:: INTEGR12 semantic presentation

REAL is non empty V50() V55() V56() V57() V61() set
NAT is non empty V21() V22() V23() V55() V56() V57() V58() V59() V60() V61() Element of K19(REAL)
K19(REAL) is set
COMPLEX is non empty V50() V55() V61() set
K20(NAT,REAL) is Relation-like V34() V35() V36() set
K19(K20(NAT,REAL)) is set
K20(NAT,COMPLEX) is Relation-like V34() set
K19(K20(NAT,COMPLEX)) is set
K20(COMPLEX,COMPLEX) is Relation-like V34() set
K19(K20(COMPLEX,COMPLEX)) is set
K20(REAL,REAL) is Relation-like V34() V35() V36() set
K19(K20(REAL,REAL)) is set
PFuncs (REAL,REAL) is set
K20(NAT,(PFuncs (REAL,REAL))) is Relation-like set
K19(K20(NAT,(PFuncs (REAL,REAL)))) is set
ExtREAL is non empty V56() set
+infty is non empty V29() ext-real positive non negative set
-infty is non empty V29() ext-real non positive negative set
0 is Relation-like non-empty empty-yielding RAT -valued V6() V7() V8() V9() empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non positive non negative V34() V35() V36() V37() V55() V56() V57() V58() V59() V60() V61() V67() bounded Element of NAT
RAT is non empty V50() V55() V56() V57() V58() V61() set
the Relation-like non-empty empty-yielding RAT -valued V6() V7() V8() V9() empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non positive non negative V34() V35() V36() V37() V55() V56() V57() V58() V59() V60() V61() bounded set is Relation-like non-empty empty-yielding RAT -valued V6() V7() V8() V9() empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non positive non negative V34() V35() V36() V37() V55() V56() V57() V58() V59() V60() V61() bounded set
[0,REAL] is set
{0,REAL} is non empty set
{0} is V9() non empty V55() V56() V57() V58() V59() V60() set
{{0,REAL},{0}} is non empty set
{+infty,-infty} is non empty V56() set
REAL \/ {+infty,-infty} is non empty V56() set
INT is non empty V50() V55() V56() V57() V58() V59() V61() set
K20(K20(COMPLEX,COMPLEX),COMPLEX) is Relation-like V34() set
K19(K20(K20(COMPLEX,COMPLEX),COMPLEX)) is set
K20(K20(REAL,REAL),REAL) is Relation-like V34() V35() V36() set
K19(K20(K20(REAL,REAL),REAL)) is set
K20(RAT,RAT) is Relation-like RAT -valued V34() V35() V36() set
K19(K20(RAT,RAT)) is set
K20(K20(RAT,RAT),RAT) is Relation-like RAT -valued V34() V35() V36() set
K19(K20(K20(RAT,RAT),RAT)) is set
K20(INT,INT) is Relation-like RAT -valued INT -valued V34() V35() V36() set
K19(K20(INT,INT)) is set
K20(K20(INT,INT),INT) is Relation-like RAT -valued INT -valued V34() V35() V36() set
K19(K20(K20(INT,INT),INT)) is set
K20(NAT,NAT) is Relation-like RAT -valued INT -valued V34() V35() V36() V37() set
K20(K20(NAT,NAT),NAT) is Relation-like RAT -valued INT -valued V34() V35() V36() V37() set
K19(K20(K20(NAT,NAT),NAT)) is set
NAT is non empty V21() V22() V23() V55() V56() V57() V58() V59() V60() V61() set
K19(NAT) is set
K19(NAT) is set
K20(COMPLEX,REAL) is Relation-like V34() V35() V36() set
K19(K20(COMPLEX,REAL)) is set
{} is Relation-like non-empty empty-yielding RAT -valued V6() V7() V8() V9() empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non positive non negative V34() V35() V36() V37() V55() V56() V57() V58() V59() V60() V61() bounded set
1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
{{},1} is non empty V55() V56() V57() V58() V59() V60() set
sin is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
dom sin is non empty V55() V56() V57() Element of K19(REAL)
cos is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
dom cos is non empty V55() V56() V57() Element of K19(REAL)
exp_R is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
2 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
#Z 2 is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
[#] REAL is V55() V56() V57() Element of K19(REAL)
dom exp_R is non empty V55() V56() V57() Element of K19(REAL)
rng exp_R is non empty V55() V56() V57() Element of K19(REAL)
right_open_halfline 0 is V55() V56() V57() Element of K19(REAL)
K363(0,+infty) is set
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : ( not b₁ <= 0 & not +infty <= b₁ ) } is set
ln is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R " is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ln is V55() V56() V57() Element of K19(REAL)
rng ln is V55() V56() V57() Element of K19(REAL)
arcsin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
PI is V28() V29() ext-real Element of REAL
PI / 2 is V28() V29() ext-real Element of REAL
K99(2) is non empty V28() V29() ext-real positive non negative set
K97(PI,K99(2)) is V28() V29() ext-real set
- (PI / 2) is V28() V29() ext-real Element of REAL
[.(- (PI / 2)),(PI / 2).] is V55() V56() V57() closed Element of K19(REAL)
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : ( - (PI / 2) <= b₁ & b₁ <= PI / 2 ) } is set
sin | [.(- (PI / 2)),(PI / 2).] is Relation-like REAL -defined [.(- (PI / 2)),(PI / 2).] -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin | [.(- (PI / 2)),(PI / 2).]) " is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- 1 is V28() V29() V30() ext-real non positive Element of REAL
].(- 1),1.[ is V55() V56() V57() open Element of K19(REAL)
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : ( not b₁ <= - 1 & not 1 <= b₁ ) } is set
arccos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
[.0,PI.] is V55() V56() V57() closed Element of K19(REAL)
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : ( 0 <= b₁ & b₁ <= PI ) } is set
cos | [.0,PI.] is Relation-like REAL -defined [.0,PI.] -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos | [.0,PI.]) " is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) is V28() V29() V30() ext-real non positive set
1 / 2 is V28() V29() ext-real non negative Element of REAL
K97(1,K99(2)) is V28() V29() ext-real non negative set
ln * arcsin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * arcsin) is V55() V56() V57() Element of K19(REAL)
ln * arccos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * arccos) is V55() V56() V57() Element of K19(REAL)
#R (1 / 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (1 / 2) is V28() V29() ext-real non positive Element of REAL
sin ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
cos ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
sin * ln is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (sin * ln) is V55() V56() V57() Element of K19(REAL)
cos * ln is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (cos * ln) is V55() V56() V57() Element of K19(REAL)
sin * exp_R is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (sin * exp_R) is non empty V55() V56() V57() Element of K19(REAL)
cos * exp_R is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (cos * exp_R) is non empty V55() V56() V57() Element of K19(REAL)
exp_R * cos is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R * cos) is non empty V55() V56() V57() Element of K19(REAL)
exp_R * sin is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R * sin) is non empty V55() V56() V57() Element of K19(REAL)
sin / cos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
cos / sin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
tan is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom tan is V55() V56() V57() Element of K19(REAL)
cot is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom cot is V55() V56() V57() Element of K19(REAL)
- cot is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) cot is Relation-like REAL -defined V6() V34() V35() V36() set
exp_R (#) tan is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) tan) is V55() V56() V57() Element of K19(REAL)
exp_R (#) cot is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) cot) is V55() V56() V57() Element of K19(REAL)
arccot is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
].0,PI.[ is V55() V56() V57() open Element of K19(REAL)
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : ( not b₁ <= 0 & not PI <= b₁ ) } is set
cot | ].0,PI.[ is Relation-like REAL -defined ].0,PI.[ -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cot | ].0,PI.[) " is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arctan is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
].(- (PI / 2)),(PI / 2).[ is V55() V56() V57() open Element of K19(REAL)
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : ( not b₁ <= - (PI / 2) & not PI / 2 <= b₁ ) } is set
tan | ].(- (PI / 2)),(PI / 2).[ is Relation-like REAL -defined ].(- (PI / 2)),(PI / 2).[ -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(tan | ].(- (PI / 2)),(PI / 2).[) " is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
ln * arccot is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * arccot) is V55() V56() V57() Element of K19(REAL)
arctan * exp_R is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (arctan * exp_R) is V55() V56() V57() Element of K19(REAL)
arccot * exp_R is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (arccot * exp_R) is V55() V56() V57() Element of K19(REAL)
arctan * ln is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (arctan * ln) is V55() V56() V57() Element of K19(REAL)
arccot * ln is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (arccot * ln) is V55() V56() V57() Element of K19(REAL)
exp_R (#) arctan is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R (#) arccot is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- 1 is V28() V29() V30() ext-real non positive V67() Element of INT
A is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A + f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(A + f1) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((A + f1) ^) is V55() V56() V57() Element of K19(REAL)
