dom (ln / (cos ^2)) is V51() V52() V53() Element of K19(REAL)
(dom (tan / (id f))) /\ (dom (ln / (cos ^2))) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
tan . Z is V28() V29() ext-real Element of REAL
(tan . Z) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97((tan . Z),K99(Z)) is set
ln . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(cos . Z) ^2 is V28() V29() ext-real Element of REAL
K97((cos . Z),(cos . Z)) is set
(ln . Z) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() set
K97((ln . Z),K99(((cos . Z) ^2))) is set
((tan . Z) / Z) + ((ln . Z) / ((cos . Z) ^2)) is V28() V29() ext-real Element of REAL
((tan / (id f)) + (ln / (cos ^2))) . Z is V28() V29() ext-real Element of REAL
(tan / (id f)) . Z is V28() V29() ext-real Element of REAL
(ln / (cos ^2)) . Z is V28() V29() ext-real Element of REAL
((tan / (id f)) . Z) + ((ln / (cos ^2)) . Z) is V28() V29() ext-real Element of REAL
(id f) . Z is V28() V29() ext-real Element of REAL
(tan . Z) / ((id f) . Z) is V28() V29() ext-real Element of REAL
K99(((id f) . Z)) is V28() set
K97((tan . Z),K99(((id f) . Z))) is set
((tan . Z) / ((id f) . Z)) + ((ln / (cos ^2)) . Z) is V28() V29() ext-real Element of REAL
((tan . Z) / Z) + ((ln / (cos ^2)) . Z) is V28() V29() ext-real Element of REAL
(cos ^2) . Z is V28() V29() ext-real Element of REAL
(ln . Z) / ((cos ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((cos ^2) . Z)) is V28() set
K97((ln . Z),K99(((cos ^2) . Z))) is set
((tan . Z) / Z) + ((ln . Z) / ((cos ^2) . Z)) is V28() V29() ext-real Element of REAL
(ln (#) tan) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln (#) tan) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((ln (#) tan) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
tan Z is V28() V29() ext-real Element of REAL
sin Z is set
sin . Z is V28() V29() ext-real Element of REAL
cos Z is set
cos . Z is V28() V29() ext-real Element of REAL
K101((sin Z),(cos Z)) is set
(tan Z) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97((tan Z),K99(Z)) is set
ln . Z is V28() V29() ext-real Element of REAL
(cos . Z) ^2 is V28() V29() ext-real Element of REAL
K97((cos . Z),(cos . Z)) is set
(ln . Z) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() set
K97((ln . Z),K99(((cos . Z) ^2))) is set
((tan Z) / Z) + ((ln . Z) / ((cos . Z) ^2)) is V28() V29() ext-real Element of REAL
tan . Z is V28() V29() ext-real Element of REAL
(tan . Z) / Z is V28() V29() ext-real Element of REAL
K97((tan . Z),K99(Z)) is set
((tan . Z) / Z) + ((ln . Z) / ((cos . Z) ^2)) is V28() V29() ext-real Element of REAL
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln (#) cot) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln (#) cot) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln (#) cot) . (upper_bound A)) - ((ln (#) cot) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln (#) cot) . (lower_bound A))) is V28() set
K96(((ln (#) cot) . (upper_bound A)),K98(((ln (#) cot) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
cot / (id f) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cot / (id f)) - (ln / (sin ^2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (ln / (sin ^2)) is V1() V6() V34() set
K98(1) (#) (ln / (sin ^2)) is V1() V6() set
(cot / (id f)) + (- (ln / (sin ^2))) is V1() V6() set
dom (cot / (id f)) is V51() V52() V53() Element of K19(REAL)
dom (ln / (sin ^2)) is V51() V52() V53() Element of K19(REAL)
(dom (cot / (id f))) /\ (dom (ln / (sin ^2))) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cot . Z is V28() V29() ext-real Element of REAL
(cot . Z) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97((cot . Z),K99(Z)) is set
ln . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is set
(ln . Z) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() set
K97((ln . Z),K99(((sin . Z) ^2))) is set
((cot . Z) / Z) - ((ln . Z) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
K98(((ln . Z) / ((sin . Z) ^2))) is V28() set
K96(((cot . Z) / Z),K98(((ln . Z) / ((sin . Z) ^2)))) is set
((cot / (id f)) - (ln / (sin ^2))) . Z is V28() V29() ext-real Element of REAL
(cot / (id f)) . Z is V28() V29() ext-real Element of REAL
(ln / (sin ^2)) . Z is V28() V29() ext-real Element of REAL
((cot / (id f)) . Z) - ((ln / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K98(((ln / (sin ^2)) . Z)) is V28() set
K96(((cot / (id f)) . Z),K98(((ln / (sin ^2)) . Z))) is set
(id f) . Z is V28() V29() ext-real Element of REAL
(cot . Z) / ((id f) . Z) is V28() V29() ext-real Element of REAL
K99(((id f) . Z)) is V28() set
K97((cot . Z),K99(((id f) . Z))) is set
((cot . Z) / ((id f) . Z)) - ((ln / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K96(((cot . Z) / ((id f) . Z)),K98(((ln / (sin ^2)) . Z))) is set
((cot . Z) / Z) - ((ln / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K96(((cot . Z) / Z),K98(((ln / (sin ^2)) . Z))) is set
(sin ^2) . Z is V28() V29() ext-real Element of REAL
(ln . Z) / ((sin ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((sin ^2) . Z)) is V28() set
K97((ln . Z),K99(((sin ^2) . Z))) is set
((cot . Z) / Z) - ((ln . Z) / ((sin ^2) . Z)) is V28() V29() ext-real Element of REAL
K98(((ln . Z) / ((sin ^2) . Z))) is V28() set
K96(((cot . Z) / Z),K98(((ln . Z) / ((sin ^2) . Z)))) is set
(ln (#) cot) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln (#) cot) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((ln (#) cot) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cot Z is V28() V29() ext-real Element of REAL
cos Z is set
cos . Z is V28() V29() ext-real Element of REAL
sin Z is set
sin . Z is V28() V29() ext-real Element of REAL
K101((cos Z),(sin Z)) is set
(cot Z) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97((cot Z),K99(Z)) is set
ln . Z is V28() V29() ext-real Element of REAL
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is set
(ln . Z) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() set
K97((ln . Z),K99(((sin . Z) ^2))) is set
((cot Z) / Z) - ((ln . Z) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
K98(((ln . Z) / ((sin . Z) ^2))) is V28() set
K96(((cot Z) / Z),K98(((ln . Z) / ((sin . Z) ^2)))) is set
cot . Z is V28() V29() ext-real Element of REAL
(cot . Z) / Z is V28() V29() ext-real Element of REAL
K97((cot . Z),K99(Z)) is set
((cot . Z) / Z) - ((ln . Z) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
K96(((cot . Z) / Z),K98(((ln . Z) / ((sin . Z) ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln (#) arctan) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln (#) arctan) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln (#) arctan) . (upper_bound A)) - ((ln (#) arctan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln (#) arctan) . (lower_bound A))) is V28() set
K96(((ln (#) arctan) . (upper_bound A)),K98(((ln (#) arctan) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
ln / (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arctan / (id Z) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arctan / (id Z)) + (ln / (f1 + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (arctan / (id Z)) is V51() V52() V53() Element of K19(REAL)
dom (ln / (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
(dom (arctan / (id Z))) /\ (dom (ln / (f1 + (#Z 2)))) is V51() V52() V53() Element of K19(REAL)
dom arctan is V51() V52() V53() Element of K19(REAL)
dom (id Z) is V51() V52() V53() Element of K19(Z)
K19(Z) is set
(id Z) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id Z)) \ ((id Z) " {0}) is V51() V52() V53() Element of K19(Z)
(dom arctan) /\ ((dom (id Z)) \ ((id Z) " {0})) is V51() V52() V53() Element of K19(Z)
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom ln) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(dom arctan) /\ (dom ln) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
(arctan . x) / x is V28() V29() ext-real Element of REAL
K99(x) is V28() set
K97((arctan . x),K99(x)) is set
ln . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(ln . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((ln . x),K99((1 + (x ^2)))) is set
((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
((arctan / (id Z)) + (ln / (f1 + (#Z 2)))) . x is V28() V29() ext-real Element of REAL
(arctan / (id Z)) . x is V28() V29() ext-real Element of REAL
(ln / (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
((arctan / (id Z)) . x) + ((ln / (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
(id Z) . x is V28() V29() ext-real Element of REAL
((id Z) . x) " is V28() V29() ext-real Element of REAL
(arctan . x) * (((id Z) . x) ") is V28() V29() ext-real Element of REAL
((arctan . x) * (((id Z) . x) ")) + ((ln / (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((f1 + (#Z 2)) . x) " is V28() V29() ext-real Element of REAL
(ln . x) * (((f1 + (#Z 2)) . x) ") is V28() V29() ext-real Element of REAL
((arctan . x) * (((id Z) . x) ")) + ((ln . x) * (((f1 + (#Z 2)) . x) ")) is V28() V29() ext-real Element of REAL
(ln . x) / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . x)) is V28() set
K97((ln . x),K99(((f1 + (#Z 2)) . x))) is set
((arctan . x) / x) + ((ln . x) / ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(ln . x) / ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + ((#Z 2) . x))) is V28() set
K97((ln . x),K99(((f1 . x) + ((#Z 2) . x)))) is set
((arctan . x) / x) + ((ln . x) / ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(ln . x) / (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . x))) is V28() set
K97((ln . x),K99((1 + ((#Z 2) . x)))) is set
((arctan . x) / x) + ((ln . x) / (1 + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
x #Z 2 is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(ln . x) / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97((ln . x),K99((1 + (x #Z 2)))) is set
((arctan . x) / x) + ((ln . x) / (1 + (x #Z 2))) is V28() V29() ext-real Element of REAL
(ln (#) arctan) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln (#) arctan) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((ln (#) arctan) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
(arctan . x) / x is V28() V29() ext-real Element of REAL
K99(x) is V28() set
K97((arctan . x),K99(x)) is set
ln . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(ln . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((ln . x),K99((1 + (x ^2)))) is set
((arctan . x) / x) + ((ln . x) / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln (#) arccot) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln (#) arccot) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln (#) arccot) . (upper_bound A)) - ((ln (#) arccot) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln (#) arccot) . (lower_bound A))) is V28() set
K96(((ln (#) arccot) . (upper_bound A)),K98(((ln (#) arccot) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
ln / (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccot / (id Z) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arccot / (id Z)) - (ln / (f1 + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (ln / (f1 + (#Z 2))) is V1() V6() V34() set
K98(1) (#) (ln / (f1 + (#Z 2))) is V1() V6() set
(arccot / (id Z)) + (- (ln / (f1 + (#Z 2)))) is V1() V6() set
dom (arccot / (id Z)) is V51() V52() V53() Element of K19(REAL)
dom (ln / (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
(dom (arccot / (id Z))) /\ (dom (ln / (f1 + (#Z 2)))) is V51() V52() V53() Element of K19(REAL)
dom arccot is V51() V52() V53() Element of K19(REAL)
dom (id Z) is V51() V52() V53() Element of K19(Z)
K19(Z) is set
(id Z) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id Z)) \ ((id Z) " {0}) is V51() V52() V53() Element of K19(Z)
(dom arccot) /\ ((dom (id Z)) \ ((id Z) " {0})) is V51() V52() V53() Element of K19(Z)
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom ln) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(dom arccot) /\ (dom ln) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
(arccot . x) / x is V28() V29() ext-real Element of REAL
K99(x) is V28() set
K97((arccot . x),K99(x)) is set
ln . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(ln . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((ln . x),K99((1 + (x ^2)))) is set
((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
K98(((ln . x) / (1 + (x ^2)))) is V28() set
K96(((arccot . x) / x),K98(((ln . x) / (1 + (x ^2))))) is set
((arccot / (id Z)) - (ln / (f1 + (#Z 2)))) . x is V28() V29() ext-real Element of REAL
(arccot / (id Z)) . x is V28() V29() ext-real Element of REAL
(ln / (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
((arccot / (id Z)) . x) - ((ln / (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K98(((ln / (f1 + (#Z 2))) . x)) is V28() set
K96(((arccot / (id Z)) . x),K98(((ln / (f1 + (#Z 2))) . x))) is set
(id Z) . x is V28() V29() ext-real Element of REAL
((id Z) . x) " is V28() V29() ext-real Element of REAL
(arccot . x) * (((id Z) . x) ") is V28() V29() ext-real Element of REAL
((arccot . x) * (((id Z) . x) ")) - ((ln / (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K96(((arccot . x) * (((id Z) . x) ")),K98(((ln / (f1 + (#Z 2))) . x))) is set
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((f1 + (#Z 2)) . x) " is V28() V29() ext-real Element of REAL
(ln . x) * (((f1 + (#Z 2)) . x) ") is V28() V29() ext-real Element of REAL
((arccot . x) * (((id Z) . x) ")) - ((ln . x) * (((f1 + (#Z 2)) . x) ")) is V28() V29() ext-real Element of REAL
K98(((ln . x) * (((f1 + (#Z 2)) . x) "))) is V28() set
K96(((arccot . x) * (((id Z) . x) ")),K98(((ln . x) * (((f1 + (#Z 2)) . x) ")))) is set
(ln . x) / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . x)) is V28() set
K97((ln . x),K99(((f1 + (#Z 2)) . x))) is set
((arccot . x) / x) - ((ln . x) / ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K98(((ln . x) / ((f1 + (#Z 2)) . x))) is V28() set
K96(((arccot . x) / x),K98(((ln . x) / ((f1 + (#Z 2)) . x)))) is set
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(ln . x) / ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + ((#Z 2) . x))) is V28() set
K97((ln . x),K99(((f1 . x) + ((#Z 2) . x)))) is set
((arccot . x) / x) - ((ln . x) / ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K98(((ln . x) / ((f1 . x) + ((#Z 2) . x)))) is V28() set
K96(((arccot . x) / x),K98(((ln . x) / ((f1 . x) + ((#Z 2) . x))))) is set
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(ln . x) / (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . x))) is V28() set
K97((ln . x),K99((1 + ((#Z 2) . x)))) is set
((arccot . x) / x) - ((ln . x) / (1 + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K98(((ln . x) / (1 + ((#Z 2) . x)))) is V28() set
K96(((arccot . x) / x),K98(((ln . x) / (1 + ((#Z 2) . x))))) is set
x #Z 2 is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(ln . x) / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97((ln . x),K99((1 + (x #Z 2)))) is set
((arccot . x) / x) - ((ln . x) / (1 + (x #Z 2))) is V28() V29() ext-real Element of REAL
K98(((ln . x) / (1 + (x #Z 2)))) is V28() set
K96(((arccot . x) / x),K98(((ln . x) / (1 + (x #Z 2))))) is set
(ln (#) arccot) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln (#) arccot) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((ln (#) arccot) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
(arccot . x) / x is V28() V29() ext-real Element of REAL
K99(x) is V28() set
K97((arccot . x),K99(x)) is set
ln . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(ln . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((ln . x),K99((1 + (x ^2)))) is set
((arccot . x) / x) - ((ln . x) / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
K98(((ln . x) / (1 + (x ^2)))) is V28() set
K96(((arccot . x) / x),K98(((ln . x) / (1 + (x ^2))))) is set
(exp_R * tan) / (cos ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(exp_R * tan) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(exp_R * tan) . (lower_bound A) is V28() V29() ext-real Element of REAL
((exp_R * tan) . (upper_bound A)) - ((exp_R * tan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((exp_R * tan) . (lower_bound A))) is V28() set
K96(((exp_R * tan) . (upper_bound A)),K98(((exp_R * tan) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom (cos ^2) is V51() V52() V53() Element of K19(REAL)
(cos ^2) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (cos ^2)) \ ((cos ^2) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (exp_R * tan)) /\ ((dom (cos ^2)) \ ((cos ^2) " {0})) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
tan . Z is V28() V29() ext-real Element of REAL
exp_R . (tan . Z) is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(cos . Z) ^2 is V28() V29() ext-real Element of REAL
K97((cos . Z),(cos . Z)) is set
(exp_R . (tan . Z)) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() set
K97((exp_R . (tan . Z)),K99(((cos . Z) ^2))) is set
((exp_R * tan) / (cos ^2)) . Z is V28() V29() ext-real Element of REAL
(exp_R * tan) . Z is V28() V29() ext-real Element of REAL
(cos ^2) . Z is V28() V29() ext-real Element of REAL
((exp_R * tan) . Z) / ((cos ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((cos ^2) . Z)) is V28() set
K97(((exp_R * tan) . Z),K99(((cos ^2) . Z))) is set
(exp_R . (tan . Z)) / ((cos ^2) . Z) is V28() V29() ext-real Element of REAL
K97((exp_R . (tan . Z)),K99(((cos ^2) . Z))) is set
(exp_R * tan) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R * tan) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((exp_R * tan) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
tan . Z is V28() V29() ext-real Element of REAL
exp_R . (tan . Z) is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(cos . Z) ^2 is V28() V29() ext-real Element of REAL
K97((cos . Z),(cos . Z)) is set
(exp_R . (tan . Z)) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() set
K97((exp_R . (tan . Z)),K99(((cos . Z) ^2))) is set
(exp_R * cot) / (sin ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((exp_R * cot) / (sin ^2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((exp_R * cot) / (sin ^2)) is V1() V6() set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(exp_R * cot) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(exp_R * cot) . (lower_bound A) is V28() V29() ext-real Element of REAL
((exp_R * cot) . (upper_bound A)) - ((exp_R * cot) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((exp_R * cot) . (lower_bound A))) is V28() set
K96(((exp_R * cot) . (upper_bound A)),K98(((exp_R * cot) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom ((exp_R * cot) / (sin ^2)) is V51() V52() V53() Element of K19(REAL)
dom (sin ^2) is V51() V52() V53() Element of K19(REAL)
(sin ^2) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (sin ^2)) \ ((sin ^2) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (exp_R * cot)) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cot . Z is V28() V29() ext-real Element of REAL
exp_R . (cot . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is set
(exp_R . (cot . Z)) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() set