f is V55() V56() V57() open Element of K19(REAL)
((A + f1) ^) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (A + f1) is V55() V56() V57() Element of K19(REAL)
(A + f1) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V28() V29() ext-real Element of REAL
((A + f1) `| f) . Z is V28() V29() ext-real Element of REAL
2 * Z is V28() V29() ext-real Element of REAL
Z is V28() V29() ext-real Element of REAL
(A + f1) . Z is V28() V29() ext-real Element of REAL
Z is V28() V29() ext-real Element of REAL
(((A + f1) ^) `| f) . Z is V28() V29() ext-real Element of REAL
2 * Z is V28() V29() ext-real Element of REAL
Z |^ 2 is V28() V29() ext-real Element of REAL
1 + (Z |^ 2) is V28() V29() ext-real Element of REAL
(1 + (Z |^ 2)) ^2 is V28() V29() ext-real Element of REAL
K97((1 + (Z |^ 2)),(1 + (Z |^ 2))) is V28() V29() ext-real set
(2 * Z) / ((1 + (Z |^ 2)) ^2) is V28() V29() ext-real Element of REAL
K99(((1 + (Z |^ 2)) ^2)) is V28() V29() ext-real set
K97((2 * Z),K99(((1 + (Z |^ 2)) ^2))) is V28() V29() ext-real set
- ((2 * Z) / ((1 + (Z |^ 2)) ^2)) is V28() V29() ext-real Element of REAL
(A + f1) . Z is V28() V29() ext-real Element of REAL
f1 . Z is V28() V29() ext-real Element of REAL
Z #Z 2 is V28() V29() ext-real Element of REAL
A . Z is V28() V29() ext-real Element of REAL
(A . Z) + (f1 . Z) is V28() V29() ext-real Element of REAL
diff (((A + f1) ^),Z) is V28() V29() ext-real Element of REAL
diff ((A + f1),Z) is V28() V29() ext-real Element of REAL
((A + f1) . Z) ^2 is V28() V29() ext-real Element of REAL
K97(((A + f1) . Z),((A + f1) . Z)) is V28() V29() ext-real set
(diff ((A + f1),Z)) / (((A + f1) . Z) ^2) is V28() V29() ext-real Element of REAL
K99((((A + f1) . Z) ^2)) is V28() V29() ext-real set
K97((diff ((A + f1),Z)),K99((((A + f1) . Z) ^2))) is V28() V29() ext-real set
- ((diff ((A + f1),Z)) / (((A + f1) . Z) ^2)) is V28() V29() ext-real Element of REAL
((A + f1) `| f) . Z is V28() V29() ext-real Element of REAL
(((A + f1) `| f) . Z) / (((A + f1) . Z) ^2) is V28() V29() ext-real Element of REAL
K97((((A + f1) `| f) . Z),K99((((A + f1) . Z) ^2))) is V28() V29() ext-real set
- ((((A + f1) `| f) . Z) / (((A + f1) . Z) ^2)) is V28() V29() ext-real Element of REAL
Z is V28() V29() ext-real Element of REAL
(((A + f1) ^) `| f) . Z is V28() V29() ext-real Element of REAL
2 * Z is V28() V29() ext-real Element of REAL
Z |^ 2 is V28() V29() ext-real Element of REAL
1 + (Z |^ 2) is V28() V29() ext-real Element of REAL
(1 + (Z |^ 2)) ^2 is V28() V29() ext-real Element of REAL
K97((1 + (Z |^ 2)),(1 + (Z |^ 2))) is V28() V29() ext-real set
(2 * Z) / ((1 + (Z |^ 2)) ^2) is V28() V29() ext-real Element of REAL
K99(((1 + (Z |^ 2)) ^2)) is V28() V29() ext-real set
K97((2 * Z),K99(((1 + (Z |^ 2)) ^2))) is V28() V29() ext-real set
- ((2 * Z) / ((1 + (Z |^ 2)) ^2)) is V28() V29() ext-real Element of REAL
].(- 1),1.[ is V55() V56() V57() open Element of K19(REAL)
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : ( not b₁ <= - 1 & not 1 <= b₁ ) } is set
- (ln * arccot) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (ln * arccot) is Relation-like REAL -defined V6() V34() V35() V36() set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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() V29() ext-real set
K96(((- (ln * arccot)) . (upper_bound A)),K98(((- (ln * arccot)) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(f + Z) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((f + Z) ^) / g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is V55() V56() V57() open Element of K19(REAL)
dom ((f + Z) ^) is V55() V56() V57() Element of K19(REAL)
dom g is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
g " {0} is V55() V56() V57() Element of K19(REAL)
(dom g) \ (g " {0}) is V55() V56() V57() Element of K19(REAL)
(dom ((f + Z) ^)) /\ ((dom g) \ (g " {0})) is V55() V56() V57() Element of K19(REAL)
dom (f + Z) is V55() V56() V57() 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
g . x is V28() V29() ext-real Element of REAL
f1 | f2 is Relation-like REAL -defined f2 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (g ^) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
g | f2 is Relation-like REAL -defined f2 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
rng (g | f2) is V55() V56() V57() Element of K19(REAL)
x is set
dom (g | f2) is V55() V56() V57() Element of K19(f2)
K19(f2) is set
y is set
(g | f2) . y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
g .: f2 is V55() V56() V57() Element of K19(REAL)
dom (- (ln * arccot)) is V55() V56() V57() Element of K19(REAL)
(- (ln * arccot)) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
((- (ln * arccot)) `| f2) . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 + (x ^2) is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
(1 + (x ^2)) * (arccot . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x ^2)) * (arccot . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x ^2)) * (arccot . x))) is V28() V29() ext-real set
K97(1,K99(((1 + (x ^2)) * (arccot . x)))) is V28() V29() ext-real set
diff ((- (ln * arccot)),x) is V28() V29() ext-real Element of REAL
diff ((ln * arccot),x) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((ln * arccot),x)) is V28() V29() ext-real Element of REAL
diff (arccot,x) is V28() V29() ext-real Element of REAL
(diff (arccot,x)) / (arccot . x) is V28() V29() ext-real Element of REAL
K99((arccot . x)) is V28() V29() ext-real set
K97((diff (arccot,x)),K99((arccot . x))) is V28() V29() ext-real set
(- 1) * ((diff (arccot,x)) / (arccot . x)) is V28() V29() ext-real Element of REAL
1 / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() V29() ext-real set
K97(1,K99((1 + (x ^2)))) is V28() V29() ext-real set
- (1 / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
(- (1 / (1 + (x ^2)))) / (arccot . x) is V28() V29() ext-real Element of REAL
K97((- (1 / (1 + (x ^2)))),K99((arccot . x))) is V28() V29() ext-real set
(- 1) * ((- (1 / (1 + (x ^2)))) / (arccot . x)) is V28() V29() ext-real Element of REAL
(1 / (1 + (x ^2))) / (arccot . x) is V28() V29() ext-real Element of REAL
K97((1 / (1 + (x ^2))),K99((arccot . x))) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 + (x ^2) is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
(1 + (x ^2)) * (arccot . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x ^2)) * (arccot . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x ^2)) * (arccot . x))) is V28() V29() ext-real set
K97(1,K99(((1 + (x ^2)) * (arccot . x)))) is V28() V29() ext-real set
(((f + Z) ^) / g) . x is V28() V29() ext-real Element of REAL
((f + Z) ^) . x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
(((f + Z) ^) . x) / (g . x) is V28() V29() ext-real Element of REAL
K99((g . x)) is V28() V29() ext-real set
K97((((f + Z) ^) . x),K99((g . x))) is V28() V29() ext-real set
(f + Z) . x is V28() V29() ext-real Element of REAL
((f + Z) . x) " is V28() V29() ext-real Element of REAL
(((f + Z) . x) ") / (g . x) is V28() V29() ext-real Element of REAL
K97((((f + Z) . x) "),K99((g . x))) is V28() V29() ext-real 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
((f . x) + (Z . x)) " is V28() V29() ext-real Element of REAL
(((f . x) + (Z . x)) ") / (g . x) is V28() V29() ext-real Element of REAL
K97((((f . x) + (Z . x)) "),K99((g . x))) is V28() V29() ext-real set
((f . x) + (Z . x)) * (g . x) is V28() V29() ext-real Element of REAL
1 / (((f . x) + (Z . x)) * (g . x)) is V28() V29() ext-real Element of REAL
K99((((f . x) + (Z . x)) * (g . x))) is V28() V29() ext-real set
K97(1,K99((((f . x) + (Z . x)) * (g . x)))) is V28() V29() ext-real set
(#Z 2) . x is V28() V29() ext-real Element of REAL
1 + ((#Z 2) . x) is V28() V29() ext-real Element of REAL
(1 + ((#Z 2) . x)) * (g . x) is V28() V29() ext-real Element of REAL
1 / ((1 + ((#Z 2) . x)) * (g . x)) is V28() V29() ext-real Element of REAL
K99(((1 + ((#Z 2) . x)) * (g . x))) is V28() V29() ext-real set
K97(1,K99(((1 + ((#Z 2) . x)) * (g . x)))) is V28() V29() ext-real set
x #Z 2 is V28() V29() ext-real Element of REAL
1 + (x #Z 2) is V28() V29() ext-real Element of REAL
(1 + (x #Z 2)) * (g . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x #Z 2)) * (g . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x #Z 2)) * (g . x))) is V28() V29() ext-real set
K97(1,K99(((1 + (x #Z 2)) * (g . x)))) is V28() V29() ext-real set
dom ((- (ln * arccot)) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- (ln * arccot)) `| f2) . x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 + (x ^2) is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
(1 + (x ^2)) * (arccot . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x ^2)) * (arccot . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x ^2)) * (arccot . x))) is V28() V29() ext-real set
K97(1,K99(((1 + (x ^2)) * (arccot . x)))) is V28() V29() ext-real set
exp_R ^2 is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R (#) exp_R is Relation-like REAL -defined V6() total V34() V35() V36() set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(arctan * exp_R) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(arctan * exp_R) . (lower_bound A) is V28() V29() ext-real Element of REAL
((arctan * exp_R) . (upper_bound A)) - ((arctan * exp_R) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((arctan * exp_R) . (lower_bound A))) is V28() V29() ext-real set