K97((exp_R . (cot . Z)),K99(((sin . Z) ^2))) is set
- ((exp_R . (cot . Z)) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
(- ((exp_R * cot) / (sin ^2))) . Z is V28() V29() ext-real Element of REAL
((exp_R * cot) / (sin ^2)) . Z is V28() V29() ext-real Element of REAL
- (((exp_R * cot) / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
(exp_R * cot) . Z is V28() V29() ext-real Element of REAL
(sin ^2) . Z is V28() V29() ext-real Element of REAL
((exp_R * cot) . Z) / ((sin ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((sin ^2) . Z)) is V28() set
K97(((exp_R * cot) . Z),K99(((sin ^2) . Z))) is set
- (((exp_R * cot) . Z) / ((sin ^2) . Z)) is V28() V29() ext-real Element of REAL
(exp_R . (cot . Z)) / ((sin ^2) . Z) is V28() V29() ext-real Element of REAL
K97((exp_R . (cot . Z)),K99(((sin ^2) . Z))) is set
- ((exp_R . (cot . Z)) / ((sin ^2) . Z)) is V28() V29() ext-real Element of REAL
(exp_R * cot) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R * cot) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((exp_R * cot) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cot . Z is V28() V29() ext-real Element of REAL
exp_R . (cot . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is set
(exp_R . (cot . Z)) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() set
K97((exp_R . (cot . Z)),K99(((sin . Z) ^2))) is set
- ((exp_R . (cot . Z)) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
- (exp_R * cot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (exp_R * cot) is V1() V6() set
A is V51() V52() V53() open Element of K19(REAL)
(- (exp_R * cot)) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (exp_R * cot)) is V51() V52() V53() Element of K19(REAL)
f1 is V28() V29() ext-real Element of REAL
sin . f1 is V28() V29() ext-real Element of REAL
dom (cos / sin) is V51() V52() V53() Element of K19(REAL)
(- 1) (#) (exp_R * cot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
((- (exp_R * cot)) `| A) . f1 is V28() V29() ext-real Element of REAL
cot . f1 is V28() V29() ext-real Element of REAL
exp_R . (cot . f1) is V28() V29() ext-real Element of REAL
sin . f1 is V28() V29() ext-real Element of REAL
(sin . f1) ^2 is V28() V29() ext-real Element of REAL
K97((sin . f1),(sin . f1)) is set
(exp_R . (cot . f1)) / ((sin . f1) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . f1) ^2)) is V28() set
K97((exp_R . (cot . f1)),K99(((sin . f1) ^2))) is set
diff ((- (exp_R * cot)),f1) is V28() V29() ext-real Element of REAL
diff ((exp_R * cot),f1) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((exp_R * cot),f1)) is V28() V29() ext-real Element of REAL
diff (exp_R,(cot . f1)) is V28() V29() ext-real Element of REAL
diff (cot,f1) is V28() V29() ext-real Element of REAL
(diff (exp_R,(cot . f1))) * (diff (cot,f1)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (exp_R,(cot . f1))) * (diff (cot,f1))) is V28() V29() ext-real Element of REAL
1 / ((sin . f1) ^2) is V28() V29() ext-real Element of REAL
K97(1,K99(((sin . f1) ^2))) is set
- (1 / ((sin . f1) ^2)) is V28() V29() ext-real Element of REAL
(diff (exp_R,(cot . f1))) * (- (1 / ((sin . f1) ^2))) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (exp_R,(cot . f1))) * (- (1 / ((sin . f1) ^2)))) is V28() V29() ext-real Element of REAL
(exp_R . (cot . f1)) * (- (1 / ((sin . f1) ^2))) is V28() V29() ext-real Element of REAL
(- 1) * ((exp_R . (cot . f1)) * (- (1 / ((sin . f1) ^2)))) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
((- (exp_R * cot)) `| A) . f1 is V28() V29() ext-real Element of REAL
cot . f1 is V28() V29() ext-real Element of REAL
exp_R . (cot . f1) is V28() V29() ext-real Element of REAL
sin . f1 is V28() V29() ext-real Element of REAL
(sin . f1) ^2 is V28() V29() ext-real Element of REAL
K97((sin . f1),(sin . f1)) is set
(exp_R . (cot . f1)) / ((sin . f1) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . f1) ^2)) is V28() set
K97((exp_R . (cot . f1)),K99(((sin . f1) ^2))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(- (exp_R * cot)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(- (exp_R * cot)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (exp_R * cot)) . (upper_bound A)) - ((- (exp_R * cot)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (exp_R * cot)) . (lower_bound A))) is V28() set
K96(((- (exp_R * cot)) . (upper_bound A)),K98(((- (exp_R * cot)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom (sin ^2) is V51() V52() V53() Element of K19(REAL)
(sin ^2) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (sin ^2)) \ ((sin ^2) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (exp_R * cot)) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cot . Z is V28() V29() ext-real Element of REAL
exp_R . (cot . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is set
(exp_R . (cot . Z)) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() set
K97((exp_R . (cot . Z)),K99(((sin . Z) ^2))) is set
((exp_R * cot) / (sin ^2)) . Z is V28() V29() ext-real Element of REAL
(exp_R * cot) . Z is V28() V29() ext-real Element of REAL
(sin ^2) . Z is V28() V29() ext-real Element of REAL
((exp_R * cot) . Z) / ((sin ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((sin ^2) . Z)) is V28() set
K97(((exp_R * cot) . Z),K99(((sin ^2) . Z))) is set
(exp_R . (cot . Z)) / ((sin ^2) . Z) is V28() V29() ext-real Element of REAL
K97((exp_R . (cot . Z)),K99(((sin ^2) . Z))) is set
(- (exp_R * cot)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- (exp_R * cot)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((- (exp_R * cot)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
cot . Z is V28() V29() ext-real Element of REAL
exp_R . (cot . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is set
(exp_R . (cot . Z)) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() set
K97((exp_R . (cot . Z)),K99(((sin . Z) ^2))) is set
cos * ln is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * ln) ^2 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * ln) (#) (cos * ln) is V1() V6() set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(tan * ln) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(tan * ln) . (lower_bound A) is V28() V29() ext-real Element of REAL
((tan * ln) . (upper_bound A)) - ((tan * ln) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((tan * ln) . (lower_bound A))) is V28() set
K96(((tan * ln) . (upper_bound A)),K98(((tan * ln) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f) (#) ((cos * ln) ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id f) (#) ((cos * ln) ^2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((id f) (#) ((cos * ln) ^2)) is V51() V52() V53() Element of K19(REAL)
dom (id f) is V51() V52() V53() Element of K19(f)
K19(f) is set
dom ((cos * ln) ^2) is V51() V52() V53() Element of K19(REAL)
(dom (id f)) /\ (dom ((cos * ln) ^2)) is V51() V52() V53() Element of K19(REAL)
dom (cos * ln) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
ln . Z is V28() V29() ext-real Element of REAL
cos . (ln . Z) is V28() V29() ext-real Element of REAL
(cos . (ln . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (ln . Z)),(cos . (ln . Z))) is set
Z * ((cos . (ln . Z)) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * ((cos . (ln . Z)) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * ((cos . (ln . Z)) ^2))) is V28() set
K97(1,K99((Z * ((cos . (ln . Z)) ^2)))) is set
(((id f) (#) ((cos * ln) ^2)) ^) . Z is V28() V29() ext-real Element of REAL
((id f) (#) ((cos * ln) ^2)) . Z is V28() V29() ext-real Element of REAL
1 / (((id f) (#) ((cos * ln) ^2)) . Z) is V28() V29() ext-real Element of REAL
K99((((id f) (#) ((cos * ln) ^2)) . Z)) is V28() set
K97(1,K99((((id f) (#) ((cos * ln) ^2)) . Z))) is set
(id f) . Z is V28() V29() ext-real Element of REAL
((cos * ln) ^2) . Z is V28() V29() ext-real Element of REAL
((id f) . Z) * (((cos * ln) ^2) . Z) is V28() V29() ext-real Element of REAL
1 / (((id f) . Z) * (((cos * ln) ^2) . Z)) is V28() V29() ext-real Element of REAL
K99((((id f) . Z) * (((cos * ln) ^2) . Z))) is V28() set
K97(1,K99((((id f) . Z) * (((cos * ln) ^2) . Z)))) is set
Z * (((cos * ln) ^2) . Z) is V28() V29() ext-real Element of REAL
1 / (Z * (((cos * ln) ^2) . Z)) is V28() V29() ext-real Element of REAL
K99((Z * (((cos * ln) ^2) . Z))) is V28() set
K97(1,K99((Z * (((cos * ln) ^2) . Z)))) is set
(cos * ln) . Z is V28() V29() ext-real Element of REAL
((cos * ln) . Z) ^2 is V28() V29() ext-real Element of REAL
K97(((cos * ln) . Z),((cos * ln) . Z)) is set
Z * (((cos * ln) . Z) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * (((cos * ln) . Z) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * (((cos * ln) . Z) ^2))) is V28() set
K97(1,K99((Z * (((cos * ln) . Z) ^2)))) is set
(tan * ln) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((tan * ln) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((tan * ln) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
ln . Z is V28() V29() ext-real Element of REAL
cos . (ln . Z) is V28() V29() ext-real Element of REAL
(cos . (ln . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (ln . Z)),(cos . (ln . Z))) is set
Z * ((cos . (ln . Z)) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * ((cos . (ln . Z)) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * ((cos . (ln . Z)) ^2))) is V28() set
K97(1,K99((Z * ((cos . (ln . Z)) ^2)))) is set
sin * ln is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * ln) ^2 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * ln) (#) (sin * ln) is V1() V6() set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(cot * ln) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(cot * ln) . (lower_bound A) is V28() V29() ext-real Element of REAL
((cot * ln) . (upper_bound A)) - ((cot * ln) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((cot * ln) . (lower_bound A))) is V28() set
K96(((cot * ln) . (upper_bound A)),K98(((cot * ln) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f) (#) ((sin * ln) ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id f) (#) ((sin * ln) ^2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (((id f) (#) ((sin * ln) ^2)) ^) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (((id f) (#) ((sin * ln) ^2)) ^) is V1() V6() set
dom (((id f) (#) ((sin * ln) ^2)) ^) is V51() V52() V53() Element of K19(REAL)
dom ((id f) (#) ((sin * ln) ^2)) is V51() V52() V53() Element of K19(REAL)
dom (id f) is V51() V52() V53() Element of K19(f)
K19(f) is set
dom ((sin * ln) ^2) is V51() V52() V53() Element of K19(REAL)
(dom (id f)) /\ (dom ((sin * ln) ^2)) is V51() V52() V53() Element of K19(REAL)
dom (sin * ln) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
ln . Z is V28() V29() ext-real Element of REAL
sin . (ln . Z) is V28() V29() ext-real Element of REAL
(sin . (ln . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (ln . Z)),(sin . (ln . Z))) is set
Z * ((sin . (ln . Z)) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * ((sin . (ln . Z)) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * ((sin . (ln . Z)) ^2))) is V28() set
K97(1,K99((Z * ((sin . (ln . Z)) ^2)))) is set
- (1 / (Z * ((sin . (ln . Z)) ^2))) is V28() V29() ext-real Element of REAL
(- (((id f) (#) ((sin * ln) ^2)) ^)) . Z is V28() V29() ext-real Element of REAL
(((id f) (#) ((sin * ln) ^2)) ^) . Z is V28() V29() ext-real Element of REAL
- ((((id f) (#) ((sin * ln) ^2)) ^) . Z) is V28() V29() ext-real Element of REAL
((id f) (#) ((sin * ln) ^2)) . Z is V28() V29() ext-real Element of REAL
1 / (((id f) (#) ((sin * ln) ^2)) . Z) is V28() V29() ext-real Element of REAL
K99((((id f) (#) ((sin * ln) ^2)) . Z)) is V28() set
K97(1,K99((((id f) (#) ((sin * ln) ^2)) . Z))) is set
- (1 / (((id f) (#) ((sin * ln) ^2)) . Z)) is V28() V29() ext-real Element of REAL
(id f) . Z is V28() V29() ext-real Element of REAL
((sin * ln) ^2) . Z is V28() V29() ext-real Element of REAL
((id f) . Z) * (((sin * ln) ^2) . Z) is V28() V29() ext-real Element of REAL
1 / (((id f) . Z) * (((sin * ln) ^2) . Z)) is V28() V29() ext-real Element of REAL
K99((((id f) . Z) * (((sin * ln) ^2) . Z))) is V28() set
K97(1,K99((((id f) . Z) * (((sin * ln) ^2) . Z)))) is set
- (1 / (((id f) . Z) * (((sin * ln) ^2) . Z))) is V28() V29() ext-real Element of REAL
Z * (((sin * ln) ^2) . Z) is V28() V29() ext-real Element of REAL
1 / (Z * (((sin * ln) ^2) . Z)) is V28() V29() ext-real Element of REAL
K99((Z * (((sin * ln) ^2) . Z))) is V28() set
K97(1,K99((Z * (((sin * ln) ^2) . Z)))) is set
- (1 / (Z * (((sin * ln) ^2) . Z))) is V28() V29() ext-real Element of REAL
(sin * ln) . Z is V28() V29() ext-real Element of REAL
((sin * ln) . Z) ^2 is V28() V29() ext-real Element of REAL
K97(((sin * ln) . Z),((sin * ln) . Z)) is set
Z * (((sin * ln) . Z) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * (((sin * ln) . Z) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * (((sin * ln) . Z) ^2))) is V28() set
K97(1,K99((Z * (((sin * ln) . Z) ^2)))) is set
- (1 / (Z * (((sin * ln) . Z) ^2))) is V28() V29() ext-real Element of REAL
(cot * ln) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((cot * ln) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((cot * ln) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
ln . Z is V28() V29() ext-real Element of REAL
sin . (ln . Z) is V28() V29() ext-real Element of REAL
(sin . (ln . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (ln . Z)),(sin . (ln . Z))) is set
Z * ((sin . (ln . Z)) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * ((sin . (ln . Z)) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * ((sin . (ln . Z)) ^2))) is V28() set
K97(1,K99((Z * ((sin . (ln . Z)) ^2)))) is set
- (1 / (Z * ((sin . (ln . Z)) ^2))) is V28() V29() ext-real Element of REAL
- (cot * ln) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (cot * ln) is V1() V6() set
A is V51() V52() V53() open Element of K19(REAL)
(- (cot * ln)) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (cot * ln)) is V51() V52() V53() Element of K19(REAL)
f1 is V28() V29() ext-real Element of REAL
f is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
ln . f1 is V28() V29() ext-real Element of REAL
sin . (ln . f1) is V28() V29() ext-real Element of REAL
dom (cos / sin) is V51() V52() V53() Element of K19(REAL)
f1 is V28() V29() ext-real Element of REAL
diff (ln,f1) is V28() V29() ext-real Element of REAL
1 / f1 is V28() V29() ext-real Element of REAL
K99(f1) is V28() set
K97(1,K99(f1)) is set
(- 1) (#) (cot * ln) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
((- (cot * ln)) `| A) . f1 is V28() V29() ext-real Element of REAL
ln . f1 is V28() V29() ext-real Element of REAL
sin . (ln . f1) is V28() V29() ext-real Element of REAL
(sin . (ln . f1)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (ln . f1)),(sin . (ln . f1))) is set
f1 * ((sin . (ln . f1)) ^2) is V28() V29() ext-real Element of REAL
1 / (f1 * ((sin . (ln . f1)) ^2)) is V28() V29() ext-real Element of REAL
K99((f1 * ((sin . (ln . f1)) ^2))) is V28() set
K97(1,K99((f1 * ((sin . (ln . f1)) ^2)))) is set
diff ((- (cot * ln)),f1) is V28() V29() ext-real Element of REAL
diff ((cot * ln),f1) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((cot * ln),f1)) is V28() V29() ext-real Element of REAL
diff (cot,(ln . f1)) is V28() V29() ext-real Element of REAL
diff (ln,f1) is V28() V29() ext-real Element of REAL
(diff (cot,(ln . f1))) * (diff (ln,f1)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (cot,(ln . f1))) * (diff (ln,f1))) is V28() V29() ext-real Element of REAL
1 / ((sin . (ln . f1)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (ln . f1)) ^2)) is V28() set
K97(1,K99(((sin . (ln . f1)) ^2))) is set
- (1 / ((sin . (ln . f1)) ^2)) is V28() V29() ext-real Element of REAL
(- (1 / ((sin . (ln . f1)) ^2))) * (diff (ln,f1)) is V28() V29() ext-real Element of REAL
(- 1) * ((- (1 / ((sin . (ln . f1)) ^2))) * (diff (ln,f1))) is V28() V29() ext-real Element of REAL
(diff (ln,f1)) / ((sin . (ln . f1)) ^2) is V28() V29() ext-real Element of REAL
K97((diff (ln,f1)),K99(((sin . (ln . f1)) ^2))) is set
- ((diff (ln,f1)) / ((sin . (ln . f1)) ^2)) is V28() V29() ext-real Element of REAL
(- 1) * (- ((diff (ln,f1)) / ((sin . (ln . f1)) ^2))) is V28() V29() ext-real Element of REAL
1 / f1 is V28() V29() ext-real Element of REAL
K99(f1) is V28() set
K97(1,K99(f1)) is set
(1 / f1) / ((sin . (ln . f1)) ^2) is V28() V29() ext-real Element of REAL
K97((1 / f1),K99(((sin . (ln . f1)) ^2))) is set
- ((1 / f1) / ((sin . (ln . f1)) ^2)) is V28() V29() ext-real Element of REAL
(- 1) * (- ((1 / f1) / ((sin . (ln . f1)) ^2))) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
((- (cot * ln)) `| A) . f1 is V28() V29() ext-real Element of REAL
ln . f1 is V28() V29() ext-real Element of REAL
sin . (ln . f1) is V28() V29() ext-real Element of REAL
(sin . (ln . f1)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (ln . f1)),(sin . (ln . f1))) is set
f1 * ((sin . (ln . f1)) ^2) is V28() V29() ext-real Element of REAL
1 / (f1 * ((sin . (ln . f1)) ^2)) is V28() V29() ext-real Element of REAL
K99((f1 * ((sin . (ln . f1)) ^2))) is V28() set
K97(1,K99((f1 * ((sin . (ln . f1)) ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(- (cot * ln)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(- (cot * ln)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (cot * ln)) . (upper_bound A)) - ((- (cot * ln)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (cot * ln)) . (lower_bound A))) is V28() set
K96(((- (cot * ln)) . (upper_bound A)),K98(((- (cot * ln)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f) (#) ((sin * ln) ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id f) (#) ((sin * ln) ^2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((id f) (#) ((sin * ln) ^2)) is V51() V52() V53() Element of K19(REAL)
dom (id f) is V51() V52() V53() Element of K19(f)
K19(f) is set
dom ((sin * ln) ^2) is V51() V52() V53() Element of K19(REAL)
(dom (id f)) /\ (dom ((sin * ln) ^2)) is V51() V52() V53() Element of K19(REAL)
dom (sin * ln) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
ln . Z is V28() V29() ext-real Element of REAL
sin . (ln . Z) is V28() V29() ext-real Element of REAL
(sin . (ln . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (ln . Z)),(sin . (ln . Z))) is set
Z * ((sin . (ln . Z)) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * ((sin . (ln . Z)) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * ((sin . (ln . Z)) ^2))) is V28() set
K97(1,K99((Z * ((sin . (ln . Z)) ^2)))) is set
(((id f) (#) ((sin * ln) ^2)) ^) . Z is V28() V29() ext-real Element of REAL
((id f) (#) ((sin * ln) ^2)) . Z is V28() V29() ext-real Element of REAL
1 / (((id f) (#) ((sin * ln) ^2)) . Z) is V28() V29() ext-real Element of REAL
K99((((id f) (#) ((sin * ln) ^2)) . Z)) is V28() set
K97(1,K99((((id f) (#) ((sin * ln) ^2)) . Z))) is set
(id f) . Z is V28() V29() ext-real Element of REAL
((sin * ln) ^2) . Z is V28() V29() ext-real Element of REAL
((id f) . Z) * (((sin * ln) ^2) . Z) is V28() V29() ext-real Element of REAL
1 / (((id f) . Z) * (((sin * ln) ^2) . Z)) is V28() V29() ext-real Element of REAL
K99((((id f) . Z) * (((sin * ln) ^2) . Z))) is V28() set
K97(1,K99((((id f) . Z) * (((sin * ln) ^2) . Z)))) is set
Z * (((sin * ln) ^2) . Z) is V28() V29() ext-real Element of REAL
1 / (Z * (((sin * ln) ^2) . Z)) is V28() V29() ext-real Element of REAL
K99((Z * (((sin * ln) ^2) . Z))) is V28() set
K97(1,K99((Z * (((sin * ln) ^2) . Z)))) is set
(sin * ln) . Z is V28() V29() ext-real Element of REAL
((sin * ln) . Z) ^2 is V28() V29() ext-real Element of REAL
K97(((sin * ln) . Z),((sin * ln) . Z)) is set
Z * (((sin * ln) . Z) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * (((sin * ln) . Z) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * (((sin * ln) . Z) ^2))) is V28() set
K97(1,K99((Z * (((sin * ln) . Z) ^2)))) is set
(- (cot * ln)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- (cot * ln)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((- (cot * ln)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
ln . Z is V28() V29() ext-real Element of REAL
sin . (ln . Z) is V28() V29() ext-real Element of REAL
(sin . (ln . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (ln . Z)),(sin . (ln . Z))) is set
Z * ((sin . (ln . Z)) ^2) is V28() V29() ext-real Element of REAL
1 / (Z * ((sin . (ln . Z)) ^2)) is V28() V29() ext-real Element of REAL
K99((Z * ((sin . (ln . Z)) ^2))) is V28() set
K97(1,K99((Z * ((sin . (ln . Z)) ^2)))) is set
cos * exp_R is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * exp_R) ^2 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * exp_R) (#) (cos * exp_R) is V1() V6() set
exp_R / ((cos * exp_R) ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(tan * exp_R) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(tan * exp_R) . (lower_bound A) is V28() V29() ext-real Element of REAL
((tan * exp_R) . (upper_bound A)) - ((tan * exp_R) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((tan * exp_R) . (lower_bound A))) is V28() set
K96(((tan * exp_R) . (upper_bound A)),K98(((tan * exp_R) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom exp_R is V51() V52() V53() Element of K19(REAL)
dom ((cos * exp_R) ^2) is V51() V52() V53() Element of K19(REAL)
((cos * exp_R) ^2) " {0} is V51() V52() V53() Element of K19(REAL)
(dom ((cos * exp_R) ^2)) \ (((cos * exp_R) ^2) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom exp_R) /\ ((dom ((cos * exp_R) ^2)) \ (((cos * exp_R) ^2) " {0})) is V51() V52() V53() Element of K19(REAL)
((cos * exp_R) ^2) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((cos * exp_R) ^2) ^) is V51() V52() V53() Element of K19(REAL)
dom (cos * exp_R) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
cos . (exp_R . Z) is V28() V29() ext-real Element of REAL
(cos . (exp_R . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (exp_R . Z)),(cos . (exp_R . Z))) is set
(exp_R . Z) / ((cos . (exp_R . Z)) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (exp_R . Z)) ^2)) is V28() set
K97((exp_R . Z),K99(((cos . (exp_R . Z)) ^2))) is set
(exp_R / ((cos * exp_R) ^2)) . Z is V28() V29() ext-real Element of REAL
((cos * exp_R) ^2) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) / (((cos * exp_R) ^2) . Z) is V28() V29() ext-real Element of REAL
K99((((cos * exp_R) ^2) . Z)) is V28() set
K97((exp_R . Z),K99((((cos * exp_R) ^2) . Z))) is set
(cos * exp_R) . Z is V28() V29() ext-real Element of REAL
((cos * exp_R) . Z) ^2 is V28() V29() ext-real Element of REAL
K97(((cos * exp_R) . Z),((cos * exp_R) . Z)) is set
(exp_R . Z) / (((cos * exp_R) . Z) ^2) is V28() V29() ext-real Element of REAL
K99((((cos * exp_R) . Z) ^2)) is V28() set
K97((exp_R . Z),K99((((cos * exp_R) . Z) ^2))) is set
(tan * exp_R) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((tan * exp_R) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((tan * exp_R) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
cos . (exp_R . Z) is V28() V29() ext-real Element of REAL
(cos . (exp_R . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (exp_R . Z)),(cos . (exp_R . Z))) is set
(exp_R . Z) / ((cos . (exp_R . Z)) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (exp_R . Z)) ^2)) is V28() set
K97((exp_R . Z),K99(((cos . (exp_R . Z)) ^2))) is set
sin * exp_R is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * exp_R) ^2 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * exp_R) (#) (sin * exp_R) is V1() V6() set
exp_R / ((sin * exp_R) ^2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (exp_R / ((sin * exp_R) ^2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (exp_R / ((sin * exp_R) ^2)) is V1() V6() set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(cot * exp_R) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(cot * exp_R) . (lower_bound A) is V28() V29() ext-real Element of REAL
((cot * exp_R) . (upper_bound A)) - ((cot * exp_R) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((cot * exp_R) . (lower_bound A))) is V28() set
K96(((cot * exp_R) . (upper_bound A)),K98(((cot * exp_R) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom (exp_R / ((sin * exp_R) ^2)) is V51() V52() V53() Element of K19(REAL)
dom exp_R is V51() V52() V53() Element of K19(REAL)
dom ((sin * exp_R) ^2) is V51() V52() V53() Element of K19(REAL)
((sin * exp_R) ^2) " {0} is V51() V52() V53() Element of K19(REAL)
(dom ((sin * exp_R) ^2)) \ (((sin * exp_R) ^2) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom exp_R) /\ ((dom ((sin * exp_R) ^2)) \ (((sin * exp_R) ^2) " {0})) is V51() V52() V53() Element of K19(REAL)
((sin * exp_R) ^2) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((sin * exp_R) ^2) ^) is V51() V52() V53() Element of K19(REAL)
dom (sin * exp_R) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
sin . (exp_R . Z) is V28() V29() ext-real Element of REAL
(sin . (exp_R . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (exp_R . Z)),(sin . (exp_R . Z))) is set
(exp_R . Z) / ((sin . (exp_R . Z)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (exp_R . Z)) ^2)) is V28() set
K97((exp_R . Z),K99(((sin . (exp_R . Z)) ^2))) is set
- ((exp_R . Z) / ((sin . (exp_R . Z)) ^2)) is V28() V29() ext-real Element of REAL
(- (exp_R / ((sin * exp_R) ^2))) . Z is V28() V29() ext-real Element of REAL
(exp_R / ((sin * exp_R) ^2)) . Z is V28() V29() ext-real Element of REAL
- ((exp_R / ((sin * exp_R) ^2)) . Z) is V28() V29() ext-real Element of REAL
((sin * exp_R) ^2) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) / (((sin * exp_R) ^2) . Z) is V28() V29() ext-real Element of REAL
K99((((sin * exp_R) ^2) . Z)) is V28() set
K97((exp_R . Z),K99((((sin * exp_R) ^2) . Z))) is set
- ((exp_R . Z) / (((sin * exp_R) ^2) . Z)) is V28() V29() ext-real Element of REAL
(sin * exp_R) . Z is V28() V29() ext-real Element of REAL
((sin * exp_R) . Z) ^2 is V28() V29() ext-real Element of REAL
K97(((sin * exp_R) . Z),((sin * exp_R) . Z)) is set
(exp_R . Z) / (((sin * exp_R) . Z) ^2) is V28() V29() ext-real Element of REAL
K99((((sin * exp_R) . Z) ^2)) is V28() set
K97((exp_R . Z),K99((((sin * exp_R) . Z) ^2))) is set
- ((exp_R . Z) / (((sin * exp_R) . Z) ^2)) is V28() V29() ext-real Element of REAL
(cot * exp_R) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((cot * exp_R) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((cot * exp_R) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
sin . (exp_R . Z) is V28() V29() ext-real Element of REAL
(sin . (exp_R . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (exp_R . Z)),(sin . (exp_R . Z))) is set
(exp_R . Z) / ((sin . (exp_R . Z)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (exp_R . Z)) ^2)) is V28() set
K97((exp_R . Z),K99(((sin . (exp_R . Z)) ^2))) is set
- ((exp_R . Z) / ((sin . (exp_R . Z)) ^2)) is V28() V29() ext-real Element of REAL
- (cot * exp_R) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (cot * exp_R) is V1() V6() set
A is V51() V52() V53() open Element of K19(REAL)
(- (cot * exp_R)) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (cot * exp_R)) is V51() V52() V53() Element of K19(REAL)
(- 1) (#) (cot * exp_R) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
exp_R . f1 is V28() V29() ext-real Element of REAL
sin . (exp_R . f1) is V28() V29() ext-real Element of REAL
dom (cos / sin) is V51() V52() V53() Element of K19(REAL)
f1 is V28() V29() ext-real Element of REAL
((- (cot * exp_R)) `| A) . f1 is V28() V29() ext-real Element of REAL
exp_R . f1 is V28() V29() ext-real Element of REAL
sin . (exp_R . f1) is V28() V29() ext-real Element of REAL
(sin . (exp_R . f1)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (exp_R . f1)),(sin . (exp_R . f1))) is set
(exp_R . f1) / ((sin . (exp_R . f1)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (exp_R . f1)) ^2)) is V28() set
K97((exp_R . f1),K99(((sin . (exp_R . f1)) ^2))) is set
diff ((- (cot * exp_R)),f1) is V28() V29() ext-real Element of REAL
diff ((cot * exp_R),f1) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((cot * exp_R),f1)) is V28() V29() ext-real Element of REAL
diff (cot,(exp_R . f1)) is V28() V29() ext-real Element of REAL
diff (exp_R,f1) is V28() V29() ext-real Element of REAL
(diff (cot,(exp_R . f1))) * (diff (exp_R,f1)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (cot,(exp_R . f1))) * (diff (exp_R,f1))) is V28() V29() ext-real Element of REAL
1 / ((sin . (exp_R . f1)) ^2) is V28() V29() ext-real Element of REAL
K97(1,K99(((sin . (exp_R . f1)) ^2))) is set
- (1 / ((sin . (exp_R . f1)) ^2)) is V28() V29() ext-real Element of REAL
(- (1 / ((sin . (exp_R . f1)) ^2))) * (diff (exp_R,f1)) is V28() V29() ext-real Element of REAL
(- 1) * ((- (1 / ((sin . (exp_R . f1)) ^2))) * (diff (exp_R,f1))) is V28() V29() ext-real Element of REAL
(diff (exp_R,f1)) / ((sin . (exp_R . f1)) ^2) is V28() V29() ext-real Element of REAL
K97((diff (exp_R,f1)),K99(((sin . (exp_R . f1)) ^2))) is set
- ((diff (exp_R,f1)) / ((sin . (exp_R . f1)) ^2)) is V28() V29() ext-real Element of REAL
(- 1) * (- ((diff (exp_R,f1)) / ((sin . (exp_R . f1)) ^2))) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
((- (cot * exp_R)) `| A) . f1 is V28() V29() ext-real Element of REAL
exp_R . f1 is V28() V29() ext-real Element of REAL
sin . (exp_R . f1) is V28() V29() ext-real Element of REAL
(sin . (exp_R . f1)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (exp_R . f1)),(sin . (exp_R . f1))) is set
(exp_R . f1) / ((sin . (exp_R . f1)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (exp_R . f1)) ^2)) is V28() set
K97((exp_R . f1),K99(((sin . (exp_R . f1)) ^2))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(- (cot * exp_R)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(- (cot * exp_R)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (cot * exp_R)) . (upper_bound A)) - ((- (cot * exp_R)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (cot * exp_R)) . (lower_bound A))) is V28() set
K96(((- (cot * exp_R)) . (upper_bound A)),K98(((- (cot * exp_R)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
dom exp_R is V51() V52() V53() Element of K19(REAL)
dom ((sin * exp_R) ^2) is V51() V52() V53() Element of K19(REAL)
((sin * exp_R) ^2) " {0} is V51() V52() V53() Element of K19(REAL)
(dom ((sin * exp_R) ^2)) \ (((sin * exp_R) ^2) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom exp_R) /\ ((dom ((sin * exp_R) ^2)) \ (((sin * exp_R) ^2) " {0})) is V51() V52() V53() Element of K19(REAL)
((sin * exp_R) ^2) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((sin * exp_R) ^2) ^) is V51() V52() V53() Element of K19(REAL)
dom (sin * exp_R) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
sin . (exp_R . Z) is V28() V29() ext-real Element of REAL
(sin . (exp_R . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (exp_R . Z)),(sin . (exp_R . Z))) is set
(exp_R . Z) / ((sin . (exp_R . Z)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (exp_R . Z)) ^2)) is V28() set
K97((exp_R . Z),K99(((sin . (exp_R . Z)) ^2))) is set
(exp_R / ((sin * exp_R) ^2)) . Z is V28() V29() ext-real Element of REAL
((sin * exp_R) ^2) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) / (((sin * exp_R) ^2) . Z) is V28() V29() ext-real Element of REAL
K99((((sin * exp_R) ^2) . Z)) is V28() set
K97((exp_R . Z),K99((((sin * exp_R) ^2) . Z))) is set
(sin * exp_R) . Z is V28() V29() ext-real Element of REAL
((sin * exp_R) . Z) ^2 is V28() V29() ext-real Element of REAL
K97(((sin * exp_R) . Z),((sin * exp_R) . Z)) is set
(exp_R . Z) / (((sin * exp_R) . Z) ^2) is V28() V29() ext-real Element of REAL
K99((((sin * exp_R) . Z) ^2)) is V28() set
K97((exp_R . Z),K99((((sin * exp_R) . Z) ^2))) is set
(- (cot * exp_R)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- (cot * exp_R)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((- (cot * exp_R)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
exp_R . Z is V28() V29() ext-real Element of REAL
sin . (exp_R . Z) is V28() V29() ext-real Element of REAL
(sin . (exp_R . Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (exp_R . Z)),(sin . (exp_R . Z))) is set
(exp_R . Z) / ((sin . (exp_R . Z)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (exp_R . Z)) ^2)) is V28() set
K97((exp_R . Z),K99(((sin . (exp_R . Z)) ^2))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
tan * ((id f) ^) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (tan * ((id f) ^)) is V51() V52() V53() Element of K19(REAL)
(tan * ((id f) ^)) . (upper_bound A) is V28() V29() ext-real Element of REAL
(tan * ((id f) ^)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((tan * ((id f) ^)) . (upper_bound A)) - ((tan * ((id f) ^)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((tan * ((id f) ^)) . (lower_bound A))) is V28() set
K96(((tan * ((id f) ^)) . (upper_bound A)),K98(((tan * ((id f) ^)) . (lower_bound A)))) is set
dom ((id f) ^) is V51() V52() V53() Element of K19(REAL)
dom (id f) is V51() V52() V53() Element of K19(f)
K19(f) is set
(id f) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id f)) \ ((id f) " {0}) is V51() V52() V53() Element of K19(f)
(dom (id f)) \ {0} is V51() V52() V53() Element of K19(f)
(tan * ((id f) ^)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((tan * ((id f) ^)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((tan * ((id f) ^)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
Z ^2 is V28() V29() ext-real Element of REAL
K97(Z,Z) is set
1 / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97(1,K99(Z)) is set
cos . (1 / Z) is V28() V29() ext-real Element of REAL
(cos . (1 / Z)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (1 / Z)),(cos . (1 / Z))) is set
(Z ^2) * ((cos . (1 / Z)) ^2) is V28() V29() ext-real Element of REAL
1 / ((Z ^2) * ((cos . (1 / Z)) ^2)) is V28() V29() ext-real Element of REAL
K99(((Z ^2) * ((cos . (1 / Z)) ^2))) is V28() set
K97(1,K99(((Z ^2) * ((cos . (1 / Z)) ^2)))) is set
- (1 / ((Z ^2) * ((cos . (1 / Z)) ^2))) is V28() V29() ext-real Element of REAL
A is V51() V52() V53() open Element of K19(REAL)
id A is V1() V4( REAL ) V4(A) V5( REAL ) V5(A) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id A) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
tan * ((id A) ^) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (tan * ((id A) ^)) is V51() V52() V53() Element of K19(REAL)
- (tan * ((id A) ^)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (tan * ((id A) ^)) is V1() V6() set
(- (tan * ((id A) ^))) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((id A) ^) is V51() V52() V53() Element of K19(REAL)
dom (id A) is V51() V52() V53() Element of K19(A)
K19(A) is set
(id A) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id A)) \ ((id A) " {0}) is V51() V52() V53() Element of K19(A)
(dom (id A)) \ {0} is V51() V52() V53() Element of K19(A)
dom (- (tan * ((id A) ^))) is V51() V52() V53() Element of K19(REAL)
(- 1) (#) (tan * ((id A) ^)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id A) ^) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V28() V29() ext-real Element of REAL
(((id A) ^) `| A) . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
1 / (f ^2) is V28() V29() ext-real Element of REAL
K99((f ^2)) is V28() set
K97(1,K99((f ^2))) is set
- (1 / (f ^2)) is V28() V29() ext-real Element of REAL
f is V28() V29() ext-real Element of REAL
((id A) ^) . f is V28() V29() ext-real Element of REAL
cos . (((id A) ^) . f) is V28() V29() ext-real Element of REAL
dom (sin / cos) is V51() V52() V53() Element of K19(REAL)
f is V28() V29() ext-real Element of REAL
((- (tan * ((id A) ^))) `| A) . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
1 / f is V28() V29() ext-real Element of REAL
K99(f) is V28() set
K97(1,K99(f)) is set
cos . (1 / f) is V28() V29() ext-real Element of REAL
(cos . (1 / f)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (1 / f)),(cos . (1 / f))) is set
(f ^2) * ((cos . (1 / f)) ^2) is V28() V29() ext-real Element of REAL
1 / ((f ^2) * ((cos . (1 / f)) ^2)) is V28() V29() ext-real Element of REAL
K99(((f ^2) * ((cos . (1 / f)) ^2))) is V28() set
K97(1,K99(((f ^2) * ((cos . (1 / f)) ^2)))) is set
((id A) ^) . f is V28() V29() ext-real Element of REAL
cos . (((id A) ^) . f) is V28() V29() ext-real Element of REAL
diff ((- (tan * ((id A) ^))),f) is V28() V29() ext-real Element of REAL
diff ((tan * ((id A) ^)),f) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((tan * ((id A) ^)),f)) is V28() V29() ext-real Element of REAL
diff (tan,(((id A) ^) . f)) is V28() V29() ext-real Element of REAL
diff (((id A) ^),f) is V28() V29() ext-real Element of REAL
(diff (tan,(((id A) ^) . f))) * (diff (((id A) ^),f)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (tan,(((id A) ^) . f))) * (diff (((id A) ^),f))) is V28() V29() ext-real Element of REAL
(cos . (((id A) ^) . f)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (((id A) ^) . f)),(cos . (((id A) ^) . f))) is set
1 / ((cos . (((id A) ^) . f)) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (((id A) ^) . f)) ^2)) is V28() set
K97(1,K99(((cos . (((id A) ^) . f)) ^2))) is set
(1 / ((cos . (((id A) ^) . f)) ^2)) * (diff (((id A) ^),f)) is V28() V29() ext-real Element of REAL
(- 1) * ((1 / ((cos . (((id A) ^) . f)) ^2)) * (diff (((id A) ^),f))) is V28() V29() ext-real Element of REAL
(id A) . f is V28() V29() ext-real Element of REAL
((id A) . f) " is V28() V29() ext-real Element of REAL
cos . (((id A) . f) ") is V28() V29() ext-real Element of REAL
(cos . (((id A) . f) ")) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (((id A) . f) ")),(cos . (((id A) . f) "))) is set
(diff (((id A) ^),f)) / ((cos . (((id A) . f) ")) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (((id A) . f) ")) ^2)) is V28() set
K97((diff (((id A) ^),f)),K99(((cos . (((id A) . f) ")) ^2))) is set
(- 1) * ((diff (((id A) ^),f)) / ((cos . (((id A) . f) ")) ^2)) is V28() V29() ext-real Element of REAL
f " is V28() V29() ext-real Element of REAL
1 * (f ") is V28() V29() ext-real Element of REAL
cos . (1 * (f ")) is V28() V29() ext-real Element of REAL
(cos . (1 * (f "))) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (1 * (f "))),(cos . (1 * (f ")))) is set