K96(((arctan * exp_R) . (upper_bound A)),K98(((arctan * exp_R) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (exp_R ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R / (f1 + (exp_R ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom (f1 + (exp_R ^2)) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(f1 + (exp_R ^2)) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom exp_R) /\ ((dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0})) is V55() V56() V57() Element of K19(REAL)
(f1 + (exp_R ^2)) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (exp_R ^2)) ^) is V55() V56() V57() Element of K19(REAL)
dom f1 is V55() V56() V57() Element of K19(REAL)
dom (exp_R ^2) is non empty V55() V56() V57() Element of K19(REAL)
(dom f1) /\ (dom (exp_R ^2)) is V55() V56() V57() Element of K19(REAL)
exp_R (#) exp_R is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) exp_R) is non empty V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
f1 . g is V28() V29() ext-real Element of REAL
0 * g is V28() V29() ext-real Element of REAL
(0 * g) + 1 is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
(f1 + (exp_R ^2)) . g is V28() V29() ext-real Element of REAL
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
(exp_R . g) ^2 is V28() V29() ext-real Element of REAL
K97((exp_R . g),(exp_R . g)) is V28() V29() ext-real set
1 + ((exp_R . g) ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / (1 + ((exp_R . g) ^2)) is V28() V29() ext-real Element of REAL
K99((1 + ((exp_R . g) ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99((1 + ((exp_R . g) ^2)))) is V28() V29() ext-real set
(exp_R / (f1 + (exp_R ^2))) . g is V28() V29() ext-real Element of REAL
(f1 + (exp_R ^2)) . g is V28() V29() ext-real Element of REAL
((f1 + (exp_R ^2)) . g) " is V28() V29() ext-real Element of REAL
(exp_R . g) * (((f1 + (exp_R ^2)) . g) ") is V28() V29() ext-real Element of REAL
f1 . g is V28() V29() ext-real Element of REAL
(exp_R ^2) . g is V28() V29() ext-real Element of REAL
(f1 . g) + ((exp_R ^2) . g) is V28() V29() ext-real Element of REAL
((f1 . g) + ((exp_R ^2) . g)) " is V28() V29() ext-real Element of REAL
(exp_R . g) * (((f1 . g) + ((exp_R ^2) . g)) ") is V28() V29() ext-real Element of REAL
(f1 . g) + ((exp_R . g) ^2) is V28() V29() ext-real Element of REAL
((f1 . g) + ((exp_R . g) ^2)) " is V28() V29() ext-real Element of REAL
(exp_R . g) * (((f1 . g) + ((exp_R . g) ^2)) ") is V28() V29() ext-real Element of REAL
(arctan * exp_R) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((arctan * exp_R) `| Z) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
((arctan * exp_R) `| Z) . g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
(exp_R . g) ^2 is V28() V29() ext-real Element of REAL
K97((exp_R . g),(exp_R . g)) is V28() V29() ext-real set
1 + ((exp_R . g) ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / (1 + ((exp_R . g) ^2)) is V28() V29() ext-real Element of REAL
K99((1 + ((exp_R . g) ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99((1 + ((exp_R . g) ^2)))) is V28() V29() ext-real set
- exp_R is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) exp_R is Relation-like REAL -defined V6() total V34() V35() V36() set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(arccot * exp_R) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(arccot * exp_R) . (lower_bound A) is V28() V29() ext-real Element of REAL
((arccot * exp_R) . (upper_bound A)) - ((arccot * exp_R) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((arccot * exp_R) . (lower_bound A))) is V28() V29() ext-real set
K96(((arccot * exp_R) . (upper_bound A)),K98(((arccot * exp_R) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (exp_R ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(- exp_R) / (f1 + (exp_R ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom (- exp_R) is non empty V55() V56() V57() Element of K19(REAL)
dom (f1 + (exp_R ^2)) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(f1 + (exp_R ^2)) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (- exp_R)) /\ ((dom (f1 + (exp_R ^2))) \ ((f1 + (exp_R ^2)) " {0})) is V55() V56() V57() Element of K19(REAL)
(f1 + (exp_R ^2)) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((f1 + (exp_R ^2)) ^) is V55() V56() V57() Element of K19(REAL)
dom f1 is V55() V56() V57() Element of K19(REAL)
dom (exp_R ^2) is non empty V55() V56() V57() Element of K19(REAL)
(dom f1) /\ (dom (exp_R ^2)) is V55() V56() V57() Element of K19(REAL)
exp_R (#) exp_R is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) exp_R) is non empty V55() V56() V57() Element of K19(REAL)
(- 1) (#) exp_R is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
g is V28() V29() ext-real Element of REAL
f1 . g is V28() V29() ext-real Element of REAL
0 * g is V28() V29() ext-real Element of REAL
(0 * g) + 1 is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
(f1 + (exp_R ^2)) . g is V28() V29() ext-real Element of REAL
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
(exp_R . g) ^2 is V28() V29() ext-real Element of REAL
K97((exp_R . g),(exp_R . g)) is V28() V29() ext-real set
1 + ((exp_R . g) ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / (1 + ((exp_R . g) ^2)) is V28() V29() ext-real Element of REAL
K99((1 + ((exp_R . g) ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99((1 + ((exp_R . g) ^2)))) is V28() V29() ext-real set
- ((exp_R . g) / (1 + ((exp_R . g) ^2))) is V28() V29() ext-real Element of REAL
((- exp_R) / (f1 + (exp_R ^2))) . g is V28() V29() ext-real Element of REAL
(- exp_R) . g is V28() V29() ext-real Element of REAL
(f1 + (exp_R ^2)) . g is V28() V29() ext-real Element of REAL
((f1 + (exp_R ^2)) . g) " is V28() V29() ext-real Element of REAL
((- exp_R) . g) * (((f1 + (exp_R ^2)) . g) ") is V28() V29() ext-real Element of REAL
- (exp_R . g) is V28() V29() ext-real Element of REAL
(- (exp_R . g)) * (((f1 + (exp_R ^2)) . g) ") is V28() V29() ext-real Element of REAL
f1 . g is V28() V29() ext-real Element of REAL
(exp_R ^2) . g is V28() V29() ext-real Element of REAL
(f1 . g) + ((exp_R ^2) . g) is V28() V29() ext-real Element of REAL
((f1 . g) + ((exp_R ^2) . g)) " is V28() V29() ext-real Element of REAL
(- (exp_R . g)) * (((f1 . g) + ((exp_R ^2) . g)) ") is V28() V29() ext-real Element of REAL
(f1 . g) + ((exp_R . g) ^2) is V28() V29() ext-real Element of REAL
((f1 . g) + ((exp_R . g) ^2)) " is V28() V29() ext-real Element of REAL
(- (exp_R . g)) * (((f1 . g) + ((exp_R . g) ^2)) ") is V28() V29() ext-real Element of REAL
(- (exp_R . g)) / (1 + ((exp_R . g) ^2)) is V28() V29() ext-real Element of REAL
K97((- (exp_R . g)),K99((1 + ((exp_R . g) ^2)))) is V28() V29() ext-real set
(arccot * exp_R) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((arccot * exp_R) `| Z) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
((arccot * exp_R) `| Z) . g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
(exp_R . g) ^2 is V28() V29() ext-real Element of REAL
K97((exp_R . g),(exp_R . g)) is V28() V29() ext-real set
1 + ((exp_R . g) ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / (1 + ((exp_R . g) ^2)) is V28() V29() ext-real Element of REAL
K99((1 + ((exp_R . g) ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99((1 + ((exp_R . g) ^2)))) is V28() V29() ext-real set
- ((exp_R . g) / (1 + ((exp_R . g) ^2))) is V28() V29() ext-real Element of REAL
exp_R (#) (sin / cos) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
cos ^2 is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
cos (#) cos is Relation-like REAL -defined V6() total V34() V35() V36() set
exp_R / (cos ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(exp_R (#) (sin / cos)) + (exp_R / (cos ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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() V29() ext-real set
K96(((exp_R (#) tan) . (upper_bound A)),K98(((exp_R (#) tan) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
dom (exp_R (#) (sin / cos)) is V55() V56() V57() Element of K19(REAL)
dom (exp_R / (cos ^2)) is V55() V56() V57() Element of K19(REAL)
(dom (exp_R (#) (sin / cos))) /\ (dom (exp_R / (cos ^2))) is V55() V56() V57() Element of K19(REAL)
dom (sin / cos) is V55() V56() V57() Element of K19(REAL)
(dom exp_R) /\ (dom (sin / cos)) is V55() V56() V57() Element of K19(REAL)
dom (cos ^2) is non empty V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(cos ^2) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (cos ^2)) \ ((cos ^2) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom exp_R) /\ ((dom (cos ^2)) \ ((cos ^2) " {0})) is V55() V56() V57() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
Z is V28() V29() ext-real Element of REAL
(cos ^2) . Z is V28() V29() ext-real Element of REAL
(cos ^2) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((cos ^2) ^) is V55() V56() V57() Element of K19(REAL)
f1 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,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
(exp_R . Z) * (sin . Z) is V28() V29() ext-real Element of REAL
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
K99((cos . Z)) is V28() V29() ext-real set
K97(((exp_R . Z) * (sin . Z)),K99((cos . Z))) is V28() V29() ext-real set
(cos . Z) ^2 is V28() V29() ext-real Element of REAL
K97((cos . Z),(cos . Z)) is V28() V29() ext-real set
(exp_R . Z) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() V29() ext-real set
K97((exp_R . Z),K99(((cos . Z) ^2))) is V28() V29() ext-real set
(((exp_R . Z) * (sin . Z)) / (cos . Z)) + ((exp_R . Z) / ((cos . Z) ^2)) is V28() V29() ext-real Element of REAL
((exp_R (#) (sin / cos)) + (exp_R / (cos ^2))) . Z is V28() V29() ext-real Element of REAL