(diff (((id A) ^),f)) / ((cos . (1 * (f "))) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (1 * (f "))) ^2)) is V28() set
K97((diff (((id A) ^),f)),K99(((cos . (1 * (f "))) ^2))) is set
(- 1) * ((diff (((id A) ^),f)) / ((cos . (1 * (f "))) ^2)) is V28() V29() ext-real Element of REAL
(((id A) ^) `| A) . f is V28() V29() ext-real Element of REAL
((((id A) ^) `| A) . f) / ((cos . (1 * (f "))) ^2) is V28() V29() ext-real Element of REAL
K97(((((id A) ^) `| A) . f),K99(((cos . (1 * (f "))) ^2))) is set
(- 1) * (((((id A) ^) `| A) . f) / ((cos . (1 * (f "))) ^2)) is V28() V29() ext-real Element of REAL
1 / (f ^2) is V28() V29() ext-real Element of REAL
K99((f ^2)) is V28() set
K97(1,K99((f ^2))) is set
- (1 / (f ^2)) is V28() V29() ext-real Element of REAL
(- (1 / (f ^2))) / ((cos . (1 * (f "))) ^2) is V28() V29() ext-real Element of REAL
K97((- (1 / (f ^2))),K99(((cos . (1 * (f "))) ^2))) is set
(- 1) * ((- (1 / (f ^2))) / ((cos . (1 * (f "))) ^2)) is V28() V29() ext-real Element of REAL
(- 1) / (f ^2) is V28() V29() ext-real Element of REAL
K97((- 1),K99((f ^2))) is set
((- 1) / (f ^2)) / ((cos . (1 / f)) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (1 / f)) ^2)) is V28() set
K97(((- 1) / (f ^2)),K99(((cos . (1 / f)) ^2))) is set
(- 1) * (((- 1) / (f ^2)) / ((cos . (1 / f)) ^2)) is V28() V29() ext-real Element of REAL
(- 1) / ((f ^2) * ((cos . (1 / f)) ^2)) is V28() V29() ext-real Element of REAL
K97((- 1),K99(((f ^2) * ((cos . (1 / f)) ^2)))) is set
(- 1) * ((- 1) / ((f ^2) * ((cos . (1 / f)) ^2))) is V28() V29() ext-real Element of REAL
f is V28() V29() ext-real Element of REAL
((- (tan * ((id A) ^))) `| A) . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
1 / f is V28() V29() ext-real Element of REAL
K99(f) is V28() set
K97(1,K99(f)) is set
cos . (1 / f) is V28() V29() ext-real Element of REAL
(cos . (1 / f)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (1 / f)),(cos . (1 / f))) is set
(f ^2) * ((cos . (1 / f)) ^2) is V28() V29() ext-real Element of REAL
1 / ((f ^2) * ((cos . (1 / f)) ^2)) is V28() V29() ext-real Element of REAL
K99(((f ^2) * ((cos . (1 / f)) ^2))) is V28() set
K97(1,K99(((f ^2) * ((cos . (1 / f)) ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
tan * ((id f) ^) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (tan * ((id f) ^)) is V51() V52() V53() Element of K19(REAL)
- (tan * ((id f) ^)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (tan * ((id f) ^)) is V1() V6() set
(- (tan * ((id f) ^))) . (upper_bound A) is V28() V29() ext-real Element of REAL
(- (tan * ((id f) ^))) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (tan * ((id f) ^))) . (upper_bound A)) - ((- (tan * ((id f) ^))) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (tan * ((id f) ^))) . (lower_bound A))) is V28() set
K96(((- (tan * ((id f) ^))) . (upper_bound A)),K98(((- (tan * ((id f) ^))) . (lower_bound A)))) is set
(- (tan * ((id f) ^))) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- (tan * ((id f) ^))) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((- (tan * ((id f) ^))) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
Z ^2 is V28() V29() ext-real Element of REAL
K97(Z,Z) is set
1 / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97(1,K99(Z)) is set
cos . (1 / Z) is V28() V29() ext-real Element of REAL
(cos . (1 / Z)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (1 / Z)),(cos . (1 / Z))) is set
(Z ^2) * ((cos . (1 / Z)) ^2) is V28() V29() ext-real Element of REAL
1 / ((Z ^2) * ((cos . (1 / Z)) ^2)) is V28() V29() ext-real Element of REAL
K99(((Z ^2) * ((cos . (1 / Z)) ^2))) is V28() set
K97(1,K99(((Z ^2) * ((cos . (1 / Z)) ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V51() V52() V53() open Element of K19(REAL)
id f is V1() V4( REAL ) V4(f) V5( REAL ) V5(f) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
cot * ((id f) ^) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (cot * ((id f) ^)) is V51() V52() V53() Element of K19(REAL)
(cot * ((id f) ^)) . (upper_bound A) is V28() V29() ext-real Element of REAL
(cot * ((id f) ^)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((cot * ((id f) ^)) . (upper_bound A)) - ((cot * ((id f) ^)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((cot * ((id f) ^)) . (lower_bound A))) is V28() set
K96(((cot * ((id f) ^)) . (upper_bound A)),K98(((cot * ((id f) ^)) . (lower_bound A)))) is set
dom ((id f) ^) is V51() V52() V53() Element of K19(REAL)
dom (id f) is V51() V52() V53() Element of K19(f)
K19(f) is set
(id f) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id f)) \ ((id f) " {0}) is V51() V52() V53() Element of K19(f)
(dom (id f)) \ {0} is V51() V52() V53() Element of K19(f)
(cot * ((id f) ^)) `| f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((cot * ((id f) ^)) `| f) is V51() V52() V53() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((cot * ((id f) ^)) `| f) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
Z ^2 is V28() V29() ext-real Element of REAL
K97(Z,Z) is set
1 / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() set
K97(1,K99(Z)) is set
sin . (1 / Z) is V28() V29() ext-real Element of REAL
(sin . (1 / Z)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (1 / Z)),(sin . (1 / Z))) is set
(Z ^2) * ((sin . (1 / Z)) ^2) is V28() V29() ext-real Element of REAL
1 / ((Z ^2) * ((sin . (1 / Z)) ^2)) is V28() V29() ext-real Element of REAL
K99(((Z ^2) * ((sin . (1 / Z)) ^2))) is V28() set
K97(1,K99(((Z ^2) * ((sin . (1 / Z)) ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln * arctan) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln * arctan) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln * arctan) . (upper_bound A)) - ((ln * arctan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln * arctan) . (lower_bound A))) is V28() set
K96(((ln * arctan) . (upper_bound A)),K98(((ln * arctan) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(f1 + (#Z 2)) (#) arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((f1 + (#Z 2)) (#) arctan) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
dom ((f1 + (#Z 2)) (#) arctan) is V51() V52() V53() Element of K19(REAL)
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
dom arctan is V51() V52() V53() Element of K19(REAL)
(dom (f1 + (#Z 2))) /\ (dom arctan) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
(1 + (x ^2)) * (arctan . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x ^2)) * (arctan . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x ^2)) * (arctan . x))) is V28() set
K97(1,K99(((1 + (x ^2)) * (arctan . x)))) is set
(((f1 + (#Z 2)) (#) arctan) ^) . x is V28() V29() ext-real Element of REAL
((f1 + (#Z 2)) (#) arctan) . x is V28() V29() ext-real Element of REAL
1 / (((f1 + (#Z 2)) (#) arctan) . x) is V28() V29() ext-real Element of REAL
K99((((f1 + (#Z 2)) (#) arctan) . x)) is V28() set
K97(1,K99((((f1 + (#Z 2)) (#) arctan) . x))) is set
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((f1 + (#Z 2)) . x) * (arctan . x) is V28() V29() ext-real Element of REAL
1 / (((f1 + (#Z 2)) . x) * (arctan . x)) is V28() V29() ext-real Element of REAL
K99((((f1 + (#Z 2)) . x) * (arctan . x))) is V28() set
K97(1,K99((((f1 + (#Z 2)) . x) * (arctan . x)))) is set
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
((f1 . x) + ((#Z 2) . x)) * (arctan . x) is V28() V29() ext-real Element of REAL
1 / (((f1 . x) + ((#Z 2) . x)) * (arctan . x)) is V28() V29() ext-real Element of REAL
K99((((f1 . x) + ((#Z 2) . x)) * (arctan . x))) is V28() set
K97(1,K99((((f1 . x) + ((#Z 2) . x)) * (arctan . x)))) is set
x #Z 2 is V28() V29() ext-real Element of REAL
(f1 . x) + (x #Z 2) is V28() V29() ext-real Element of REAL
((f1 . x) + (x #Z 2)) * (arctan . x) is V28() V29() ext-real Element of REAL
1 / (((f1 . x) + (x #Z 2)) * (arctan . x)) is V28() V29() ext-real Element of REAL
K99((((f1 . x) + (x #Z 2)) * (arctan . x))) is V28() set
K97(1,K99((((f1 . x) + (x #Z 2)) * (arctan . x)))) is set
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(1 + (x #Z 2)) * (arctan . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x #Z 2)) * (arctan . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x #Z 2)) * (arctan . x))) is V28() set
K97(1,K99(((1 + (x #Z 2)) * (arctan . x)))) is set
(ln * arctan) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln * arctan) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((ln * arctan) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
(1 + (x ^2)) * (arctan . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x ^2)) * (arctan . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x ^2)) * (arctan . x))) is V28() set
K97(1,K99(((1 + (x ^2)) * (arctan . x)))) is set
A is V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() Element of NAT
A - 1 is V28() V29() V30() ext-real Element of REAL
K96(A,K98(1)) is V29() V30() set
#Z (A - 1) is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A - 1)) * arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
#Z A is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * arctan) is V51() V52() V53() Element of K19(REAL)
f1 is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
((#Z A) * arctan) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
((#Z A) * arctan) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((#Z A) * arctan) . (upper_bound f1)) - (((#Z A) * arctan) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((#Z A) * arctan) . (lower_bound f1))) is V28() set
K96((((#Z A) * arctan) . (upper_bound f1)),K98((((#Z A) * arctan) . (lower_bound f1)))) is set
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | f1 is V1() V4( REAL ) V4(f1) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,f1) is V28() V29() ext-real Element of REAL
Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((#Z (A - 1)) * arctan) / (Z + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) (((#Z (A - 1)) * arctan) / (Z + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V51() V52() V53() open Element of K19(REAL)
dom (((#Z (A - 1)) * arctan) / (Z + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
dom ((#Z (A - 1)) * arctan) is V51() V52() V53() Element of K19(REAL)
dom (Z + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(Z + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (Z + (#Z 2))) \ ((Z + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom ((#Z (A - 1)) * arctan)) /\ ((dom (Z + (#Z 2))) \ ((Z + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(Z + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((Z + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
f . h is V28() V29() ext-real Element of REAL
arctan . h is V28() V29() ext-real Element of REAL
(arctan . h) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arctan . h) #Z (A - 1)) is V28() V29() ext-real Element of REAL
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
1 + (h ^2) is V28() V29() ext-real Element of REAL
(A * ((arctan . h) #Z (A - 1))) / (1 + (h ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (h ^2))) is V28() set
K97((A * ((arctan . h) #Z (A - 1))),K99((1 + (h ^2)))) is set
(A (#) (((#Z (A - 1)) * arctan) / (Z + (#Z 2)))) . h is V28() V29() ext-real Element of REAL
(((#Z (A - 1)) * arctan) / (Z + (#Z 2))) . h is V28() V29() ext-real Element of REAL
A * ((((#Z (A - 1)) * arctan) / (Z + (#Z 2))) . h) is V28() V29() ext-real Element of REAL
((#Z (A - 1)) * arctan) . h is V28() V29() ext-real Element of REAL
(Z + (#Z 2)) . h is V28() V29() ext-real Element of REAL
((Z + (#Z 2)) . h) " is V28() V29() ext-real Element of REAL
(((#Z (A - 1)) * arctan) . h) * (((Z + (#Z 2)) . h) ") is V28() V29() ext-real Element of REAL
A * ((((#Z (A - 1)) * arctan) . h) * (((Z + (#Z 2)) . h) ")) is V28() V29() ext-real Element of REAL
A * (((#Z (A - 1)) * arctan) . h) is V28() V29() ext-real Element of REAL
(A * (((#Z (A - 1)) * arctan) . h)) / ((Z + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K99(((Z + (#Z 2)) . h)) is V28() set
K97((A * (((#Z (A - 1)) * arctan) . h)),K99(((Z + (#Z 2)) . h))) is set
(#Z (A - 1)) . (arctan . h) is V28() V29() ext-real Element of REAL
A * ((#Z (A - 1)) . (arctan . h)) is V28() V29() ext-real Element of REAL
(A * ((#Z (A - 1)) . (arctan . h))) / ((Z + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K97((A * ((#Z (A - 1)) . (arctan . h))),K99(((Z + (#Z 2)) . h))) is set
(A * ((arctan . h) #Z (A - 1))) / ((Z + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K97((A * ((arctan . h) #Z (A - 1))),K99(((Z + (#Z 2)) . h))) is set
Z . h is V28() V29() ext-real Element of REAL
(#Z 2) . h is V28() V29() ext-real Element of REAL
(Z . h) + ((#Z 2) . h) is V28() V29() ext-real Element of REAL
(A * ((arctan . h) #Z (A - 1))) / ((Z . h) + ((#Z 2) . h)) is V28() V29() ext-real Element of REAL
K99(((Z . h) + ((#Z 2) . h))) is V28() set
K97((A * ((arctan . h) #Z (A - 1))),K99(((Z . h) + ((#Z 2) . h)))) is set
1 + ((#Z 2) . h) is V28() V29() ext-real Element of REAL
(A * ((arctan . h) #Z (A - 1))) / (1 + ((#Z 2) . h)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . h))) is V28() set
K97((A * ((arctan . h) #Z (A - 1))),K99((1 + ((#Z 2) . h)))) is set
h #Z 2 is V28() V29() ext-real Element of REAL
1 + (h #Z 2) is V28() V29() ext-real Element of REAL
(A * ((arctan . h) #Z (A - 1))) / (1 + (h #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (h #Z 2))) is V28() set
K97((A * ((arctan . h) #Z (A - 1))),K99((1 + (h #Z 2)))) is set
((#Z A) * arctan) `| x is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#Z A) * arctan) `| x) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
(((#Z A) * arctan) `| x) . h is V28() V29() ext-real Element of REAL
f . h is V28() V29() ext-real Element of REAL
arctan . h is V28() V29() ext-real Element of REAL
(arctan . h) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arctan . h) #Z (A - 1)) is V28() V29() ext-real Element of REAL
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
1 + (h ^2) is V28() V29() ext-real Element of REAL
(A * ((arctan . h) #Z (A - 1))) / (1 + (h ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (h ^2))) is V28() set
K97((A * ((arctan . h) #Z (A - 1))),K99((1 + (h ^2)))) is set
A is V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() Element of NAT
A - 1 is V28() V29() V30() ext-real Element of REAL
K96(A,K98(1)) is V29() V30() set
#Z (A - 1) is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A - 1)) * arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
#Z A is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * arccot) is V51() V52() V53() Element of K19(REAL)
f1 is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
((#Z A) * arccot) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
((#Z A) * arccot) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((#Z A) * arccot) . (upper_bound f1)) - (((#Z A) * arccot) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((#Z A) * arccot) . (lower_bound f1))) is V28() set
K96((((#Z A) * arccot) . (upper_bound f1)),K98((((#Z A) * arccot) . (lower_bound f1)))) is set
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((#Z (A - 1)) * arccot) / (f + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2)))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2)))) is V1() V6() set
Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V51() V52() V53() Element of K19(REAL)
Z | f1 is V1() V4( REAL ) V4(f1) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (Z,f1) is V28() V29() ext-real Element of REAL
x is V51() V52() V53() open Element of K19(REAL)
dom (A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2)))) is V51() V52() V53() Element of K19(REAL)
dom (((#Z (A - 1)) * arccot) / (f + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
dom ((#Z (A - 1)) * arccot) is V51() V52() V53() Element of K19(REAL)
dom (f + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom ((#Z (A - 1)) * arccot)) /\ ((dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
Z . h is V28() V29() ext-real Element of REAL
arccot . h is V28() V29() ext-real Element of REAL
(arccot . h) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccot . h) #Z (A - 1)) is V28() V29() ext-real Element of REAL
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
1 + (h ^2) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / (1 + (h ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (h ^2))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99((1 + (h ^2)))) is set
- ((A * ((arccot . h) #Z (A - 1))) / (1 + (h ^2))) is V28() V29() ext-real Element of REAL
(- (A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2))))) . h is V28() V29() ext-real Element of REAL
(A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2)))) . h is V28() V29() ext-real Element of REAL
- ((A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2)))) . h) is V28() V29() ext-real Element of REAL
(((#Z (A - 1)) * arccot) / (f + (#Z 2))) . h is V28() V29() ext-real Element of REAL
A * ((((#Z (A - 1)) * arccot) / (f + (#Z 2))) . h) is V28() V29() ext-real Element of REAL
- (A * ((((#Z (A - 1)) * arccot) / (f + (#Z 2))) . h)) is V28() V29() ext-real Element of REAL
((#Z (A - 1)) * arccot) . h is V28() V29() ext-real Element of REAL
(f + (#Z 2)) . h is V28() V29() ext-real Element of REAL
((f + (#Z 2)) . h) " is V28() V29() ext-real Element of REAL
(((#Z (A - 1)) * arccot) . h) * (((f + (#Z 2)) . h) ") is V28() V29() ext-real Element of REAL
A * ((((#Z (A - 1)) * arccot) . h) * (((f + (#Z 2)) . h) ")) is V28() V29() ext-real Element of REAL
- (A * ((((#Z (A - 1)) * arccot) . h) * (((f + (#Z 2)) . h) "))) is V28() V29() ext-real Element of REAL
A * (((#Z (A - 1)) * arccot) . h) is V28() V29() ext-real Element of REAL
(A * (((#Z (A - 1)) * arccot) . h)) / ((f + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K99(((f + (#Z 2)) . h)) is V28() set
K97((A * (((#Z (A - 1)) * arccot) . h)),K99(((f + (#Z 2)) . h))) is set
- ((A * (((#Z (A - 1)) * arccot) . h)) / ((f + (#Z 2)) . h)) is V28() V29() ext-real Element of REAL
(#Z (A - 1)) . (arccot . h) is V28() V29() ext-real Element of REAL
A * ((#Z (A - 1)) . (arccot . h)) is V28() V29() ext-real Element of REAL
(A * ((#Z (A - 1)) . (arccot . h))) / ((f + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K97((A * ((#Z (A - 1)) . (arccot . h))),K99(((f + (#Z 2)) . h))) is set
- ((A * ((#Z (A - 1)) . (arccot . h))) / ((f + (#Z 2)) . h)) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / ((f + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K97((A * ((arccot . h) #Z (A - 1))),K99(((f + (#Z 2)) . h))) is set
- ((A * ((arccot . h) #Z (A - 1))) / ((f + (#Z 2)) . h)) is V28() V29() ext-real Element of REAL
f . h is V28() V29() ext-real Element of REAL
(#Z 2) . h is V28() V29() ext-real Element of REAL
(f . h) + ((#Z 2) . h) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / ((f . h) + ((#Z 2) . h)) is V28() V29() ext-real Element of REAL
K99(((f . h) + ((#Z 2) . h))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99(((f . h) + ((#Z 2) . h)))) is set
- ((A * ((arccot . h) #Z (A - 1))) / ((f . h) + ((#Z 2) . h))) is V28() V29() ext-real Element of REAL