(exp_R (#) (sin / cos)) . Z is V28() V29() ext-real Element of REAL
(exp_R / (cos ^2)) . Z is V28() V29() ext-real Element of REAL
((exp_R (#) (sin / cos)) . Z) + ((exp_R / (cos ^2)) . 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 . Z) * ((sin / cos) . Z)) + ((exp_R / (cos ^2)) . 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 . Z) * ((sin . Z) * ((cos . Z) "))) + ((exp_R / (cos ^2)) . Z) is V28() V29() ext-real Element of REAL
(cos ^2) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) / ((cos ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((cos ^2) . Z)) is V28() V29() ext-real set
K97((exp_R . Z),K99(((cos ^2) . Z))) is V28() V29() ext-real set
(((exp_R . Z) * (sin . Z)) / (cos . Z)) + ((exp_R . Z) / ((cos ^2) . Z)) is V28() V29() ext-real Element of REAL
(exp_R (#) tan) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R (#) tan) `| f) is V55() V56() V57() 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
exp_R . Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) * (sin . Z) is V28() V29() ext-real Element of REAL
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
K99((cos . Z)) is V28() V29() ext-real set
K97(((exp_R . Z) * (sin . Z)),K99((cos . Z))) is V28() V29() ext-real set
(cos . Z) ^2 is V28() V29() ext-real Element of REAL
K97((cos . Z),(cos . Z)) is V28() V29() ext-real set
(exp_R . Z) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() V29() ext-real set
K97((exp_R . Z),K99(((cos . Z) ^2))) is V28() V29() ext-real set
(((exp_R . Z) * (sin . Z)) / (cos . Z)) + ((exp_R . Z) / ((cos . Z) ^2)) is V28() V29() ext-real Element of REAL
exp_R (#) (cos / sin) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
sin ^2 is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
sin (#) sin is Relation-like REAL -defined V6() total V34() V35() V36() set
exp_R / (sin ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(exp_R (#) (cos / sin)) - (exp_R / (sin ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (exp_R / (sin ^2)) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (exp_R / (sin ^2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(exp_R (#) (cos / sin)) + (- (exp_R / (sin ^2))) is Relation-like REAL -defined V6() V34() V35() V36() set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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() V29() ext-real set
K96(((exp_R (#) cot) . (upper_bound A)),K98(((exp_R (#) cot) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
dom (exp_R (#) (cos / sin)) is V55() V56() V57() Element of K19(REAL)
dom (exp_R / (sin ^2)) is V55() V56() V57() Element of K19(REAL)
(dom (exp_R (#) (cos / sin))) /\ (dom (exp_R / (sin ^2))) is V55() V56() V57() Element of K19(REAL)
dom (cos / sin) is V55() V56() V57() Element of K19(REAL)
(dom exp_R) /\ (dom (cos / sin)) is V55() V56() V57() Element of K19(REAL)
dom (sin ^2) is non empty V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(sin ^2) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (sin ^2)) \ ((sin ^2) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom exp_R) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) is V55() V56() V57() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
Z is V28() V29() ext-real Element of REAL
(sin ^2) . Z is V28() V29() ext-real Element of REAL
(sin ^2) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin ^2) ^) is V55() V56() V57() Element of K19(REAL)
f1 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,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
(exp_R . Z) * (cos . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
((exp_R . Z) * (cos . Z)) / (sin . Z) is V28() V29() ext-real Element of REAL
K99((sin . Z)) is V28() V29() ext-real set
K97(((exp_R . Z) * (cos . Z)),K99((sin . Z))) is V28() V29() ext-real set
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is V28() V29() ext-real set
(exp_R . Z) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() V29() ext-real set
K97((exp_R . Z),K99(((sin . Z) ^2))) is V28() V29() ext-real set
(((exp_R . Z) * (cos . Z)) / (sin . Z)) - ((exp_R . Z) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
K98(((exp_R . Z) / ((sin . Z) ^2))) is V28() V29() ext-real set
K96((((exp_R . Z) * (cos . Z)) / (sin . Z)),K98(((exp_R . Z) / ((sin . Z) ^2)))) is V28() V29() ext-real set
((exp_R (#) (cos / sin)) - (exp_R / (sin ^2))) . Z is V28() V29() ext-real Element of REAL
(exp_R (#) (cos / sin)) . Z is V28() V29() ext-real Element of REAL
(exp_R / (sin ^2)) . Z is V28() V29() ext-real Element of REAL
((exp_R (#) (cos / sin)) . Z) - ((exp_R / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K98(((exp_R / (sin ^2)) . Z)) is V28() V29() ext-real set
K96(((exp_R (#) (cos / sin)) . Z),K98(((exp_R / (sin ^2)) . Z))) is V28() V29() ext-real set
(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 . Z) * ((cos / sin) . Z)) - ((exp_R / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K96(((exp_R . Z) * ((cos / sin) . Z)),K98(((exp_R / (sin ^2)) . Z))) is V28() V29() ext-real set
(sin . Z) " is V28() V29() ext-real Element of REAL
(cos . Z) * ((sin . Z) ") is V28() V29() ext-real Element of REAL
(exp_R . Z) * ((cos . Z) * ((sin . Z) ")) is V28() V29() ext-real Element of REAL
((exp_R . Z) * ((cos . Z) * ((sin . Z) "))) - ((exp_R / (sin ^2)) . Z) is V28() V29() ext-real Element of REAL
K96(((exp_R . Z) * ((cos . Z) * ((sin . Z) "))),K98(((exp_R / (sin ^2)) . Z))) is V28() V29() ext-real set
(sin ^2) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) / ((sin ^2) . Z) is V28() V29() ext-real Element of REAL
K99(((sin ^2) . Z)) is V28() V29() ext-real set
K97((exp_R . Z),K99(((sin ^2) . Z))) is V28() V29() ext-real set
(((exp_R . Z) * (cos . Z)) / (sin . Z)) - ((exp_R . Z) / ((sin ^2) . Z)) is V28() V29() ext-real Element of REAL
K98(((exp_R . Z) / ((sin ^2) . Z))) is V28() V29() ext-real set
K96((((exp_R . Z) * (cos . Z)) / (sin . Z)),K98(((exp_R . Z) / ((sin ^2) . Z)))) is V28() V29() ext-real set
(exp_R (#) cot) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R (#) cot) `| f) is V55() V56() V57() 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
exp_R . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) * (cos . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
((exp_R . Z) * (cos . Z)) / (sin . Z) is V28() V29() ext-real Element of REAL
K99((sin . Z)) is V28() V29() ext-real set
K97(((exp_R . Z) * (cos . Z)),K99((sin . Z))) is V28() V29() ext-real set
(sin . Z) ^2 is V28() V29() ext-real Element of REAL
K97((sin . Z),(sin . Z)) is V28() V29() ext-real set
(exp_R . Z) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() V29() ext-real set
K97((exp_R . Z),K99(((sin . Z) ^2))) is V28() V29() ext-real set
(((exp_R . Z) * (cos . Z)) / (sin . Z)) - ((exp_R . Z) / ((sin . Z) ^2)) is V28() V29() ext-real Element of REAL
K98(((exp_R . Z) / ((sin . Z) ^2))) is V28() V29() ext-real set
K96((((exp_R . Z) * (cos . Z)) / (sin . Z)),K98(((exp_R . Z) / ((sin . Z) ^2)))) is V28() V29() ext-real set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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() V29() ext-real set
K96(((exp_R (#) arctan) . (upper_bound A)),K98(((exp_R (#) arctan) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R / (f1 + (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom (exp_R (#) arctan) is V55() V56() V57() Element of K19(REAL)
dom (exp_R / (f1 + (#Z 2))) is V55() V56() V57() Element of K19(REAL)
(dom (exp_R (#) arctan)) /\ (dom (exp_R / (f1 + (#Z 2)))) is V55() V56() V57() Element of K19(REAL)
(f1 + (#Z 2)) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R (#) ((f1 + (#Z 2)) ^) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) ((f1 + (#Z 2)) ^)) is V55() V56() V57() Element of K19(REAL)
dom ((f1 + (#Z 2)) ^) is V55() V56() V57() Element of K19(REAL)
(dom exp_R) /\ (dom ((f1 + (#Z 2)) ^)) is V55() V56() V57() Element of K19(REAL)
dom (f1 + (#Z 2)) is V55() V56() V57() Element of K19(REAL)
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
arctan . g is V28() V29() ext-real Element of REAL
(exp_R . g) * (arctan . g) is V28() V29() ext-real Element of REAL
g ^2 is V28() V29() ext-real Element of REAL
K97(g,g) is V28() V29() ext-real set
1 + (g ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / (1 + (g ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (g ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99((1 + (g ^2)))) is V28() V29() ext-real set
((exp_R . g) * (arctan . g)) + ((exp_R . g) / (1 + (g ^2))) is V28() V29() ext-real Element of REAL
((exp_R (#) arctan) + (exp_R / (f1 + (#Z 2)))) . g is V28() V29() ext-real Element of REAL
(exp_R (#) arctan) . g is V28() V29() ext-real Element of REAL
(exp_R / (f1 + (#Z 2))) . g is V28() V29() ext-real Element of REAL
((exp_R (#) arctan) . g) + ((exp_R / (f1 + (#Z 2))) . g) is V28() V29() ext-real Element of REAL
((exp_R . g) * (arctan . g)) + ((exp_R / (f1 + (#Z 2))) . g) is V28() V29() ext-real Element of REAL
(f1 + (#Z 2)) . g is V28() V29() ext-real Element of REAL
(exp_R . g) / ((f1 + (#Z 2)) . g) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . g)) is V28() V29() ext-real set
K97((exp_R . g),K99(((f1 + (#Z 2)) . g))) is V28() V29() ext-real set
((exp_R . g) * (arctan . g)) + ((exp_R . g) / ((f1 + (#Z 2)) . g)) is V28() V29() ext-real Element of REAL
f1 . g is V28() V29() ext-real Element of REAL
(#Z 2) . g is V28() V29() ext-real Element of REAL
(f1 . g) + ((#Z 2) . g) is V28() V29() ext-real Element of REAL
(exp_R . g) / ((f1 . g) + ((#Z 2) . g)) is V28() V29() ext-real Element of REAL
K99(((f1 . g) + ((#Z 2) . g))) is V28() V29() ext-real set
K97((exp_R . g),K99(((f1 . g) + ((#Z 2) . g)))) is V28() V29() ext-real set
((exp_R . g) * (arctan . g)) + ((exp_R . g) / ((f1 . g) + ((#Z 2) . g))) is V28() V29() ext-real Element of REAL
g #Z 2 is V28() V29() ext-real Element of REAL
(f1 . g) + (g #Z 2) is V28() V29() ext-real Element of REAL
(exp_R . g) / ((f1 . g) + (g #Z 2)) is V28() V29() ext-real Element of REAL
K99(((f1 . g) + (g #Z 2))) is V28() V29() ext-real set
K97((exp_R . g),K99(((f1 . g) + (g #Z 2)))) is V28() V29() ext-real set
((exp_R . g) * (arctan . g)) + ((exp_R . g) / ((f1 . g) + (g #Z 2))) is V28() V29() ext-real Element of REAL
(f1 . g) + (g ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / ((f1 . g) + (g ^2)) is V28() V29() ext-real Element of REAL
K99(((f1 . g) + (g ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99(((f1 . g) + (g ^2)))) is V28() V29() ext-real set
((exp_R . g) * (arctan . g)) + ((exp_R . g) / ((f1 . g) + (g ^2))) is V28() V29() ext-real Element of REAL
(exp_R (#) arctan) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R (#) arctan) `| Z) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
((exp_R (#) arctan) `| Z) . g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
arctan . g is V28() V29() ext-real Element of REAL
(exp_R . g) * (arctan . g) is V28() V29() ext-real Element of REAL
g ^2 is V28() V29() ext-real Element of REAL
K97(g,g) is V28() V29() ext-real set
1 + (g ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / (1 + (g ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (g ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99((1 + (g ^2)))) is V28() V29() ext-real set
((exp_R . g) * (arctan . g)) + ((exp_R . g) / (1 + (g ^2))) is V28() V29() ext-real Element of REAL
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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() V29() ext-real set
K96(((exp_R (#) arccot) . (upper_bound A)),K98(((exp_R (#) arccot) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 + (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R / (f1 + (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(exp_R (#) arccot) - (exp_R / (f1 + (#Z 2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (exp_R / (f1 + (#Z 2))) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (exp_R / (f1 + (#Z 2))) is Relation-like REAL -defined V6() V34() V35() V36() set
(exp_R (#) arccot) + (- (exp_R / (f1 + (#Z 2)))) is Relation-like REAL -defined V6() V34() V35() V36() set
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom (exp_R (#) arccot) is V55() V56() V57() Element of K19(REAL)
dom (exp_R / (f1 + (#Z 2))) is V55() V56() V57() Element of K19(REAL)
(dom (exp_R (#) arccot)) /\ (dom (exp_R / (f1 + (#Z 2)))) is V55() V56() V57() Element of K19(REAL)
(f1 + (#Z 2)) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
exp_R (#) ((f1 + (#Z 2)) ^) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (exp_R (#) ((f1 + (#Z 2)) ^)) is V55() V56() V57() Element of K19(REAL)
dom ((f1 + (#Z 2)) ^) is V55() V56() V57() Element of K19(REAL)
(dom exp_R) /\ (dom ((f1 + (#Z 2)) ^)) is V55() V56() V57() Element of K19(REAL)
dom (f1 + (#Z 2)) is V55() V56() V57() Element of K19(REAL)
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
arccot . g is V28() V29() ext-real Element of REAL
(exp_R . g) * (arccot . g) is V28() V29() ext-real Element of REAL
g ^2 is V28() V29() ext-real Element of REAL
K97(g,g) is V28() V29() ext-real set
1 + (g ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / (1 + (g ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (g ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99((1 + (g ^2)))) is V28() V29() ext-real set
((exp_R . g) * (arccot . g)) - ((exp_R . g) / (1 + (g ^2))) is V28() V29() ext-real Element of REAL
K98(((exp_R . g) / (1 + (g ^2)))) is V28() V29() ext-real set
K96(((exp_R . g) * (arccot . g)),K98(((exp_R . g) / (1 + (g ^2))))) is V28() V29() ext-real set
((exp_R (#) arccot) - (exp_R / (f1 + (#Z 2)))) . g is V28() V29() ext-real Element of REAL
(exp_R (#) arccot) . g is V28() V29() ext-real Element of REAL
(exp_R / (f1 + (#Z 2))) . g is V28() V29() ext-real Element of REAL
((exp_R (#) arccot) . g) - ((exp_R / (f1 + (#Z 2))) . g) is V28() V29() ext-real Element of REAL
K98(((exp_R / (f1 + (#Z 2))) . g)) is V28() V29() ext-real set
K96(((exp_R (#) arccot) . g),K98(((exp_R / (f1 + (#Z 2))) . g))) is V28() V29() ext-real set
((exp_R . g) * (arccot . g)) - ((exp_R / (f1 + (#Z 2))) . g) is V28() V29() ext-real Element of REAL
K96(((exp_R . g) * (arccot . g)),K98(((exp_R / (f1 + (#Z 2))) . g))) is V28() V29() ext-real set
(f1 + (#Z 2)) . g is V28() V29() ext-real Element of REAL
(exp_R . g) / ((f1 + (#Z 2)) . g) is V28() V29() ext-real Element of REAL
K99(((f1 + (#Z 2)) . g)) is V28() V29() ext-real set
K97((exp_R . g),K99(((f1 + (#Z 2)) . g))) is V28() V29() ext-real set
((exp_R . g) * (arccot . g)) - ((exp_R . g) / ((f1 + (#Z 2)) . g)) is V28() V29() ext-real Element of REAL
K98(((exp_R . g) / ((f1 + (#Z 2)) . g))) is V28() V29() ext-real set
K96(((exp_R . g) * (arccot . g)),K98(((exp_R . g) / ((f1 + (#Z 2)) . g)))) is V28() V29() ext-real set
f1 . g is V28() V29() ext-real Element of REAL
(#Z 2) . g is V28() V29() ext-real Element of REAL
(f1 . g) + ((#Z 2) . g) is V28() V29() ext-real Element of REAL
(exp_R . g) / ((f1 . g) + ((#Z 2) . g)) is V28() V29() ext-real Element of REAL
K99(((f1 . g) + ((#Z 2) . g))) is V28() V29() ext-real set
K97((exp_R . g),K99(((f1 . g) + ((#Z 2) . g)))) is V28() V29() ext-real set
((exp_R . g) * (arccot . g)) - ((exp_R . g) / ((f1 . g) + ((#Z 2) . g))) is V28() V29() ext-real Element of REAL
K98(((exp_R . g) / ((f1 . g) + ((#Z 2) . g)))) is V28() V29() ext-real set
K96(((exp_R . g) * (arccot . g)),K98(((exp_R . g) / ((f1 . g) + ((#Z 2) . g))))) is V28() V29() ext-real set
g #Z 2 is V28() V29() ext-real Element of REAL
(f1 . g) + (g #Z 2) is V28() V29() ext-real Element of REAL
(exp_R . g) / ((f1 . g) + (g #Z 2)) is V28() V29() ext-real Element of REAL
K99(((f1 . g) + (g #Z 2))) is V28() V29() ext-real set
K97((exp_R . g),K99(((f1 . g) + (g #Z 2)))) is V28() V29() ext-real set
((exp_R . g) * (arccot . g)) - ((exp_R . g) / ((f1 . g) + (g #Z 2))) is V28() V29() ext-real Element of REAL
K98(((exp_R . g) / ((f1 . g) + (g #Z 2)))) is V28() V29() ext-real set
K96(((exp_R . g) * (arccot . g)),K98(((exp_R . g) / ((f1 . g) + (g #Z 2))))) is V28() V29() ext-real set
(f1 . g) + (g ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / ((f1 . g) + (g ^2)) is V28() V29() ext-real Element of REAL
K99(((f1 . g) + (g ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99(((f1 . g) + (g ^2)))) is V28() V29() ext-real set
((exp_R . g) * (arccot . g)) - ((exp_R . g) / ((f1 . g) + (g ^2))) is V28() V29() ext-real Element of REAL
K98(((exp_R . g) / ((f1 . g) + (g ^2)))) is V28() V29() ext-real set
K96(((exp_R . g) * (arccot . g)),K98(((exp_R . g) / ((f1 . g) + (g ^2))))) is V28() V29() ext-real set
(exp_R (#) arccot) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R (#) arccot) `| Z) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
((exp_R (#) arccot) `| Z) . g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
exp_R . g is V28() V29() ext-real Element of REAL
arccot . g is V28() V29() ext-real Element of REAL
(exp_R . g) * (arccot . g) is V28() V29() ext-real Element of REAL
g ^2 is V28() V29() ext-real Element of REAL
K97(g,g) is V28() V29() ext-real set
1 + (g ^2) is V28() V29() ext-real Element of REAL
(exp_R . g) / (1 + (g ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (g ^2))) is V28() V29() ext-real set
K97((exp_R . g),K99((1 + (g ^2)))) is V28() V29() ext-real set
((exp_R . g) * (arccot . g)) - ((exp_R . g) / (1 + (g ^2))) is V28() V29() ext-real Element of REAL
K98(((exp_R . g) / (1 + (g ^2)))) is V28() V29() ext-real set
K96(((exp_R . g) * (arccot . g)),K98(((exp_R . g) / (1 + (g ^2))))) is V28() V29() ext-real set
(exp_R * sin) (#) cos is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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() V29() ext-real set
K96(((exp_R * sin) . (upper_bound A)),K98(((exp_R * sin) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
(dom (exp_R * sin)) /\ (dom cos) is V55() V56() V57() Element of K19(REAL)
f1 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,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 . (sin . Z) is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(exp_R . (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
(exp_R * sin) . Z is V28() V29() ext-real Element of REAL
((exp_R * sin) . Z) * (cos . Z) is V28() V29() ext-real Element of REAL
(exp_R * sin) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((exp_R * sin) `| f) is V55() V56() V57() 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
sin . Z is V28() V29() ext-real Element of REAL
exp_R . (sin . Z) is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
(exp_R . (sin . Z)) * (cos . Z) is V28() V29() ext-real Element of REAL
(exp_R * cos) (#) sin is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
- (exp_R * cos) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (exp_R * cos) is Relation-like REAL -defined V6() total V34() V35() V36() set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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() V29() ext-real set
K96(((- (exp_R * cos)) . (upper_bound A)),K98(((- (exp_R * cos)) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