1 + ((#Z 2) . h) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / (1 + ((#Z 2) . h)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . h))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99((1 + ((#Z 2) . h)))) is set
- ((A * ((arccot . h) #Z (A - 1))) / (1 + ((#Z 2) . h))) is V28() V29() ext-real Element of REAL
h #Z 2 is V28() V29() ext-real Element of REAL
1 + (h #Z 2) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / (1 + (h #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (h #Z 2))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99((1 + (h #Z 2)))) is set
- ((A * ((arccot . h) #Z (A - 1))) / (1 + (h #Z 2))) is V28() V29() ext-real Element of REAL
((#Z A) * arccot) `| x is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#Z A) * arccot) `| x) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
(((#Z A) * arccot) `| x) . h is V28() V29() ext-real Element of REAL
Z . h is V28() V29() ext-real Element of REAL
arccot . h is V28() V29() ext-real Element of REAL
(arccot . h) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccot . h) #Z (A - 1)) is V28() V29() ext-real Element of REAL
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
1 + (h ^2) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / (1 + (h ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (h ^2))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99((1 + (h ^2)))) is set
- ((A * ((arccot . h) #Z (A - 1))) / (1 + (h ^2))) is V28() V29() ext-real Element of REAL
A is V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() Element of NAT
#Z A is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * arccot) is V51() V52() V53() Element of K19(REAL)
- ((#Z A) * arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((#Z A) * arccot) is V1() V6() set
A - 1 is V28() V29() V30() ext-real Element of REAL
K96(A,K98(1)) is V29() V30() set
f1 is V51() V52() V53() open Element of K19(REAL)
(- ((#Z A) * arccot)) `| f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- ((#Z A) * arccot)) is V51() V52() V53() Element of K19(REAL)
(- 1) (#) ((#Z A) * arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V28() V29() ext-real Element of REAL
((- ((#Z A) * arccot)) `| f1) . f is V28() V29() ext-real Element of REAL
arccot . f is V28() V29() ext-real Element of REAL
(arccot . f) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccot . f) #Z (A - 1)) is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
1 + (f ^2) is V28() V29() ext-real Element of REAL
(A * ((arccot . f) #Z (A - 1))) / (1 + (f ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f ^2))) is V28() set
K97((A * ((arccot . f) #Z (A - 1))),K99((1 + (f ^2)))) is set
diff ((- ((#Z A) * arccot)),f) is V28() V29() ext-real Element of REAL
diff (((#Z A) * arccot),f) is V28() V29() ext-real Element of REAL
(- 1) * (diff (((#Z A) * arccot),f)) is V28() V29() ext-real Element of REAL
diff (arccot,f) is V28() V29() ext-real Element of REAL
(A * ((arccot . f) #Z (A - 1))) * (diff (arccot,f)) is V28() V29() ext-real Element of REAL
(- 1) * ((A * ((arccot . f) #Z (A - 1))) * (diff (arccot,f))) is V28() V29() ext-real Element of REAL
1 / (1 + (f ^2)) is V28() V29() ext-real Element of REAL
K97(1,K99((1 + (f ^2)))) is set
- (1 / (1 + (f ^2))) is V28() V29() ext-real Element of REAL
(A * ((arccot . f) #Z (A - 1))) * (- (1 / (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * ((A * ((arccot . f) #Z (A - 1))) * (- (1 / (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
f is V28() V29() ext-real Element of REAL
((- ((#Z A) * arccot)) `| f1) . f is V28() V29() ext-real Element of REAL
arccot . f is V28() V29() ext-real Element of REAL
(arccot . f) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccot . f) #Z (A - 1)) is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
1 + (f ^2) is V28() V29() ext-real Element of REAL
(A * ((arccot . f) #Z (A - 1))) / (1 + (f ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f ^2))) is V28() set
K97((A * ((arccot . f) #Z (A - 1))),K99((1 + (f ^2)))) is set
A is V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() Element of NAT
A - 1 is V28() V29() V30() ext-real Element of REAL
K96(A,K98(1)) is V29() V30() set
#Z (A - 1) is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A - 1)) * arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
#Z A is V1() V4( REAL ) V5( REAL ) V6() V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * arccot) is V51() V52() V53() Element of K19(REAL)
- ((#Z A) * arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((#Z A) * arccot) is V1() V6() set
f1 is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
(- ((#Z A) * arccot)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
(- ((#Z A) * arccot)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
((- ((#Z A) * arccot)) . (upper_bound f1)) - ((- ((#Z A) * arccot)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98(((- ((#Z A) * arccot)) . (lower_bound f1))) is V28() set
K96(((- ((#Z A) * arccot)) . (upper_bound f1)),K98(((- ((#Z A) * arccot)) . (lower_bound f1)))) is set
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((#Z (A - 1)) * arccot) / (f + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V51() V52() V53() Element of K19(REAL)
Z | f1 is V1() V4( REAL ) V4(f1) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (Z,f1) is V28() V29() ext-real Element of REAL
x is V51() V52() V53() open Element of K19(REAL)
dom (((#Z (A - 1)) * arccot) / (f + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
dom ((#Z (A - 1)) * arccot) is V51() V52() V53() Element of K19(REAL)
dom (f + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom ((#Z (A - 1)) * arccot)) /\ ((dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
Z . h is V28() V29() ext-real Element of REAL
arccot . h is V28() V29() ext-real Element of REAL
(arccot . h) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccot . h) #Z (A - 1)) is V28() V29() ext-real Element of REAL
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
1 + (h ^2) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / (1 + (h ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (h ^2))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99((1 + (h ^2)))) is set
(A (#) (((#Z (A - 1)) * arccot) / (f + (#Z 2)))) . h is V28() V29() ext-real Element of REAL
(((#Z (A - 1)) * arccot) / (f + (#Z 2))) . h is V28() V29() ext-real Element of REAL
A * ((((#Z (A - 1)) * arccot) / (f + (#Z 2))) . h) is V28() V29() ext-real Element of REAL
((#Z (A - 1)) * arccot) . h is V28() V29() ext-real Element of REAL
(f + (#Z 2)) . h is V28() V29() ext-real Element of REAL
((f + (#Z 2)) . h) " is V28() V29() ext-real Element of REAL
(((#Z (A - 1)) * arccot) . h) * (((f + (#Z 2)) . h) ") is V28() V29() ext-real Element of REAL
A * ((((#Z (A - 1)) * arccot) . h) * (((f + (#Z 2)) . h) ")) is V28() V29() ext-real Element of REAL
A * (((#Z (A - 1)) * arccot) . h) is V28() V29() ext-real Element of REAL
(A * (((#Z (A - 1)) * arccot) . h)) / ((f + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K99(((f + (#Z 2)) . h)) is V28() set
K97((A * (((#Z (A - 1)) * arccot) . h)),K99(((f + (#Z 2)) . h))) is set
(#Z (A - 1)) . (arccot . h) is V28() V29() ext-real Element of REAL
A * ((#Z (A - 1)) . (arccot . h)) is V28() V29() ext-real Element of REAL
(A * ((#Z (A - 1)) . (arccot . h))) / ((f + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K97((A * ((#Z (A - 1)) . (arccot . h))),K99(((f + (#Z 2)) . h))) is set
(A * ((arccot . h) #Z (A - 1))) / ((f + (#Z 2)) . h) is V28() V29() ext-real Element of REAL
K97((A * ((arccot . h) #Z (A - 1))),K99(((f + (#Z 2)) . h))) is set
f . h is V28() V29() ext-real Element of REAL
(#Z 2) . h is V28() V29() ext-real Element of REAL
(f . h) + ((#Z 2) . h) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / ((f . h) + ((#Z 2) . h)) is V28() V29() ext-real Element of REAL
K99(((f . h) + ((#Z 2) . h))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99(((f . h) + ((#Z 2) . h)))) is set
1 + ((#Z 2) . h) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / (1 + ((#Z 2) . h)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . h))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99((1 + ((#Z 2) . h)))) is set
h #Z 2 is V28() V29() ext-real Element of REAL
1 + (h #Z 2) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / (1 + (h #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (h #Z 2))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99((1 + (h #Z 2)))) is set
(- ((#Z A) * arccot)) `| x is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- ((#Z A) * arccot)) `| x) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
((- ((#Z A) * arccot)) `| x) . h is V28() V29() ext-real Element of REAL
Z . h is V28() V29() ext-real Element of REAL
arccot . h is V28() V29() ext-real Element of REAL
(arccot . h) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccot . h) #Z (A - 1)) is V28() V29() ext-real Element of REAL
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
1 + (h ^2) is V28() V29() ext-real Element of REAL
(A * ((arccot . h) #Z (A - 1))) / (1 + (h ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (h ^2))) is V28() set
K97((A * ((arccot . h) #Z (A - 1))),K99((1 + (h ^2)))) is set
dom ((#Z 2) * arctan) is V51() V52() V53() Element of K19(REAL)
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
((1 / 2) (#) ((#Z 2) * arctan)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
((1 / 2) (#) ((#Z 2) * arctan)) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((1 / 2) (#) ((#Z 2) * arctan)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arctan)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((1 / 2) (#) ((#Z 2) * arctan)) . (lower_bound A))) is V28() set
K96((((1 / 2) (#) ((#Z 2) * arctan)) . (upper_bound A)),K98((((1 / 2) (#) ((#Z 2) * arctan)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arctan / (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
dom arctan is V51() V52() V53() Element of K19(REAL)
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom arctan) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(arctan . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((arctan . x),K99((1 + (x ^2)))) is set
(arctan / (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
(arctan . x) / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . x)) is V28() set
K97((arctan . x),K99(((f1 + (#Z 2)) . x))) is set
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(arctan . x) / ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + ((#Z 2) . x))) is V28() set
K97((arctan . x),K99(((f1 . x) + ((#Z 2) . x)))) is set
x #Z 2 is V28() V29() ext-real Element of REAL
(f1 . x) + (x #Z 2) is V28() V29() ext-real Element of REAL
(arctan . x) / ((f1 . x) + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + (x #Z 2))) is V28() set
K97((arctan . x),K99(((f1 . x) + (x #Z 2)))) is set
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(arctan . x) / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97((arctan . x),K99((1 + (x #Z 2)))) is set
((1 / 2) (#) ((#Z 2) * arctan)) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((1 / 2) (#) ((#Z 2) * arctan)) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((1 / 2) (#) ((#Z 2) * arctan)) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(arctan . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((arctan . x),K99((1 + (x ^2)))) is set
dom ((#Z 2) * arccot) is V51() V52() V53() Element of K19(REAL)
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
((1 / 2) (#) ((#Z 2) * arccot)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
((1 / 2) (#) ((#Z 2) * arccot)) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((1 / 2) (#) ((#Z 2) * arccot)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arccot)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((1 / 2) (#) ((#Z 2) * arccot)) . (lower_bound A))) is V28() set
K96((((1 / 2) (#) ((#Z 2) * arccot)) . (upper_bound A)),K98((((1 / 2) (#) ((#Z 2) * arccot)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccot / (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (arccot / (f1 + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (arccot / (f1 + (#Z 2))) is V1() V6() set
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
dom (arccot / (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
dom arccot is V51() V52() V53() Element of K19(REAL)
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom arccot) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(arccot . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((arccot . x),K99((1 + (x ^2)))) is set
- ((arccot . x) / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
(- (arccot / (f1 + (#Z 2)))) . x is V28() V29() ext-real Element of REAL
(arccot / (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
- ((arccot / (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
(arccot . x) / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . x)) is V28() set
K97((arccot . x),K99(((f1 + (#Z 2)) . x))) is set
- ((arccot . x) / ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(arccot . x) / ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + ((#Z 2) . x))) is V28() set
K97((arccot . x),K99(((f1 . x) + ((#Z 2) . x)))) is set
- ((arccot . x) / ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
x #Z 2 is V28() V29() ext-real Element of REAL
(f1 . x) + (x #Z 2) is V28() V29() ext-real Element of REAL
(arccot . x) / ((f1 . x) + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + (x #Z 2))) is V28() set
K97((arccot . x),K99(((f1 . x) + (x #Z 2)))) is set
- ((arccot . x) / ((f1 . x) + (x #Z 2))) is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(arccot . x) / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97((arccot . x),K99((1 + (x #Z 2)))) is set
- ((arccot . x) / (1 + (x #Z 2))) is V28() V29() ext-real Element of REAL
((1 / 2) (#) ((#Z 2) * arccot)) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((1 / 2) (#) ((#Z 2) * arccot)) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((1 / 2) (#) ((#Z 2) * arccot)) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(arccot . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((arccot . x),K99((1 + (x ^2)))) is set
- ((arccot . x) / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
- ((1 / 2) (#) ((#Z 2) * arccot)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((1 / 2) (#) ((#Z 2) * arccot)) is V1() V6() set
A is V51() V52() V53() open Element of K19(REAL)
(- ((1 / 2) (#) ((#Z 2) * arccot))) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- ((1 / 2) (#) ((#Z 2) * arccot))) is V51() V52() V53() Element of K19(REAL)
(- 1) (#) ((1 / 2) (#) ((#Z 2) * arccot)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((#Z 2) * arccot) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
2 - 1 is V28() V29() V30() ext-real Element of REAL
K96(2,K98(1)) is V29() V30() set
f1 is V28() V29() ext-real Element of REAL
(((#Z 2) * arccot) `| A) . f1 is V28() V29() ext-real Element of REAL
arccot . f1 is V28() V29() ext-real Element of REAL
(arccot . f1) #Z (2 - 1) is V28() V29() ext-real Element of REAL
2 * ((arccot . f1) #Z (2 - 1)) is V28() V29() ext-real Element of REAL
f1 ^2 is V28() V29() ext-real Element of REAL
K97(f1,f1) is set
1 + (f1 ^2) is V28() V29() ext-real Element of REAL
(2 * ((arccot . f1) #Z (2 - 1))) / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f1 ^2))) is V28() set
K97((2 * ((arccot . f1) #Z (2 - 1))),K99((1 + (f1 ^2)))) is set
- ((2 * ((arccot . f1) #Z (2 - 1))) / (1 + (f1 ^2))) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| A) . f1 is V28() V29() ext-real Element of REAL
arccot . f1 is V28() V29() ext-real Element of REAL
f1 ^2 is V28() V29() ext-real Element of REAL
K97(f1,f1) is set
1 + (f1 ^2) is V28() V29() ext-real Element of REAL
(arccot . f1) / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f1 ^2))) is V28() set
K97((arccot . f1),K99((1 + (f1 ^2)))) is set
diff ((- ((1 / 2) (#) ((#Z 2) * arccot))),f1) is V28() V29() ext-real Element of REAL
diff (((1 / 2) (#) ((#Z 2) * arccot)),f1) is V28() V29() ext-real Element of REAL
(- 1) * (diff (((1 / 2) (#) ((#Z 2) * arccot)),f1)) is V28() V29() ext-real Element of REAL
diff (((#Z 2) * arccot),f1) is V28() V29() ext-real Element of REAL
(1 / 2) * (diff (((#Z 2) * arccot),f1)) is V28() V29() ext-real Element of REAL
(- 1) * ((1 / 2) * (diff (((#Z 2) * arccot),f1))) is V28() V29() ext-real Element of REAL
(((#Z 2) * arccot) `| A) . f1 is V28() V29() ext-real Element of REAL
(1 / 2) * ((((#Z 2) * arccot) `| A) . f1) is V28() V29() ext-real Element of REAL
(- 1) * ((1 / 2) * ((((#Z 2) * arccot) `| A) . f1)) is V28() V29() ext-real Element of REAL
(arccot . f1) #Z (2 - 1) is V28() V29() ext-real Element of REAL
2 * ((arccot . f1) #Z (2 - 1)) is V28() V29() ext-real Element of REAL
(2 * ((arccot . f1) #Z (2 - 1))) / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K97((2 * ((arccot . f1) #Z (2 - 1))),K99((1 + (f1 ^2)))) is set
- ((2 * ((arccot . f1) #Z (2 - 1))) / (1 + (f1 ^2))) is V28() V29() ext-real Element of REAL
(1 / 2) * (- ((2 * ((arccot . f1) #Z (2 - 1))) / (1 + (f1 ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * ((1 / 2) * (- ((2 * ((arccot . f1) #Z (2 - 1))) / (1 + (f1 ^2))))) is V28() V29() ext-real Element of REAL
(arccot . f1) #Z 1 is V28() V29() ext-real Element of REAL
2 * ((arccot . f1) #Z 1) is V28() V29() ext-real Element of REAL
(2 * ((arccot . f1) #Z 1)) / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K97((2 * ((arccot . f1) #Z 1)),K99((1 + (f1 ^2)))) is set
(1 / 2) * ((2 * ((arccot . f1) #Z 1)) / (1 + (f1 ^2))) is V28() V29() ext-real Element of REAL
- ((1 / 2) * ((2 * ((arccot . f1) #Z 1)) / (1 + (f1 ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * (- ((1 / 2) * ((2 * ((arccot . f1) #Z 1)) / (1 + (f1 ^2))))) is V28() V29() ext-real Element of REAL
2 * (arccot . f1) is V28() V29() ext-real Element of REAL
(2 * (arccot . f1)) / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K97((2 * (arccot . f1)),K99((1 + (f1 ^2)))) is set
(1 / 2) * ((2 * (arccot . f1)) / (1 + (f1 ^2))) is V28() V29() ext-real Element of REAL
- ((1 / 2) * ((2 * (arccot . f1)) / (1 + (f1 ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * (- ((1 / 2) * ((2 * (arccot . f1)) / (1 + (f1 ^2))))) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| A) . f1 is V28() V29() ext-real Element of REAL
arccot . f1 is V28() V29() ext-real Element of REAL
f1 ^2 is V28() V29() ext-real Element of REAL
K97(f1,f1) is set
1 + (f1 ^2) is V28() V29() ext-real Element of REAL
(arccot . f1) / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f1 ^2))) is V28() set
K97((arccot . f1),K99((1 + (f1 ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(- ((1 / 2) (#) ((#Z 2) * arccot))) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- ((1 / 2) (#) ((#Z 2) * arccot))) . (upper_bound A)) - ((- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A))) is V28() set
K96(((- ((1 / 2) (#) ((#Z 2) * arccot))) . (upper_bound A)),K98(((- ((1 / 2) (#) ((#Z 2) * arccot))) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccot / (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
dom arccot is V51() V52() V53() Element of K19(REAL)
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom arccot) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(arccot . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((arccot . x),K99((1 + (x ^2)))) is set
(arccot / (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
(arccot . x) / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . x)) is V28() set
K97((arccot . x),K99(((f1 + (#Z 2)) . x))) is set
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(arccot . x) / ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + ((#Z 2) . x))) is V28() set
K97((arccot . x),K99(((f1 . x) + ((#Z 2) . x)))) is set
x #Z 2 is V28() V29() ext-real Element of REAL
(f1 . x) + (x #Z 2) is V28() V29() ext-real Element of REAL
(arccot . x) / ((f1 . x) + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + (x #Z 2))) is V28() set
K97((arccot . x),K99(((f1 . x) + (x #Z 2)))) is set
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(arccot . x) / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97((arccot . x),K99((1 + (x #Z 2)))) is set
(- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(arccot . x) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((arccot . x),K99((1 + (x ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) / (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arctan + ((id Z) / (f1 + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) (#) arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) (#) arctan) . (upper_bound A) is V28() V29() ext-real Element of REAL
((id Z) (#) arctan) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((id Z) (#) arctan) . (upper_bound A)) - (((id Z) (#) arctan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((id Z) (#) arctan) . (lower_bound A))) is V28() set
K96((((id Z) (#) arctan) . (upper_bound A)),K98((((id Z) (#) arctan) . (lower_bound A)))) is set
dom arctan is V51() V52() V53() Element of K19(REAL)
dom ((id Z) / (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
(dom arctan) /\ (dom ((id Z) / (f1 + (#Z 2)))) is V51() V52() V53() Element of K19(REAL)
dom (id Z) is V51() V52() V53() Element of K19(Z)
K19(Z) is set
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (id Z)) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
x / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97(x,K99((1 + (x ^2)))) is set
(arctan . x) + (x / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
(arctan + ((id Z) / (f1 + (#Z 2)))) . x is V28() V29() ext-real Element of REAL
((id Z) / (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
(arctan . x) + (((id Z) / (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
(id Z) . x is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((id Z) . x) / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . x)) is V28() set
K97(((id Z) . x),K99(((f1 + (#Z 2)) . x))) is set
(arctan . x) + (((id Z) . x) / ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
x / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K97(x,K99(((f1 + (#Z 2)) . x))) is set
(arctan . x) + (x / ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
x / ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + ((#Z 2) . x))) is V28() set
K97(x,K99(((f1 . x) + ((#Z 2) . x)))) is set
(arctan . x) + (x / ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
x / (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . x))) is V28() set
K97(x,K99((1 + ((#Z 2) . x)))) is set
(arctan . x) + (x / (1 + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
x #Z 2 is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
x / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97(x,K99((1 + (x #Z 2)))) is set
(arctan . x) + (x / (1 + (x #Z 2))) is V28() V29() ext-real Element of REAL
((id Z) (#) arctan) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id Z) (#) arctan) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((id Z) (#) arctan) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
x / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97(x,K99((1 + (x ^2)))) is set
(arctan . x) + (x / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) / (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccot - ((id Z) / (f1 + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((id Z) / (f1 + (#Z 2))) is V1() V6() V34() set
K98(1) (#) ((id Z) / (f1 + (#Z 2))) is V1() V6() set
arccot + (- ((id Z) / (f1 + (#Z 2)))) is V1() V6() set
(id Z) (#) arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) (#) arccot) . (upper_bound A) is V28() V29() ext-real Element of REAL
((id Z) (#) arccot) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((id Z) (#) arccot) . (upper_bound A)) - (((id Z) (#) arccot) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((id Z) (#) arccot) . (lower_bound A))) is V28() set
K96((((id Z) (#) arccot) . (upper_bound A)),K98((((id Z) (#) arccot) . (lower_bound A)))) is set
dom arccot is V51() V52() V53() Element of K19(REAL)
dom ((id Z) / (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
(dom arccot) /\ (dom ((id Z) / (f1 + (#Z 2)))) is V51() V52() V53() Element of K19(REAL)
dom (id Z) is V51() V52() V53() Element of K19(Z)
K19(Z) is set
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (id Z)) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f1 + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
x / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97(x,K99((1 + (x ^2)))) is set
(arccot . x) - (x / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
K98((x / (1 + (x ^2)))) is V28() set
K96((arccot . x),K98((x / (1 + (x ^2))))) is set
(arccot - ((id Z) / (f1 + (#Z 2)))) . x is V28() V29() ext-real Element of REAL
((id Z) / (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
(arccot . x) - (((id Z) / (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K98((((id Z) / (f1 + (#Z 2))) . x)) is V28() set
K96((arccot . x),K98((((id Z) / (f1 + (#Z 2))) . x))) is set
(id Z) . x is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((id Z) . x) / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . x)) is V28() set
K97(((id Z) . x),K99(((f1 + (#Z 2)) . x))) is set
(arccot . x) - (((id Z) . x) / ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K98((((id Z) . x) / ((f1 + (#Z 2)) . x))) is V28() set
K96((arccot . x),K98((((id Z) . x) / ((f1 + (#Z 2)) . x)))) is set
x / ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K97(x,K99(((f1 + (#Z 2)) . x))) is set
(arccot . x) - (x / ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K98((x / ((f1 + (#Z 2)) . x))) is V28() set
K96((arccot . x),K98((x / ((f1 + (#Z 2)) . x)))) is set
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
x / ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f1 . x) + ((#Z 2) . x))) is V28() set
K97(x,K99(((f1 . x) + ((#Z 2) . x)))) is set
(arccot . x) - (x / ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K98((x / ((f1 . x) + ((#Z 2) . x)))) is V28() set
K96((arccot . x),K98((x / ((f1 . x) + ((#Z 2) . x))))) is set
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
x / (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . x))) is V28() set
K97(x,K99((1 + ((#Z 2) . x)))) is set
(arccot . x) - (x / (1 + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K98((x / (1 + ((#Z 2) . x)))) is V28() set
K96((arccot . x),K98((x / (1 + ((#Z 2) . x))))) is set
x #Z 2 is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
x / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97(x,K99((1 + (x #Z 2)))) is set
(arccot . x) - (x / (1 + (x #Z 2))) is V28() V29() ext-real Element of REAL
K98((x / (1 + (x #Z 2)))) is V28() set
K96((arccot . x),K98((x / (1 + (x #Z 2))))) is set
((id Z) (#) arccot) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id Z) (#) arccot) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((id Z) (#) arccot) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
x / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97(x,K99((1 + (x ^2)))) is set
(arccot . x) - (x / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
K98((x / (1 + (x ^2)))) is V28() set
K96((arccot . x),K98((x / (1 + (x ^2))))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(exp_R * arctan) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(exp_R * arctan) . (lower_bound A) is V28() V29() ext-real Element of REAL
((exp_R * arctan) . (upper_bound A)) - ((exp_R * arctan) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((exp_R * arctan) . (lower_bound A))) is V28() set
K96(((exp_R * arctan) . (upper_bound A)),K98(((exp_R * arctan) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(exp_R * arctan) / (f + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V51() V52() V53() open Element of K19(REAL)
dom (f + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (exp_R * arctan)) /\ ((dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
exp_R . (arctan . x) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(exp_R . (arctan . x)) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((exp_R . (arctan . x)),K99((1 + (x ^2)))) is set
((exp_R * arctan) / (f + (#Z 2))) . x is V28() V29() ext-real Element of REAL
(exp_R * arctan) . x is V28() V29() ext-real Element of REAL
(f + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((exp_R * arctan) . x) / ((f + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f + (#Z 2)) . x)) is V28() set
K97(((exp_R * arctan) . x),K99(((f + (#Z 2)) . x))) is set
(exp_R . (arctan . x)) / ((f + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K97((exp_R . (arctan . x)),K99(((f + (#Z 2)) . x))) is set
f . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(exp_R . (arctan . x)) / ((f . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f . x) + ((#Z 2) . x))) is V28() set
K97((exp_R . (arctan . x)),K99(((f . x) + ((#Z 2) . x)))) is set
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(exp_R . (arctan . x)) / (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . x))) is V28() set
K97((exp_R . (arctan . x)),K99((1 + ((#Z 2) . x)))) is set
x #Z 2 is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(exp_R . (arctan . x)) / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97((exp_R . (arctan . x)),K99((1 + (x #Z 2)))) is set
(exp_R * arctan) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R * arctan) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((exp_R * arctan) `| Z) . x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
exp_R . (arctan . x) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(exp_R . (arctan . x)) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((exp_R . (arctan . x)),K99((1 + (x ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(exp_R * arccot) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(exp_R * arccot) . (lower_bound A) is V28() V29() ext-real Element of REAL
((exp_R * arccot) . (upper_bound A)) - ((exp_R * arccot) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((exp_R * arccot) . (lower_bound A))) is V28() set
K96(((exp_R * arccot) . (upper_bound A)),K98(((exp_R * arccot) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(exp_R * arccot) / (f + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((exp_R * arccot) / (f + (#Z 2))) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((exp_R * arccot) / (f + (#Z 2))) is V1() V6() set
Z is V51() V52() V53() open Element of K19(REAL)
dom ((exp_R * arccot) / (f + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
dom (f + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (exp_R * arccot)) /\ ((dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
exp_R . (arccot . x) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((exp_R . (arccot . x)),K99((1 + (x ^2)))) is set
- ((exp_R . (arccot . x)) / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
(- ((exp_R * arccot) / (f + (#Z 2)))) . x is V28() V29() ext-real Element of REAL
((exp_R * arccot) / (f + (#Z 2))) . x is V28() V29() ext-real Element of REAL
- (((exp_R * arccot) / (f + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
(exp_R * arccot) . x is V28() V29() ext-real Element of REAL
(f + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((exp_R * arccot) . x) / ((f + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f + (#Z 2)) . x)) is V28() set
K97(((exp_R * arccot) . x),K99(((f + (#Z 2)) . x))) is set
- (((exp_R * arccot) . x) / ((f + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / ((f + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K97((exp_R . (arccot . x)),K99(((f + (#Z 2)) . x))) is set
- ((exp_R . (arccot . x)) / ((f + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / ((f . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f . x) + ((#Z 2) . x))) is V28() set
K97((exp_R . (arccot . x)),K99(((f . x) + ((#Z 2) . x)))) is set
- ((exp_R . (arccot . x)) / ((f . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . x))) is V28() set
K97((exp_R . (arccot . x)),K99((1 + ((#Z 2) . x)))) is set
- ((exp_R . (arccot . x)) / (1 + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
x #Z 2 is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97((exp_R . (arccot . x)),K99((1 + (x #Z 2)))) is set
- ((exp_R . (arccot . x)) / (1 + (x #Z 2))) is V28() V29() ext-real Element of REAL
(exp_R * arccot) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R * arccot) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((exp_R * arccot) `| Z) . x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
exp_R . (arccot . x) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((exp_R . (arccot . x)),K99((1 + (x ^2)))) is set
- ((exp_R . (arccot . x)) / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
- (exp_R * arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (exp_R * arccot) is V1() V6() set
A is V51() V52() V53() open Element of K19(REAL)
(- (exp_R * arccot)) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (exp_R * arccot)) is V51() V52() V53() Element of K19(REAL)
(- 1) (#) (exp_R * arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
((- (exp_R * arccot)) `| A) . f1 is V28() V29() ext-real Element of REAL
arccot . f1 is V28() V29() ext-real Element of REAL
exp_R . (arccot . f1) is V28() V29() ext-real Element of REAL
f1 ^2 is V28() V29() ext-real Element of REAL
K97(f1,f1) is set
1 + (f1 ^2) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . f1)) / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f1 ^2))) is V28() set
K97((exp_R . (arccot . f1)),K99((1 + (f1 ^2)))) is set
diff ((- (exp_R * arccot)),f1) is V28() V29() ext-real Element of REAL
diff ((exp_R * arccot),f1) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((exp_R * arccot),f1)) is V28() V29() ext-real Element of REAL
diff (exp_R,(arccot . f1)) is V28() V29() ext-real Element of REAL
diff (arccot,f1) is V28() V29() ext-real Element of REAL
(diff (exp_R,(arccot . f1))) * (diff (arccot,f1)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (exp_R,(arccot . f1))) * (diff (arccot,f1))) is V28() V29() ext-real Element of REAL
arccot `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arccot `| A) . f1 is V28() V29() ext-real Element of REAL
(diff (exp_R,(arccot . f1))) * ((arccot `| A) . f1) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (exp_R,(arccot . f1))) * ((arccot `| A) . f1)) is V28() V29() ext-real Element of REAL
1 / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K97(1,K99((1 + (f1 ^2)))) is set
- (1 / (1 + (f1 ^2))) is V28() V29() ext-real Element of REAL
(diff (exp_R,(arccot . f1))) * (- (1 / (1 + (f1 ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (exp_R,(arccot . f1))) * (- (1 / (1 + (f1 ^2))))) is V28() V29() ext-real Element of REAL
(diff (exp_R,(arccot . f1))) * (1 / (1 + (f1 ^2))) is V28() V29() ext-real Element of REAL
- ((diff (exp_R,(arccot . f1))) * (1 / (1 + (f1 ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * (- ((diff (exp_R,(arccot . f1))) * (1 / (1 + (f1 ^2))))) is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
((- (exp_R * arccot)) `| A) . f1 is V28() V29() ext-real Element of REAL
arccot . f1 is V28() V29() ext-real Element of REAL
exp_R . (arccot . f1) is V28() V29() ext-real Element of REAL
f1 ^2 is V28() V29() ext-real Element of REAL
K97(f1,f1) is set
1 + (f1 ^2) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . f1)) / (1 + (f1 ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f1 ^2))) is V28() set
K97((exp_R . (arccot . f1)),K99((1 + (f1 ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(- (exp_R * arccot)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(- (exp_R * arccot)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (exp_R * arccot)) . (upper_bound A)) - ((- (exp_R * arccot)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (exp_R * arccot)) . (lower_bound A))) is V28() set
K96(((- (exp_R * arccot)) . (upper_bound A)),K98(((- (exp_R * arccot)) . (lower_bound A)))) is set
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V51() V52() V53() Element of K19(REAL)
f1 | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(exp_R * arccot) / (f + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V51() V52() V53() open Element of K19(REAL)
dom (f + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (exp_R * arccot)) /\ ((dom (f + (#Z 2))) \ ((f + (#Z 2)) " {0})) is V51() V52() V53() Element of K19(REAL)
(f + (#Z 2)) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f + (#Z 2)) ^) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
exp_R . (arccot . x) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((exp_R . (arccot . x)),K99((1 + (x ^2)))) is set
((exp_R * arccot) / (f + (#Z 2))) . x is V28() V29() ext-real Element of REAL
(exp_R * arccot) . x is V28() V29() ext-real Element of REAL
(f + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((exp_R * arccot) . x) / ((f + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K99(((f + (#Z 2)) . x)) is V28() set
K97(((exp_R * arccot) . x),K99(((f + (#Z 2)) . x))) is set
(exp_R . (arccot . x)) / ((f + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
K97((exp_R . (arccot . x)),K99(((f + (#Z 2)) . x))) is set
f . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / ((f . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99(((f . x) + ((#Z 2) . x))) is V28() set
K97((exp_R . (arccot . x)),K99(((f . x) + ((#Z 2) . x)))) is set
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . x))) is V28() set
K97((exp_R . (arccot . x)),K99((1 + ((#Z 2) . x)))) is set
x #Z 2 is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (x #Z 2))) is V28() set
K97((exp_R . (arccot . x)),K99((1 + (x #Z 2)))) is set
(- (exp_R * arccot)) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- (exp_R * arccot)) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- (exp_R * arccot)) `| Z) . x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
exp_R . (arccot . x) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
(exp_R . (arccot . x)) / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() set