(dom (exp_R * cos)) /\ (dom sin) is V55() V56() V57() Element of K19(REAL)
f1 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (exp_R * cos)) is non empty V55() V56() V57() Element of K19(REAL)
(- 1) (#) (exp_R * cos) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(- (exp_R * cos)) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V28() V29() ext-real Element of REAL
((- (exp_R * cos)) `| f) . Z is V28() V29() ext-real Element of REAL
cos . Z is V28() V29() ext-real Element of REAL
exp_R . (cos . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(exp_R . (cos . Z)) * (sin . Z) is V28() V29() ext-real Element of REAL
diff ((- (exp_R * cos)),Z) is V28() V29() ext-real Element of REAL
diff ((exp_R * cos),Z) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((exp_R * cos),Z)) is V28() V29() ext-real Element of REAL
diff (exp_R,(cos . Z)) is V28() V29() ext-real Element of REAL
diff (cos,Z) is V28() V29() ext-real Element of REAL
(diff (exp_R,(cos . Z))) * (diff (cos,Z)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (exp_R,(cos . Z))) * (diff (cos,Z))) is V28() V29() ext-real Element of REAL
- (sin . Z) is V28() V29() ext-real Element of REAL
(diff (exp_R,(cos . Z))) * (- (sin . Z)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (exp_R,(cos . Z))) * (- (sin . Z))) is V28() V29() ext-real Element of REAL
(exp_R . (cos . Z)) * (- (sin . Z)) is V28() V29() ext-real Element of REAL
(- 1) * ((exp_R . (cos . Z)) * (- (sin . Z))) is V28() V29() ext-real Element of 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 . (cos . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(exp_R . (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
(exp_R * cos) . Z is V28() V29() ext-real Element of REAL
((exp_R * cos) . Z) * (sin . Z) is V28() V29() ext-real Element of REAL
dom ((- (exp_R * cos)) `| f) is V55() V56() V57() 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
cos . Z is V28() V29() ext-real Element of REAL
exp_R . (cos . Z) is V28() V29() ext-real Element of REAL
sin . Z is V28() V29() ext-real Element of REAL
(exp_R . (cos . Z)) * (sin . Z) is V28() V29() ext-real Element of REAL
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(sin * ln) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(sin * ln) . (lower_bound A) is V28() V29() ext-real Element of REAL
((sin * ln) . (upper_bound A)) - ((sin * ln) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((sin * ln) . (lower_bound A))) is V28() V29() ext-real set
K96(((sin * ln) . (upper_bound A)),K98(((sin * ln) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
id f is Relation-like REAL -defined f -defined REAL -valued f -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * ln) (#) ((id f) ^) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * ln) / (id f) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((cos * ln) / (id f)) is V55() V56() V57() Element of K19(REAL)
dom ((id f) ^) is V55() V56() V57() Element of K19(REAL)
(dom (cos * ln)) /\ (dom ((id f) ^)) is V55() V56() V57() Element of K19(REAL)
Z is set
ln . Z is V28() V29() ext-real Element of REAL
f1 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,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)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() V29() ext-real set
K97((cos . (ln . Z)),K99(Z)) is V28() V29() ext-real set
((cos * ln) (#) ((id f) ^)) . Z is V28() V29() ext-real Element of REAL
((cos * ln) / (id f)) . Z is V28() V29() ext-real Element of REAL
(cos * ln) . Z is V28() V29() ext-real Element of REAL
(id f) . Z is V28() V29() ext-real Element of REAL
((id f) . Z) " is V28() V29() ext-real Element of REAL
((cos * ln) . Z) * (((id f) . Z) ") is V28() V29() ext-real Element of REAL
((cos * ln) . Z) / Z is V28() V29() ext-real Element of REAL
K97(((cos * ln) . Z),K99(Z)) is V28() V29() ext-real set
(sin * ln) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin * ln) `| f) is V55() V56() V57() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((sin * 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)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() V29() ext-real set
K97((cos . (ln . Z)),K99(Z)) is V28() V29() ext-real set
{ b₁ where b₁ is V28() V29() ext-real Element of REAL : not b₁ <= 0 } is set
- (cos * ln) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (cos * ln) is Relation-like REAL -defined V6() V34() V35() V36() set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(- (cos * ln)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(- (cos * ln)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (cos * ln)) . (upper_bound A)) - ((- (cos * ln)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (cos * ln)) . (lower_bound A))) is V28() V29() ext-real set
K96(((- (cos * ln)) . (upper_bound A)),K98(((- (cos * ln)) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
id f is Relation-like REAL -defined f -defined REAL -valued f -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * ln) (#) ((id f) ^) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * ln) / (id f) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin * ln) / (id f)) is V55() V56() V57() Element of K19(REAL)
dom ((id f) ^) is V55() V56() V57() Element of K19(REAL)
(dom (sin * ln)) /\ (dom ((id f) ^)) is V55() V56() V57() Element of K19(REAL)
Z is set
ln . Z is V28() V29() ext-real Element of REAL
f1 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (cos * ln)) is V55() V56() V57() Element of K19(REAL)
(- 1) (#) (cos * ln) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(- (cos * ln)) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V28() V29() ext-real Element of REAL
((- (cos * ln)) `| f) . 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)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() V29() ext-real set
K97((sin . (ln . Z)),K99(Z)) is V28() V29() ext-real set
diff ((- (cos * ln)),Z) is V28() V29() ext-real Element of REAL
diff ((cos * ln),Z) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((cos * ln),Z)) is V28() V29() ext-real Element of REAL
diff (cos,(ln . Z)) is V28() V29() ext-real Element of REAL
diff (ln,Z) is V28() V29() ext-real Element of REAL
(diff (cos,(ln . Z))) * (diff (ln,Z)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (cos,(ln . Z))) * (diff (ln,Z))) is V28() V29() ext-real Element of REAL
- (sin . (ln . Z)) is V28() V29() ext-real Element of REAL
(- (sin . (ln . Z))) * (diff (ln,Z)) is V28() V29() ext-real Element of REAL
(- 1) * ((- (sin . (ln . Z))) * (diff (ln,Z))) is V28() V29() ext-real Element of REAL
1 / Z is V28() V29() ext-real Element of REAL
K97(1,K99(Z)) is V28() V29() ext-real set
(- (sin . (ln . Z))) * (1 / Z) is V28() V29() ext-real Element of REAL
(- 1) * ((- (sin . (ln . Z))) * (1 / Z)) is V28() V29() ext-real Element of 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)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() V29() ext-real set
K97((sin . (ln . Z)),K99(Z)) is V28() V29() ext-real set
((sin * ln) (#) ((id f) ^)) . Z is V28() V29() ext-real Element of REAL
((sin * ln) / (id f)) . Z is V28() V29() ext-real Element of REAL
(sin * ln) . Z is V28() V29() ext-real Element of REAL
(id f) . Z is V28() V29() ext-real Element of REAL
((id f) . Z) " is V28() V29() ext-real Element of REAL
((sin * ln) . Z) * (((id f) . Z) ") is V28() V29() ext-real Element of REAL
((sin * ln) . Z) / Z is V28() V29() ext-real Element of REAL
K97(((sin * ln) . Z),K99(Z)) is V28() V29() ext-real set
dom ((- (cos * ln)) `| f) is V55() V56() V57() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((- (cos * 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)) / Z is V28() V29() ext-real Element of REAL
K99(Z) is V28() V29() ext-real set
K97((sin . (ln . Z)),K99(Z)) is V28() V29() ext-real set
exp_R (#) (cos * exp_R) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(sin * exp_R) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(sin * exp_R) . (lower_bound A) is V28() V29() ext-real Element of REAL
((sin * exp_R) . (upper_bound A)) - ((sin * exp_R) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((sin * exp_R) . (lower_bound A))) is V28() V29() ext-real set
K96(((sin * exp_R) . (upper_bound A)),K98(((sin * exp_R) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
(dom exp_R) /\ (dom (cos * exp_R)) is V55() V56() V57() Element of K19(REAL)
Z is set
exp_R . Z is V28() V29() ext-real Element of REAL
f1 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,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
(exp_R . Z) * (cos . (exp_R . Z)) is V28() V29() ext-real Element of REAL
(exp_R (#) (cos * exp_R)) . Z is V28() V29() ext-real Element of REAL
(cos * exp_R) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) * ((cos * exp_R) . Z) is V28() V29() ext-real Element of REAL
(sin * exp_R) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin * exp_R) `| f) is V55() V56() V57() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((sin * 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
(exp_R . Z) * (cos . (exp_R . Z)) is V28() V29() ext-real Element of REAL
exp_R (#) (sin * exp_R) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
- (cos * exp_R) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (cos * exp_R) is Relation-like REAL -defined V6() total V34() V35() V36() set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(- (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
(- (cos * exp_R)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (cos * exp_R)) . (upper_bound A)) - ((- (cos * exp_R)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (cos * exp_R)) . (lower_bound A))) is V28() V29() ext-real set