K97((exp_R . (arccot . x)),K99((1 + (x ^2)))) is set
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
ln * (f1 + f) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * (f1 + f)) is V51() V52() V53() Element of K19(REAL)
(1 / 2) (#) (ln * (f1 + f)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((1 / 2) (#) (ln * (f1 + f))) . (upper_bound A) is V28() V29() ext-real Element of REAL
((1 / 2) (#) (ln * (f1 + f))) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((1 / 2) (#) (ln * (f1 + f))) . (upper_bound A)) - (((1 / 2) (#) (ln * (f1 + f))) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((1 / 2) (#) (ln * (f1 + f))) . (lower_bound A))) is V28() set
K96((((1 / 2) (#) (ln * (f1 + f))) . (upper_bound A)),K98((((1 / 2) (#) (ln * (f1 + f))) . (lower_bound A)))) is set
Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V51() V52() V53() Element of K19(REAL)
Z | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (Z,A) is V28() V29() ext-real Element of REAL
x is V51() V52() V53() open Element of K19(REAL)
id x is V1() V4( REAL ) V4(x) V5( REAL ) V5(x) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id x) / (f1 + f) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((1 / 2) (#) (ln * (f1 + f))) is V51() V52() V53() Element of K19(REAL)
dom (id x) is V51() V52() V53() Element of K19(x)
K19(x) is set
dom (f1 + f) is V51() V52() V53() Element of K19(REAL)
(f1 + f) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (f1 + f)) \ ((f1 + f) " {0}) is V51() V52() V53() Element of K19(REAL)
(dom (id x)) /\ ((dom (f1 + f)) \ ((f1 + f) " {0})) is V51() V52() V53() Element of K19(REAL)
(f1 + f) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + f) ^) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
Z . h is V28() V29() ext-real Element of REAL
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
1 + (h ^2) is V28() V29() ext-real Element of REAL
h / (1 + (h ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (h ^2))) is V28() set
K97(h,K99((1 + (h ^2)))) is set
((id x) / (f1 + f)) . h is V28() V29() ext-real Element of REAL
(id x) . h is V28() V29() ext-real Element of REAL
(f1 + f) . h is V28() V29() ext-real Element of REAL
((id x) . h) / ((f1 + f) . h) is V28() V29() ext-real Element of REAL
K99(((f1 + f) . h)) is V28() set
K97(((id x) . h),K99(((f1 + f) . h))) is set
h / ((f1 + f) . h) is V28() V29() ext-real Element of REAL
K97(h,K99(((f1 + f) . h))) is set
f1 . h is V28() V29() ext-real Element of REAL
f . h is V28() V29() ext-real Element of REAL
(f1 . h) + (f . h) is V28() V29() ext-real Element of REAL
h / ((f1 . h) + (f . h)) is V28() V29() ext-real Element of REAL
K99(((f1 . h) + (f . h))) is V28() set
K97(h,K99(((f1 . h) + (f . h)))) is set
(#Z 2) . h is V28() V29() ext-real Element of REAL
1 + ((#Z 2) . h) is V28() V29() ext-real Element of REAL
h / (1 + ((#Z 2) . h)) is V28() V29() ext-real Element of REAL
K99((1 + ((#Z 2) . h))) is V28() set
K97(h,K99((1 + ((#Z 2) . h)))) is set
h #Z 2 is V28() V29() ext-real Element of REAL
1 + (h #Z 2) is V28() V29() ext-real Element of REAL
h / (1 + (h #Z 2)) is V28() V29() ext-real Element of REAL
K99((1 + (h #Z 2))) is V28() set
K97(h,K99((1 + (h #Z 2)))) is set
((1 / 2) (#) (ln * (f1 + f))) `| x is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((1 / 2) (#) (ln * (f1 + f))) `| x) is V51() V52() V53() Element of K19(REAL)
h is V28() V29() ext-real Element of REAL
(((1 / 2) (#) (ln * (f1 + f))) `| x) . h is V28() V29() ext-real Element of REAL
Z . h is V28() V29() ext-real Element of REAL
h ^2 is V28() V29() ext-real Element of REAL
K97(h,h) is set
1 + (h ^2) is V28() V29() ext-real Element of REAL
h / (1 + (h ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (h ^2))) is V28() set
K97(h,K99((1 + (h ^2)))) is set
A is V28() V29() ext-real Element of REAL
A / 2 is V28() V29() ext-real Element of REAL
K97(A,K99(2)) is set
f1 is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
ln * (f + Z) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * (f + Z)) is V51() V52() V53() Element of K19(REAL)
A (#) (f + Z) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(A / 2) (#) (ln * (f + Z)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((A / 2) (#) (ln * (f + Z))) . (upper_bound f1) is V28() V29() ext-real Element of REAL
((A / 2) (#) (ln * (f + Z))) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((A / 2) (#) (ln * (f + Z))) . (upper_bound f1)) - (((A / 2) (#) (ln * (f + Z))) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((A / 2) (#) (ln * (f + Z))) . (lower_bound f1))) is V28() set
K96((((A / 2) (#) (ln * (f + Z))) . (upper_bound f1)),K98((((A / 2) (#) (ln * (f + Z))) . (lower_bound f1)))) is set
x is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom x is V51() V52() V53() Element of K19(REAL)
x | f1 is V1() V4( REAL ) V4(f1) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (x,f1) is V28() V29() ext-real Element of REAL
h is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z 2) * h is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) / (A (#) (f + Z)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((A / 2) (#) (ln * (f + Z))) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
h . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() set
K97(x,K99(A)) is set
x is V28() V29() ext-real Element of REAL
x . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() set
K97(x,K99(A)) is set
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is set
1 + ((x / A) ^2) is V28() V29() ext-real Element of REAL
A * (1 + ((x / A) ^2)) is V28() V29() ext-real Element of REAL
x / (A * (1 + ((x / A) ^2))) is V28() V29() ext-real Element of REAL
K99((A * (1 + ((x / A) ^2)))) is V28() set
K97(x,K99((A * (1 + ((x / A) ^2))))) is set
dom (f + Z) is V51() V52() V53() Element of K19(REAL)
dom f is V51() V52() V53() Element of K19(REAL)
dom Z is V51() V52() V53() Element of K19(REAL)
(dom f) /\ (dom Z) is V51() V52() V53() Element of K19(REAL)
((id Z) / (A (#) (f + Z))) . x is V28() V29() ext-real Element of REAL
(id Z) . x is V28() V29() ext-real Element of REAL
(A (#) (f + Z)) . x is V28() V29() ext-real Element of REAL
((id Z) . x) / ((A (#) (f + Z)) . x) is V28() V29() ext-real Element of REAL
K99(((A (#) (f + Z)) . x)) is V28() set
K97(((id Z) . x),K99(((A (#) (f + Z)) . x))) is set
x / ((A (#) (f + Z)) . x) is V28() V29() ext-real Element of REAL
K97(x,K99(((A (#) (f + Z)) . x))) is set
(f + Z) . x is V28() V29() ext-real Element of REAL
A * ((f + Z) . x) is V28() V29() ext-real Element of REAL
x / (A * ((f + Z) . x)) is V28() V29() ext-real Element of REAL
K99((A * ((f + Z) . x))) is V28() set
K97(x,K99((A * ((f + Z) . x)))) is set
f . x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
(f . x) + (Z . x) is V28() V29() ext-real Element of REAL
A * ((f . x) + (Z . x)) is V28() V29() ext-real Element of REAL
x / (A * ((f . x) + (Z . x))) is V28() V29() ext-real Element of REAL
K99((A * ((f . x) + (Z . x)))) is V28() set
K97(x,K99((A * ((f . x) + (Z . x))))) is set
1 + (Z . x) is V28() V29() ext-real Element of REAL
A * (1 + (Z . x)) is V28() V29() ext-real Element of REAL
x / (A * (1 + (Z . x))) is V28() V29() ext-real Element of REAL
K99((A * (1 + (Z . x)))) is V28() set
K97(x,K99((A * (1 + (Z . x))))) is set
h . x is V28() V29() ext-real Element of REAL
(#Z 2) . (h . x) is V28() V29() ext-real Element of REAL
1 + ((#Z 2) . (h . x)) is V28() V29() ext-real Element of REAL
A * (1 + ((#Z 2) . (h . x))) is V28() V29() ext-real Element of REAL
x / (A * (1 + ((#Z 2) . (h . x)))) is V28() V29() ext-real Element of REAL
K99((A * (1 + ((#Z 2) . (h . x))))) is V28() set
K97(x,K99((A * (1 + ((#Z 2) . (h . x)))))) is set
(h . x) #Z 2 is V28() V29() ext-real Element of REAL
1 + ((h . x) #Z 2) is V28() V29() ext-real Element of REAL
A * (1 + ((h . x) #Z 2)) is V28() V29() ext-real Element of REAL
x / (A * (1 + ((h . x) #Z 2))) is V28() V29() ext-real Element of REAL
K99((A * (1 + ((h . x) #Z 2)))) is V28() set
K97(x,K99((A * (1 + ((h . x) #Z 2))))) is set
(h . x) ^2 is V28() V29() ext-real Element of REAL
K97((h . x),(h . x)) is set
1 + ((h . x) ^2) is V28() V29() ext-real Element of REAL
A * (1 + ((h . x) ^2)) is V28() V29() ext-real Element of REAL
x / (A * (1 + ((h . x) ^2))) is V28() V29() ext-real Element of REAL
K99((A * (1 + ((h . x) ^2)))) is V28() set
K97(x,K99((A * (1 + ((h . x) ^2))))) is set
((A / 2) (#) (ln * (f + Z))) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((A / 2) (#) (ln * (f + Z))) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((A / 2) (#) (ln * (f + Z))) `| Z) . x is V28() V29() ext-real Element of REAL
x . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() set
K97(x,K99(A)) is set
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is set
1 + ((x / A) ^2) is V28() V29() ext-real Element of REAL
A * (1 + ((x / A) ^2)) is V28() V29() ext-real Element of REAL
x / (A * (1 + ((x / A) ^2))) is V28() V29() ext-real Element of REAL
K99((A * (1 + ((x / A) ^2)))) is V28() set
K97(x,K99((A * (1 + ((x / A) ^2))))) is set
A is V51() V52() V53() open Element of K19(REAL)
id A is V1() V4( REAL ) V4(A) V5( REAL ) V5(A) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id A) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id A) ^) (#) arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id A) ^) (#) arctan) is V51() V52() V53() Element of K19(REAL)
- (((id A) ^) (#) arctan) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (((id A) ^) (#) arctan) is V1() V6() set
(- (((id A) ^) (#) arctan)) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (((id A) ^) (#) arctan)) is V51() V52() V53() Element of K19(REAL)
f is V28() V29() ext-real Element of REAL
(id A) . f is V28() V29() ext-real Element of REAL
dom ((id A) ^) is V51() V52() V53() Element of K19(REAL)
dom arctan is V51() V52() V53() Element of K19(REAL)
(dom ((id A) ^)) /\ (dom arctan) is V51() V52() V53() Element of K19(REAL)
dom (id A) is V51() V52() V53() Element of K19(A)
K19(A) is set
(id A) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id A)) \ ((id A) " {0}) is V51() V52() V53() Element of K19(A)
(dom (id A)) \ {0} is V51() V52() V53() Element of K19(A)
((id A) ^) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V28() V29() ext-real Element of REAL
(((id A) ^) `| A) . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
1 / (f ^2) is V28() V29() ext-real Element of REAL
K99((f ^2)) is V28() set
K97(1,K99((f ^2))) is set
- (1 / (f ^2)) is V28() V29() ext-real Element of REAL
(- 1) (#) (((id A) ^) (#) arctan) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V28() V29() ext-real Element of REAL
((- (((id A) ^) (#) arctan)) `| A) . f is V28() V29() ext-real Element of REAL
arctan . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
(arctan . f) / (f ^2) is V28() V29() ext-real Element of REAL
K99((f ^2)) is V28() set
K97((arctan . f),K99((f ^2))) is set
1 + (f ^2) is V28() V29() ext-real Element of REAL
f * (1 + (f ^2)) is V28() V29() ext-real Element of REAL
1 / (f * (1 + (f ^2))) is V28() V29() ext-real Element of REAL
K99((f * (1 + (f ^2)))) is V28() set
K97(1,K99((f * (1 + (f ^2))))) is set
((arctan . f) / (f ^2)) - (1 / (f * (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
K98((1 / (f * (1 + (f ^2))))) is V28() set
K96(((arctan . f) / (f ^2)),K98((1 / (f * (1 + (f ^2)))))) is set
diff ((- (((id A) ^) (#) arctan)),f) is V28() V29() ext-real Element of REAL
diff ((((id A) ^) (#) arctan),f) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((((id A) ^) (#) arctan),f)) is V28() V29() ext-real Element of REAL
diff (((id A) ^),f) is V28() V29() ext-real Element of REAL
(arctan . f) * (diff (((id A) ^),f)) is V28() V29() ext-real Element of REAL
((id A) ^) . f is V28() V29() ext-real Element of REAL
diff (arctan,f) is V28() V29() ext-real Element of REAL
(((id A) ^) . f) * (diff (arctan,f)) is V28() V29() ext-real Element of REAL
((arctan . f) * (diff (((id A) ^),f))) + ((((id A) ^) . f) * (diff (arctan,f))) is V28() V29() ext-real Element of REAL
(- 1) * (((arctan . f) * (diff (((id A) ^),f))) + ((((id A) ^) . f) * (diff (arctan,f)))) is V28() V29() ext-real Element of REAL
(((id A) ^) `| A) . f is V28() V29() ext-real Element of REAL
(arctan . f) * ((((id A) ^) `| A) . f) is V28() V29() ext-real Element of REAL
((arctan . f) * ((((id A) ^) `| A) . f)) + ((((id A) ^) . f) * (diff (arctan,f))) is V28() V29() ext-real Element of REAL
(- 1) * (((arctan . f) * ((((id A) ^) `| A) . f)) + ((((id A) ^) . f) * (diff (arctan,f)))) is V28() V29() ext-real Element of REAL
1 / (f ^2) is V28() V29() ext-real Element of REAL
K97(1,K99((f ^2))) is set
- (1 / (f ^2)) is V28() V29() ext-real Element of REAL
(arctan . f) * (- (1 / (f ^2))) is V28() V29() ext-real Element of REAL
((arctan . f) * (- (1 / (f ^2)))) + ((((id A) ^) . f) * (diff (arctan,f))) is V28() V29() ext-real Element of REAL
(- 1) * (((arctan . f) * (- (1 / (f ^2)))) + ((((id A) ^) . f) * (diff (arctan,f)))) is V28() V29() ext-real Element of REAL
(arctan . f) * (1 / (f ^2)) is V28() V29() ext-real Element of REAL
- ((arctan . f) * (1 / (f ^2))) is V28() V29() ext-real Element of REAL
arctan `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arctan `| A) . f is V28() V29() ext-real Element of REAL
(((id A) ^) . f) * ((arctan `| A) . f) is V28() V29() ext-real Element of REAL
(- ((arctan . f) * (1 / (f ^2)))) + ((((id A) ^) . f) * ((arctan `| A) . f)) is V28() V29() ext-real Element of REAL
(- 1) * ((- ((arctan . f) * (1 / (f ^2)))) + ((((id A) ^) . f) * ((arctan `| A) . f))) is V28() V29() ext-real Element of REAL
(arctan . f) * 1 is V28() V29() ext-real Element of REAL
((arctan . f) * 1) / (f ^2) is V28() V29() ext-real Element of REAL
K97(((arctan . f) * 1),K99((f ^2))) is set
- (((arctan . f) * 1) / (f ^2)) is V28() V29() ext-real Element of REAL
1 / (1 + (f ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f ^2))) is V28() set
K97(1,K99((1 + (f ^2)))) is set
(((id A) ^) . f) * (1 / (1 + (f ^2))) is V28() V29() ext-real Element of REAL
(- (((arctan . f) * 1) / (f ^2))) + ((((id A) ^) . f) * (1 / (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * ((- (((arctan . f) * 1) / (f ^2))) + ((((id A) ^) . f) * (1 / (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
- ((arctan . f) / (f ^2)) is V28() V29() ext-real Element of REAL
(id A) . f is V28() V29() ext-real Element of REAL
((id A) . f) " is V28() V29() ext-real Element of REAL
(((id A) . f) ") * (1 / (1 + (f ^2))) is V28() V29() ext-real Element of REAL
(- ((arctan . f) / (f ^2))) + ((((id A) . f) ") * (1 / (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * ((- ((arctan . f) / (f ^2))) + ((((id A) . f) ") * (1 / (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
1 / f is V28() V29() ext-real Element of REAL
K99(f) is V28() set
K97(1,K99(f)) is set
(1 / f) * (1 / (1 + (f ^2))) is V28() V29() ext-real Element of REAL
(- ((arctan . f) / (f ^2))) + ((1 / f) * (1 / (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * ((- ((arctan . f) / (f ^2))) + ((1 / f) * (1 / (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
1 * 1 is V28() V29() V30() ext-real Element of REAL
(1 * 1) / (f * (1 + (f ^2))) is V28() V29() ext-real Element of REAL
K97((1 * 1),K99((f * (1 + (f ^2))))) is set
(- ((arctan . f) / (f ^2))) + ((1 * 1) / (f * (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * ((- ((arctan . f) / (f ^2))) + ((1 * 1) / (f * (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
f is V28() V29() ext-real Element of REAL
((- (((id A) ^) (#) arctan)) `| A) . f is V28() V29() ext-real Element of REAL
arctan . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
(arctan . f) / (f ^2) is V28() V29() ext-real Element of REAL
K99((f ^2)) is V28() set
K97((arctan . f),K99((f ^2))) is set
1 + (f ^2) is V28() V29() ext-real Element of REAL
f * (1 + (f ^2)) is V28() V29() ext-real Element of REAL
1 / (f * (1 + (f ^2))) is V28() V29() ext-real Element of REAL
K99((f * (1 + (f ^2)))) is V28() set
K97(1,K99((f * (1 + (f ^2))))) is set
((arctan . f) / (f ^2)) - (1 / (f * (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
K98((1 / (f * (1 + (f ^2))))) is V28() set
K96(((arctan . f) / (f ^2)),K98((1 / (f * (1 + (f ^2)))))) is set
A is V51() V52() V53() open Element of K19(REAL)
id A is V1() V4( REAL ) V4(A) V5( REAL ) V5(A) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id A) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id A) ^) (#) arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id A) ^) (#) arccot) is V51() V52() V53() Element of K19(REAL)
- (((id A) ^) (#) arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (((id A) ^) (#) arccot) is V1() V6() set
(- (((id A) ^) (#) arccot)) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (((id A) ^) (#) arccot)) is V51() V52() V53() Element of K19(REAL)
f is V28() V29() ext-real Element of REAL
(id A) . f is V28() V29() ext-real Element of REAL
dom ((id A) ^) is V51() V52() V53() Element of K19(REAL)
dom arccot is V51() V52() V53() Element of K19(REAL)
(dom ((id A) ^)) /\ (dom arccot) is V51() V52() V53() Element of K19(REAL)
dom (id A) is V51() V52() V53() Element of K19(A)
K19(A) is set
(id A) " {0} is V51() V52() V53() Element of K19(REAL)
(dom (id A)) \ ((id A) " {0}) is V51() V52() V53() Element of K19(A)
(dom (id A)) \ {0} is V51() V52() V53() Element of K19(A)
((id A) ^) `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V28() V29() ext-real Element of REAL
(((id A) ^) `| A) . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
1 / (f ^2) is V28() V29() ext-real Element of REAL
K99((f ^2)) is V28() set
K97(1,K99((f ^2))) is set
- (1 / (f ^2)) is V28() V29() ext-real Element of REAL
(- 1) (#) (((id A) ^) (#) arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V28() V29() ext-real Element of REAL
((- (((id A) ^) (#) arccot)) `| A) . f is V28() V29() ext-real Element of REAL
arccot . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
(arccot . f) / (f ^2) is V28() V29() ext-real Element of REAL
K99((f ^2)) is V28() set
K97((arccot . f),K99((f ^2))) is set
1 + (f ^2) is V28() V29() ext-real Element of REAL
f * (1 + (f ^2)) is V28() V29() ext-real Element of REAL
1 / (f * (1 + (f ^2))) is V28() V29() ext-real Element of REAL
K99((f * (1 + (f ^2)))) is V28() set
K97(1,K99((f * (1 + (f ^2))))) is set
((arccot . f) / (f ^2)) + (1 / (f * (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
diff ((- (((id A) ^) (#) arccot)),f) is V28() V29() ext-real Element of REAL
diff ((((id A) ^) (#) arccot),f) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((((id A) ^) (#) arccot),f)) is V28() V29() ext-real Element of REAL
diff (((id A) ^),f) is V28() V29() ext-real Element of REAL
(arccot . f) * (diff (((id A) ^),f)) is V28() V29() ext-real Element of REAL
((id A) ^) . f is V28() V29() ext-real Element of REAL
diff (arccot,f) is V28() V29() ext-real Element of REAL
(((id A) ^) . f) * (diff (arccot,f)) is V28() V29() ext-real Element of REAL
((arccot . f) * (diff (((id A) ^),f))) + ((((id A) ^) . f) * (diff (arccot,f))) is V28() V29() ext-real Element of REAL