K96(((- (cos * exp_R)) . (upper_bound A)),K98(((- (cos * exp_R)) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
(dom exp_R) /\ (dom (sin * exp_R)) is V55() V56() V57() Element of K19(REAL)
Z is set
exp_R . Z is V28() V29() ext-real Element of REAL
f1 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (cos * exp_R)) is non empty V55() V56() V57() Element of K19(REAL)
(- 1) (#) (cos * exp_R) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(- (cos * exp_R)) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is V28() V29() ext-real Element of REAL
((- (cos * exp_R)) `| f) . 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
(exp_R . Z) * (sin . (exp_R . Z)) is V28() V29() ext-real Element of REAL
diff ((- (cos * exp_R)),Z) is V28() V29() ext-real Element of REAL
diff ((cos * exp_R),Z) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((cos * exp_R),Z)) is V28() V29() ext-real Element of REAL
diff (cos,(exp_R . Z)) is V28() V29() ext-real Element of REAL
diff (exp_R,Z) is V28() V29() ext-real Element of REAL
(diff (cos,(exp_R . Z))) * (diff (exp_R,Z)) is V28() V29() ext-real Element of REAL
(- 1) * ((diff (cos,(exp_R . Z))) * (diff (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))) * (diff (exp_R,Z)) is V28() V29() ext-real Element of REAL
(- 1) * ((- (sin . (exp_R . Z))) * (diff (exp_R,Z))) is V28() V29() ext-real Element of REAL
(- (sin . (exp_R . Z))) * (exp_R . Z) is V28() V29() ext-real Element of REAL
(- 1) * ((- (sin . (exp_R . Z))) * (exp_R . Z)) is V28() V29() ext-real Element of 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
(exp_R . Z) * (sin . (exp_R . Z)) is V28() V29() ext-real Element of REAL
(exp_R (#) (sin * exp_R)) . Z is V28() V29() ext-real Element of REAL
(sin * exp_R) . Z is V28() V29() ext-real Element of REAL
(exp_R . Z) * ((sin * exp_R) . Z) is V28() V29() ext-real Element of REAL
dom ((- (cos * exp_R)) `| f) is V55() V56() V57() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((- (cos * 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
(exp_R . Z) * (sin . (exp_R . Z)) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
A / 2 is V28() V29() ext-real Element of REAL
K97(A,K99(2)) is V28() V29() ext-real set
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
ln * (f + Z) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * (f + Z)) is V55() V56() V57() Element of K19(REAL)
(A / 2) (#) (ln * (f + Z)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z 2) * g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arctan * g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f2 is V55() V56() V57() Element of K19(REAL)
integral (f2,f1) is V28() V29() ext-real Element of REAL
x is V55() V56() V57() open Element of K19(REAL)
id x is Relation-like REAL -defined x -defined REAL -valued x -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id x) (#) (arctan * g) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((A / 2) (#) (ln * (f + Z))) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) ((A / 2) (#) (ln * (f + Z))) is Relation-like REAL -defined V6() V34() V35() V36() set
((id x) (#) (arctan * g)) + (- ((A / 2) (#) (ln * (f + Z)))) is Relation-like REAL -defined V6() V34() V35() V36() set
(((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) . (upper_bound f1) is V28() V29() ext-real Element of REAL
(((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) . (lower_bound f1) is V28() V29() ext-real Element of REAL
((((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) . (upper_bound f1)) - ((((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98(((((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) . (lower_bound f1))) is V28() V29() ext-real set
K96(((((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) . (upper_bound f1)),K98(((((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) . (lower_bound f1)))) is V28() V29() ext-real set
dom (id x) is V55() V56() V57() Element of K19(x)
K19(x) is set
(dom (id x)) /\ (dom f2) is V55() V56() V57() Element of K19(REAL)
dom ((id x) (#) (arctan * g)) is V55() V56() V57() Element of K19(REAL)
dom ((A / 2) (#) (ln * (f + Z))) is V55() V56() V57() Element of K19(REAL)
(dom ((id x) (#) (arctan * g))) /\ (dom ((A / 2) (#) (ln * (f + Z)))) is V55() V56() V57() Element of K19(REAL)
dom (((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) is V55() V56() V57() Element of K19(REAL)
1 / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(1,K99(A)) is V28() V29() ext-real set
y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
(1 / A) * y is V28() V29() ext-real Element of REAL
((1 / A) * y) + 0 is V28() V29() ext-real Element of REAL
y / A is V28() V29() ext-real Element of REAL
K97(y,K99(A)) is V28() V29() ext-real set
y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
(1 / A) * y is V28() V29() ext-real Element of REAL
((1 / A) * y) + 0 is V28() V29() ext-real Element of REAL
f2 | x is Relation-like REAL -defined x -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
y / A is V28() V29() ext-real Element of REAL
K97(y,K99(A)) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
c₁₀ is V28() V29() ext-real Element of REAL
g . c₁₀ is V28() V29() ext-real Element of REAL
c₁₀ / A is V28() V29() ext-real Element of REAL
K97(c₁₀,K99(A)) is V28() V29() ext-real set
y is V28() V29() ext-real Element of REAL
f2 . y is V28() V29() ext-real Element of REAL
y / A is V28() V29() ext-real Element of REAL
K97(y,K99(A)) is V28() V29() ext-real set
arctan . (y / A) is V28() V29() ext-real Element of REAL
(arctan * g) . y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
arctan . (g . y) is V28() V29() ext-real Element of REAL
(((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) `| x is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) `| x) is V55() V56() V57() Element of K19(REAL)
y is V28() V29() ext-real Element of REAL
((((id x) (#) (arctan * g)) - ((A / 2) (#) (ln * (f + Z)))) `| x) . y is V28() V29() ext-real Element of REAL
f2 . y is V28() V29() ext-real Element of REAL
y / A is V28() V29() ext-real Element of REAL
K97(y,K99(A)) is V28() V29() ext-real set
arctan . (y / A) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
A / 2 is V28() V29() ext-real Element of REAL
K97(A,K99(2)) is V28() V29() ext-real set
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
ln * (f + Z) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (ln * (f + Z)) is V55() V56() V57() Element of K19(REAL)
(A / 2) (#) (ln * (f + Z)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z 2) * g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccot * g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f2 is V55() V56() V57() Element of K19(REAL)
integral (f2,f1) is V28() V29() ext-real Element of REAL
x is V55() V56() V57() open Element of K19(REAL)
id x is Relation-like REAL -defined x -defined REAL -valued x -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id x) (#) (arccot * g) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) . (upper_bound f1) is V28() V29() ext-real Element of REAL
(((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) . (lower_bound f1) is V28() V29() ext-real Element of REAL
((((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) . (upper_bound f1)) - ((((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98(((((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) . (lower_bound f1))) is V28() V29() ext-real set
K96(((((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) . (upper_bound f1)),K98(((((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) . (lower_bound f1)))) is V28() V29() ext-real set
dom (id x) is V55() V56() V57() Element of K19(x)
K19(x) is set
(dom (id x)) /\ (dom f2) is V55() V56() V57() Element of K19(REAL)
dom ((id x) (#) (arccot * g)) is V55() V56() V57() Element of K19(REAL)
dom ((A / 2) (#) (ln * (f + Z))) is V55() V56() V57() Element of K19(REAL)
(dom ((id x) (#) (arccot * g))) /\ (dom ((A / 2) (#) (ln * (f + Z)))) is V55() V56() V57() Element of K19(REAL)
dom (((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) is V55() V56() V57() Element of K19(REAL)
y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
y / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(y,K99(A)) is V28() V29() ext-real set
1 / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(1,K99(A)) is V28() V29() ext-real set
y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
(1 / A) * y is V28() V29() ext-real Element of REAL
((1 / A) * y) + 0 is V28() V29() ext-real Element of REAL
y / A is V28() V29() ext-real Element of REAL
K97(y,K99(A)) is V28() V29() ext-real set
y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
(1 / A) * y is V28() V29() ext-real Element of REAL
((1 / A) * y) + 0 is V28() V29() ext-real Element of REAL
f2 | x is Relation-like REAL -defined x -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
y is V28() V29() ext-real Element of REAL
f . y is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K97(x,K99(A)) is V28() V29() ext-real set
y is V28() V29() ext-real Element of REAL
f2 . y is V28() V29() ext-real Element of REAL
y / A is V28() V29() ext-real Element of REAL
K97(y,K99(A)) is V28() V29() ext-real set
arccot . (y / A) is V28() V29() ext-real Element of REAL
(arccot * g) . y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
arccot . (g . y) is V28() V29() ext-real Element of REAL