(- 1) * (((arccot . f) * (diff (((id A) ^),f))) + ((((id A) ^) . f) * (diff (arccot,f)))) is V28() V29() ext-real Element of REAL
(((id A) ^) `| A) . f is V28() V29() ext-real Element of REAL
(arccot . f) * ((((id A) ^) `| A) . f) is V28() V29() ext-real Element of REAL
((arccot . f) * ((((id A) ^) `| A) . f)) + ((((id A) ^) . f) * (diff (arccot,f))) is V28() V29() ext-real Element of REAL
(- 1) * (((arccot . f) * ((((id A) ^) `| A) . f)) + ((((id A) ^) . f) * (diff (arccot,f)))) is V28() V29() ext-real Element of REAL
1 / (f ^2) is V28() V29() ext-real Element of REAL
K97(1,K99((f ^2))) is set
- (1 / (f ^2)) is V28() V29() ext-real Element of REAL
(arccot . f) * (- (1 / (f ^2))) is V28() V29() ext-real Element of REAL
((arccot . f) * (- (1 / (f ^2)))) + ((((id A) ^) . f) * (diff (arccot,f))) is V28() V29() ext-real Element of REAL
(- 1) * (((arccot . f) * (- (1 / (f ^2)))) + ((((id A) ^) . f) * (diff (arccot,f)))) is V28() V29() ext-real Element of REAL
(arccot . f) * (1 / (f ^2)) is V28() V29() ext-real Element of REAL
- ((arccot . f) * (1 / (f ^2))) is V28() V29() ext-real Element of REAL
arccot `| A is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arccot `| A) . f is V28() V29() ext-real Element of REAL
(((id A) ^) . f) * ((arccot `| A) . f) is V28() V29() ext-real Element of REAL
(- ((arccot . f) * (1 / (f ^2)))) + ((((id A) ^) . f) * ((arccot `| A) . f)) is V28() V29() ext-real Element of REAL
(- 1) * ((- ((arccot . f) * (1 / (f ^2)))) + ((((id A) ^) . f) * ((arccot `| A) . f))) is V28() V29() ext-real Element of REAL
1 / (1 + (f ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (f ^2))) is V28() set
K97(1,K99((1 + (f ^2)))) is set
- (1 / (1 + (f ^2))) is V28() V29() ext-real Element of REAL
(((id A) ^) . f) * (- (1 / (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
(- ((arccot . f) * (1 / (f ^2)))) + ((((id A) ^) . f) * (- (1 / (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
(- 1) * ((- ((arccot . f) * (1 / (f ^2)))) + ((((id A) ^) . f) * (- (1 / (1 + (f ^2)))))) is V28() V29() ext-real Element of REAL
(arccot . f) * 1 is V28() V29() ext-real Element of REAL
((arccot . f) * 1) / (f ^2) is V28() V29() ext-real Element of REAL
K97(((arccot . f) * 1),K99((f ^2))) is set
- (((arccot . f) * 1) / (f ^2)) is V28() V29() ext-real Element of REAL
(((id A) ^) . f) * (1 / (1 + (f ^2))) is V28() V29() ext-real Element of REAL
(- (((arccot . f) * 1) / (f ^2))) - ((((id A) ^) . f) * (1 / (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
K98(((((id A) ^) . f) * (1 / (1 + (f ^2))))) is V28() set
K96((- (((arccot . f) * 1) / (f ^2))),K98(((((id A) ^) . f) * (1 / (1 + (f ^2)))))) is set
(- 1) * ((- (((arccot . f) * 1) / (f ^2))) - ((((id A) ^) . f) * (1 / (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
- ((arccot . f) / (f ^2)) is V28() V29() ext-real Element of REAL
(id A) . f is V28() V29() ext-real Element of REAL
((id A) . f) " is V28() V29() ext-real Element of REAL
(((id A) . f) ") * (1 / (1 + (f ^2))) is V28() V29() ext-real Element of REAL
(- ((arccot . f) / (f ^2))) - ((((id A) . f) ") * (1 / (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
K98(((((id A) . f) ") * (1 / (1 + (f ^2))))) is V28() set
K96((- ((arccot . f) / (f ^2))),K98(((((id A) . f) ") * (1 / (1 + (f ^2)))))) is set
(- 1) * ((- ((arccot . f) / (f ^2))) - ((((id A) . f) ") * (1 / (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
1 / f is V28() V29() ext-real Element of REAL
K99(f) is V28() set
K97(1,K99(f)) is set
(1 / f) * (1 / (1 + (f ^2))) is V28() V29() ext-real Element of REAL
(- ((arccot . f) / (f ^2))) - ((1 / f) * (1 / (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
K98(((1 / f) * (1 / (1 + (f ^2))))) is V28() set
K96((- ((arccot . f) / (f ^2))),K98(((1 / f) * (1 / (1 + (f ^2)))))) is set
(- 1) * ((- ((arccot . f) / (f ^2))) - ((1 / f) * (1 / (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
1 * 1 is V28() V29() V30() ext-real Element of REAL
(1 * 1) / (f * (1 + (f ^2))) is V28() V29() ext-real Element of REAL
K97((1 * 1),K99((f * (1 + (f ^2))))) is set
(- ((arccot . f) / (f ^2))) - ((1 * 1) / (f * (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
K98(((1 * 1) / (f * (1 + (f ^2))))) is V28() set
K96((- ((arccot . f) / (f ^2))),K98(((1 * 1) / (f * (1 + (f ^2)))))) is set
(- 1) * ((- ((arccot . f) / (f ^2))) - ((1 * 1) / (f * (1 + (f ^2))))) is V28() V29() ext-real Element of REAL
f is V28() V29() ext-real Element of REAL
((- (((id A) ^) (#) arccot)) `| A) . f is V28() V29() ext-real Element of REAL
arccot . f is V28() V29() ext-real Element of REAL
f ^2 is V28() V29() ext-real Element of REAL
K97(f,f) is set
(arccot . f) / (f ^2) is V28() V29() ext-real Element of REAL
K99((f ^2)) is V28() set
K97((arccot . f),K99((f ^2))) is set
1 + (f ^2) is V28() V29() ext-real Element of REAL
f * (1 + (f ^2)) is V28() V29() ext-real Element of REAL
1 / (f * (1 + (f ^2))) is V28() V29() ext-real Element of REAL
K99((f * (1 + (f ^2)))) is V28() set
K97(1,K99((f * (1 + (f ^2))))) is set
((arccot . f) / (f ^2)) + (1 / (f * (1 + (f ^2)))) is V28() V29() ext-real Element of REAL
arctan / (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) (#) (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) (#) (f1 + (#Z 2))) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (((id Z) (#) (f1 + (#Z 2))) ^) is V1() V6() V34() set
K98(1) (#) (((id Z) (#) (f1 + (#Z 2))) ^) is V1() V6() set
(arctan / (#Z 2)) + (- (((id Z) (#) (f1 + (#Z 2))) ^)) is V1() V6() set
(id Z) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) ^) (#) arctan is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id Z) ^) (#) arctan) is V51() V52() V53() Element of K19(REAL)
- (((id Z) ^) (#) arctan) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (((id Z) ^) (#) arctan) is V1() V6() set
(- (((id Z) ^) (#) arctan)) . (upper_bound A) is V28() V29() ext-real Element of REAL
(- (((id Z) ^) (#) arctan)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (((id Z) ^) (#) arctan)) . (upper_bound A)) - ((- (((id Z) ^) (#) arctan)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (((id Z) ^) (#) arctan)) . (lower_bound A))) is V28() set
K96(((- (((id Z) ^) (#) arctan)) . (upper_bound A)),K98(((- (((id Z) ^) (#) arctan)) . (lower_bound A)))) is set
dom (arctan / (#Z 2)) is V51() V52() V53() Element of K19(REAL)
dom (((id Z) (#) (f1 + (#Z 2))) ^) is V51() V52() V53() Element of K19(REAL)
(dom (arctan / (#Z 2))) /\ (dom (((id Z) (#) (f1 + (#Z 2))) ^)) is V51() V52() V53() Element of K19(REAL)
dom ((id Z) (#) (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
dom (id Z) is V51() V52() V53() Element of K19(Z)
K19(Z) is set
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(dom (id Z)) /\ (dom (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
(arctan . x) / (x ^2) is V28() V29() ext-real Element of REAL
K99((x ^2)) is V28() set
K97((arctan . x),K99((x ^2))) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
x * (1 + (x ^2)) is V28() V29() ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V28() V29() ext-real Element of REAL
K99((x * (1 + (x ^2)))) is V28() set
K97(1,K99((x * (1 + (x ^2))))) is set
((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) is V28() V29() ext-real Element of REAL
K98((1 / (x * (1 + (x ^2))))) is V28() set
K96(((arctan . x) / (x ^2)),K98((1 / (x * (1 + (x ^2)))))) is set
((arctan / (#Z 2)) - (((id Z) (#) (f1 + (#Z 2))) ^)) . x is V28() V29() ext-real Element of REAL
(arctan / (#Z 2)) . x is V28() V29() ext-real Element of REAL
(((id Z) (#) (f1 + (#Z 2))) ^) . x is V28() V29() ext-real Element of REAL
((arctan / (#Z 2)) . x) - ((((id Z) (#) (f1 + (#Z 2))) ^) . x) is V28() V29() ext-real Element of REAL
K98(((((id Z) (#) (f1 + (#Z 2))) ^) . x)) is V28() set
K96(((arctan / (#Z 2)) . x),K98(((((id Z) (#) (f1 + (#Z 2))) ^) . x))) is set
((id Z) (#) (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
1 / (((id Z) (#) (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K99((((id Z) (#) (f1 + (#Z 2))) . x)) is V28() set
K97(1,K99((((id Z) (#) (f1 + (#Z 2))) . x))) is set
((arctan / (#Z 2)) . x) - (1 / (((id Z) (#) (f1 + (#Z 2))) . x)) is V28() V29() ext-real Element of REAL
K98((1 / (((id Z) (#) (f1 + (#Z 2))) . x))) is V28() set
K96(((arctan / (#Z 2)) . x),K98((1 / (((id Z) (#) (f1 + (#Z 2))) . x)))) is set
(id Z) . x is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((id Z) . x) * ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
1 / (((id Z) . x) * ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K99((((id Z) . x) * ((f1 + (#Z 2)) . x))) is V28() set
K97(1,K99((((id Z) . x) * ((f1 + (#Z 2)) . x)))) is set
((arctan / (#Z 2)) . x) - (1 / (((id Z) . x) * ((f1 + (#Z 2)) . x))) is V28() V29() ext-real Element of REAL
K98((1 / (((id Z) . x) * ((f1 + (#Z 2)) . x)))) is V28() set
K96(((arctan / (#Z 2)) . x),K98((1 / (((id Z) . x) * ((f1 + (#Z 2)) . x))))) is set
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
((id Z) . x) * ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
1 / (((id Z) . x) * ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K99((((id Z) . x) * ((f1 . x) + ((#Z 2) . x)))) is V28() set
K97(1,K99((((id Z) . x) * ((f1 . x) + ((#Z 2) . x))))) is set
((arctan / (#Z 2)) . x) - (1 / (((id Z) . x) * ((f1 . x) + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
K98((1 / (((id Z) . x) * ((f1 . x) + ((#Z 2) . x))))) is V28() set
K96(((arctan / (#Z 2)) . x),K98((1 / (((id Z) . x) * ((f1 . x) + ((#Z 2) . x)))))) is set
x * ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
1 / (x * ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K99((x * ((f1 . x) + ((#Z 2) . x)))) is V28() set
K97(1,K99((x * ((f1 . x) + ((#Z 2) . x))))) is set
((arctan / (#Z 2)) . x) - (1 / (x * ((f1 . x) + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
K98((1 / (x * ((f1 . x) + ((#Z 2) . x))))) is V28() set
K96(((arctan / (#Z 2)) . x),K98((1 / (x * ((f1 . x) + ((#Z 2) . x)))))) is set
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
x * (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
1 / (x * (1 + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K99((x * (1 + ((#Z 2) . x)))) is V28() set
K97(1,K99((x * (1 + ((#Z 2) . x))))) is set
((arctan / (#Z 2)) . x) - (1 / (x * (1 + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
K98((1 / (x * (1 + ((#Z 2) . x))))) is V28() set
K96(((arctan / (#Z 2)) . x),K98((1 / (x * (1 + ((#Z 2) . x)))))) is set
(arctan . x) / ((#Z 2) . x) is V28() V29() ext-real Element of REAL
K99(((#Z 2) . x)) is V28() set
K97((arctan . x),K99(((#Z 2) . x))) is set
((arctan . x) / ((#Z 2) . x)) - (1 / (x * (1 + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
K96(((arctan . x) / ((#Z 2) . x)),K98((1 / (x * (1 + ((#Z 2) . x)))))) is set
x #Z 2 is V28() V29() ext-real Element of REAL
(arctan . x) / (x #Z 2) is V28() V29() ext-real Element of REAL
K99((x #Z 2)) is V28() set
K97((arctan . x),K99((x #Z 2))) is set
((arctan . x) / (x #Z 2)) - (1 / (x * (1 + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
K96(((arctan . x) / (x #Z 2)),K98((1 / (x * (1 + ((#Z 2) . x)))))) is set
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
x * (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
1 / (x * (1 + (x #Z 2))) is V28() V29() ext-real Element of REAL
K99((x * (1 + (x #Z 2)))) is V28() set
K97(1,K99((x * (1 + (x #Z 2))))) is set
((arctan . x) / (x #Z 2)) - (1 / (x * (1 + (x #Z 2)))) is V28() V29() ext-real Element of REAL
K98((1 / (x * (1 + (x #Z 2))))) is V28() set
K96(((arctan . x) / (x #Z 2)),K98((1 / (x * (1 + (x #Z 2)))))) is set
((arctan . x) / (x ^2)) - (1 / (x * (1 + (x #Z 2)))) is V28() V29() ext-real Element of REAL
K96(((arctan . x) / (x ^2)),K98((1 / (x * (1 + (x #Z 2)))))) is set
(- (((id Z) ^) (#) arctan)) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- (((id Z) ^) (#) arctan)) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- (((id Z) ^) (#) arctan)) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arctan . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
(arctan . x) / (x ^2) is V28() V29() ext-real Element of REAL
K99((x ^2)) is V28() set
K97((arctan . x),K99((x ^2))) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
x * (1 + (x ^2)) is V28() V29() ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V28() V29() ext-real Element of REAL
K99((x * (1 + (x ^2)))) is V28() set
K97(1,K99((x * (1 + (x ^2))))) is set
((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2)))) is V28() V29() ext-real Element of REAL
K98((1 / (x * (1 + (x ^2))))) is V28() set
K96(((arctan . x) / (x ^2)),K98((1 / (x * (1 + (x ^2)))))) is set
arccot / (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V51() V52() V53() closed_interval V76() Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
f1 is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V51() V52() V53() Element of K19(REAL)
f | A is V1() V4( REAL ) V4(A) V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V51() V52() V53() open Element of K19(REAL)
id Z is V1() V4( REAL ) V4(Z) V5( REAL ) V5(Z) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) (#) (f1 + (#Z 2)) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) (#) (f1 + (#Z 2))) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arccot / (#Z 2)) + (((id Z) (#) (f1 + (#Z 2))) ^) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) ^ is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) ^) (#) arccot is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id Z) ^) (#) arccot) is V51() V52() V53() Element of K19(REAL)
- (((id Z) ^) (#) arccot) is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (((id Z) ^) (#) arccot) is V1() V6() set
(- (((id Z) ^) (#) arccot)) . (upper_bound A) is V28() V29() ext-real Element of REAL
(- (((id Z) ^) (#) arccot)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (((id Z) ^) (#) arccot)) . (upper_bound A)) - ((- (((id Z) ^) (#) arccot)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (((id Z) ^) (#) arccot)) . (lower_bound A))) is V28() set
K96(((- (((id Z) ^) (#) arccot)) . (upper_bound A)),K98(((- (((id Z) ^) (#) arccot)) . (lower_bound A)))) is set
dom (arccot / (#Z 2)) is V51() V52() V53() Element of K19(REAL)
dom (((id Z) (#) (f1 + (#Z 2))) ^) is V51() V52() V53() Element of K19(REAL)
(dom (arccot / (#Z 2))) /\ (dom (((id Z) (#) (f1 + (#Z 2))) ^)) is V51() V52() V53() Element of K19(REAL)
dom ((id Z) (#) (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
dom (id Z) is V51() V52() V53() Element of K19(Z)
K19(Z) is set
dom (f1 + (#Z 2)) is V51() V52() V53() Element of K19(REAL)
(dom (id Z)) /\ (dom (f1 + (#Z 2))) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
(arccot . x) / (x ^2) is V28() V29() ext-real Element of REAL
K99((x ^2)) is V28() set
K97((arccot . x),K99((x ^2))) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
x * (1 + (x ^2)) is V28() V29() ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V28() V29() ext-real Element of REAL
K99((x * (1 + (x ^2)))) is V28() set
K97(1,K99((x * (1 + (x ^2))))) is set
((arccot . x) / (x ^2)) + (1 / (x * (1 + (x ^2)))) is V28() V29() ext-real Element of REAL
((arccot / (#Z 2)) + (((id Z) (#) (f1 + (#Z 2))) ^)) . x is V28() V29() ext-real Element of REAL
(arccot / (#Z 2)) . x is V28() V29() ext-real Element of REAL
(((id Z) (#) (f1 + (#Z 2))) ^) . x is V28() V29() ext-real Element of REAL
((arccot / (#Z 2)) . x) + ((((id Z) (#) (f1 + (#Z 2))) ^) . x) is V28() V29() ext-real Element of REAL
((id Z) (#) (f1 + (#Z 2))) . x is V28() V29() ext-real Element of REAL
1 / (((id Z) (#) (f1 + (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K99((((id Z) (#) (f1 + (#Z 2))) . x)) is V28() set
K97(1,K99((((id Z) (#) (f1 + (#Z 2))) . x))) is set
((arccot / (#Z 2)) . x) + (1 / (((id Z) (#) (f1 + (#Z 2))) . x)) is V28() V29() ext-real Element of REAL
(id Z) . x is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . x is V28() V29() ext-real Element of REAL
((id Z) . x) * ((f1 + (#Z 2)) . x) is V28() V29() ext-real Element of REAL
1 / (((id Z) . x) * ((f1 + (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K99((((id Z) . x) * ((f1 + (#Z 2)) . x))) is V28() set
K97(1,K99((((id Z) . x) * ((f1 + (#Z 2)) . x)))) is set
((arccot / (#Z 2)) . x) + (1 / (((id Z) . x) * ((f1 + (#Z 2)) . x))) is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(f1 . x) + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
((id Z) . x) * ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
1 / (((id Z) . x) * ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K99((((id Z) . x) * ((f1 . x) + ((#Z 2) . x)))) is V28() set
K97(1,K99((((id Z) . x) * ((f1 . x) + ((#Z 2) . x))))) is set
((arccot / (#Z 2)) . x) + (1 / (((id Z) . x) * ((f1 . x) + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
x * ((f1 . x) + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
1 / (x * ((f1 . x) + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K99((x * ((f1 . x) + ((#Z 2) . x)))) is V28() set
K97(1,K99((x * ((f1 . x) + ((#Z 2) . x))))) is set
((arccot / (#Z 2)) . x) + (1 / (x * ((f1 . x) + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
x * (1 + ((#Z 2) . x)) is V28() V29() ext-real Element of REAL
1 / (x * (1 + ((#Z 2) . x))) is V28() V29() ext-real Element of REAL
K99((x * (1 + ((#Z 2) . x)))) is V28() set
K97(1,K99((x * (1 + ((#Z 2) . x))))) is set
((arccot / (#Z 2)) . x) + (1 / (x * (1 + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
(arccot . x) / ((#Z 2) . x) is V28() V29() ext-real Element of REAL
K99(((#Z 2) . x)) is V28() set
K97((arccot . x),K99(((#Z 2) . x))) is set
((arccot . x) / ((#Z 2) . x)) + (1 / (x * (1 + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
x #Z 2 is V28() V29() ext-real Element of REAL
(arccot . x) / (x #Z 2) is V28() V29() ext-real Element of REAL
K99((x #Z 2)) is V28() set
K97((arccot . x),K99((x #Z 2))) is set
((arccot . x) / (x #Z 2)) + (1 / (x * (1 + ((#Z 2) . x)))) is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
x * (1 + (x #Z 2)) is V28() V29() ext-real Element of REAL
1 / (x * (1 + (x #Z 2))) is V28() V29() ext-real Element of REAL
K99((x * (1 + (x #Z 2)))) is V28() set
K97(1,K99((x * (1 + (x #Z 2))))) is set
((arccot . x) / (x #Z 2)) + (1 / (x * (1 + (x #Z 2)))) is V28() V29() ext-real Element of REAL
((arccot . x) / (x ^2)) + (1 / (x * (1 + (x #Z 2)))) is V28() V29() ext-real Element of REAL
(- (((id Z) ^) (#) arccot)) `| Z is V1() V4( REAL ) V5( REAL ) V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((- (((id Z) ^) (#) arccot)) `| Z) is V51() V52() V53() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- (((id Z) ^) (#) arccot)) `| Z) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is set
(arccot . x) / (x ^2) is V28() V29() ext-real Element of REAL
K99((x ^2)) is V28() set
K97((arccot . x),K99((x ^2))) is set
1 + (x ^2) is V28() V29() ext-real Element of REAL
x * (1 + (x ^2)) is V28() V29() ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V28() V29() ext-real Element of REAL
K99((x * (1 + (x ^2)))) is V28() set
K97(1,K99((x * (1 + (x ^2))))) is set
((arccot . x) / (x ^2)) + (1 / (x * (1 + (x ^2)))) is V28() V29() ext-real Element of REAL