(((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) `| x is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) `| x) is V55() V56() V57() Element of K19(REAL)
y is V28() V29() ext-real Element of REAL
((((id x) (#) (arccot * g)) + ((A / 2) (#) (ln * (f + Z)))) `| x) . y is V28() V29() ext-real Element of REAL
f2 . y is V28() V29() ext-real Element of REAL
y / A is V28() V29() ext-real Element of REAL
K97(y,K99(A)) is V28() V29() ext-real set
arccot . (y / A) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
f | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,f1) is V28() V29() ext-real Element of REAL
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arctan * Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z ^2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z (#) Z is Relation-like REAL -defined V6() V34() V35() V36() set
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g + (Z ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) (g + (Z ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is V55() V56() V57() open Element of K19(REAL)
id f2 is Relation-like REAL -defined f2 -defined REAL -valued f2 -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f2) / (A (#) (g + (Z ^2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arctan * Z) + ((id f2) / (A (#) (g + (Z ^2)))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f2) (#) (arctan * Z) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id f2) (#) (arctan * Z)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
((id f2) (#) (arctan * Z)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((id f2) (#) (arctan * Z)) . (upper_bound f1)) - (((id f2) (#) (arctan * Z)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((id f2) (#) (arctan * Z)) . (lower_bound f1))) is V28() V29() ext-real set
K96((((id f2) (#) (arctan * Z)) . (upper_bound f1)),K98((((id f2) (#) (arctan * Z)) . (lower_bound f1)))) is V28() V29() ext-real set
dom (arctan * Z) is V55() V56() V57() Element of K19(REAL)
dom ((id f2) / (A (#) (g + (Z ^2)))) is V55() V56() V57() Element of K19(REAL)
(dom (arctan * Z)) /\ (dom ((id f2) / (A (#) (g + (Z ^2))))) is V55() V56() V57() Element of K19(REAL)
dom (id f2) is V55() V56() V57() Element of K19(f2)
K19(f2) is set
(dom (id f2)) /\ (dom (arctan * Z)) is V55() V56() V57() Element of K19(REAL)
dom ((id f2) (#) (arctan * Z)) is V55() V56() V57() Element of K19(REAL)
dom (A (#) (g + (Z ^2))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(A (#) (g + (Z ^2))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (A (#) (g + (Z ^2)))) \ ((A (#) (g + (Z ^2))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (id f2)) /\ ((dom (A (#) (g + (Z ^2)))) \ ((A (#) (g + (Z ^2))) " {0})) is V55() V56() V57() Element of K19(REAL)
(A (#) (g + (Z ^2))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((A (#) (g + (Z ^2))) ^) is V55() V56() V57() Element of K19(REAL)
dom (g + (Z ^2)) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
arctan . (x / A) is V28() V29() ext-real Element of REAL
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is V28() V29() ext-real 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() V29() ext-real set
K97(x,K99((A * (1 + ((x / A) ^2))))) is V28() V29() ext-real set
(arctan . (x / A)) + (x / (A * (1 + ((x / A) ^2)))) is V28() V29() ext-real Element of REAL
((arctan * Z) + ((id f2) / (A (#) (g + (Z ^2))))) . x is V28() V29() ext-real Element of REAL
(arctan * Z) . x is V28() V29() ext-real Element of REAL
((id f2) / (A (#) (g + (Z ^2)))) . x is V28() V29() ext-real Element of REAL
((arctan * Z) . x) + (((id f2) / (A (#) (g + (Z ^2)))) . x) is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
arctan . (Z . x) is V28() V29() ext-real Element of REAL
(arctan . (Z . x)) + (((id f2) / (A (#) (g + (Z ^2)))) . x) is V28() V29() ext-real Element of REAL
(id f2) . x is V28() V29() ext-real Element of REAL
(A (#) (g + (Z ^2))) . x is V28() V29() ext-real Element of REAL
((id f2) . x) / ((A (#) (g + (Z ^2))) . x) is V28() V29() ext-real Element of REAL
K99(((A (#) (g + (Z ^2))) . x)) is V28() V29() ext-real set
K97(((id f2) . x),K99(((A (#) (g + (Z ^2))) . x))) is V28() V29() ext-real set
(arctan . (Z . x)) + (((id f2) . x) / ((A (#) (g + (Z ^2))) . x)) is V28() V29() ext-real Element of REAL
x / ((A (#) (g + (Z ^2))) . x) is V28() V29() ext-real Element of REAL
K97(x,K99(((A (#) (g + (Z ^2))) . x))) is V28() V29() ext-real set
(arctan . (Z . x)) + (x / ((A (#) (g + (Z ^2))) . x)) is V28() V29() ext-real Element of REAL
(g + (Z ^2)) . x is V28() V29() ext-real Element of REAL
A * ((g + (Z ^2)) . x) is V28() V29() ext-real Element of REAL
x / (A * ((g + (Z ^2)) . x)) is V28() V29() ext-real Element of REAL
K99((A * ((g + (Z ^2)) . x))) is V28() V29() ext-real set
K97(x,K99((A * ((g + (Z ^2)) . x)))) is V28() V29() ext-real set
(arctan . (Z . x)) + (x / (A * ((g + (Z ^2)) . x))) is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
(Z ^2) . x is V28() V29() ext-real Element of REAL
(g . x) + ((Z ^2) . x) is V28() V29() ext-real Element of REAL
A * ((g . x) + ((Z ^2) . x)) is V28() V29() ext-real Element of REAL
x / (A * ((g . x) + ((Z ^2) . x))) is V28() V29() ext-real Element of REAL
K99((A * ((g . x) + ((Z ^2) . x)))) is V28() V29() ext-real set
K97(x,K99((A * ((g . x) + ((Z ^2) . x))))) is V28() V29() ext-real set
(arctan . (Z . x)) + (x / (A * ((g . x) + ((Z ^2) . x)))) is V28() V29() ext-real Element of REAL
(Z . x) ^2 is V28() V29() ext-real Element of REAL
K97((Z . x),(Z . x)) is V28() V29() ext-real set
(g . x) + ((Z . x) ^2) is V28() V29() ext-real Element of REAL
A * ((g . x) + ((Z . x) ^2)) is V28() V29() ext-real Element of REAL
x / (A * ((g . x) + ((Z . x) ^2))) is V28() V29() ext-real Element of REAL
K99((A * ((g . x) + ((Z . x) ^2)))) is V28() V29() ext-real set
K97(x,K99((A * ((g . x) + ((Z . x) ^2))))) is V28() V29() ext-real set
(arctan . (Z . x)) + (x / (A * ((g . x) + ((Z . x) ^2)))) is V28() V29() ext-real Element of REAL
(arctan . (x / A)) + (x / (A * ((g . x) + ((Z . x) ^2)))) is V28() V29() ext-real Element of REAL
1 + ((Z . x) ^2) is V28() V29() ext-real Element of REAL
A * (1 + ((Z . x) ^2)) is V28() V29() ext-real Element of REAL
x / (A * (1 + ((Z . x) ^2))) is V28() V29() ext-real Element of REAL
K99((A * (1 + ((Z . x) ^2)))) is V28() V29() ext-real set
K97(x,K99((A * (1 + ((Z . x) ^2))))) is V28() V29() ext-real set
(arctan . (x / A)) + (x / (A * (1 + ((Z . x) ^2)))) is V28() V29() ext-real Element of REAL
((id f2) (#) (arctan * Z)) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id f2) (#) (arctan * Z)) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((id f2) (#) (arctan * Z)) `| f2) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
arctan . (x / A) is V28() V29() ext-real Element of REAL
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is V28() V29() ext-real 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() V29() ext-real set
K97(x,K99((A * (1 + ((x / A) ^2))))) is V28() V29() ext-real set
(arctan . (x / A)) + (x / (A * (1 + ((x / A) ^2)))) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
f | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,f1) is V28() V29() ext-real Element of REAL
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccot * Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z ^2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z (#) Z is Relation-like REAL -defined V6() V34() V35() V36() set
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g + (Z ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) (g + (Z ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is V55() V56() V57() open Element of K19(REAL)
id f2 is Relation-like REAL -defined f2 -defined REAL -valued f2 -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f2) / (A (#) (g + (Z ^2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arccot * Z) - ((id f2) / (A (#) (g + (Z ^2)))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((id f2) / (A (#) (g + (Z ^2)))) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) ((id f2) / (A (#) (g + (Z ^2)))) is Relation-like REAL -defined V6() V34() V35() V36() set
(arccot * Z) + (- ((id f2) / (A (#) (g + (Z ^2))))) is Relation-like REAL -defined V6() V34() V35() V36() set
(id f2) (#) (arccot * Z) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id f2) (#) (arccot * Z)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
((id f2) (#) (arccot * Z)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((id f2) (#) (arccot * Z)) . (upper_bound f1)) - (((id f2) (#) (arccot * Z)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((id f2) (#) (arccot * Z)) . (lower_bound f1))) is V28() V29() ext-real set
K96((((id f2) (#) (arccot * Z)) . (upper_bound f1)),K98((((id f2) (#) (arccot * Z)) . (lower_bound f1)))) is V28() V29() ext-real set
dom (arccot * Z) is V55() V56() V57() Element of K19(REAL)
dom ((id f2) / (A (#) (g + (Z ^2)))) is V55() V56() V57() Element of K19(REAL)
(dom (arccot * Z)) /\ (dom ((id f2) / (A (#) (g + (Z ^2))))) is V55() V56() V57() Element of K19(REAL)
dom (id f2) is V55() V56() V57() Element of K19(f2)
K19(f2) is set
(dom (id f2)) /\ (dom (arccot * Z)) is V55() V56() V57() Element of K19(REAL)
dom ((id f2) (#) (arccot * Z)) is V55() V56() V57() Element of K19(REAL)
dom (A (#) (g + (Z ^2))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(A (#) (g + (Z ^2))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (A (#) (g + (Z ^2)))) \ ((A (#) (g + (Z ^2))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (id f2)) /\ ((dom (A (#) (g + (Z ^2)))) \ ((A (#) (g + (Z ^2))) " {0})) is V55() V56() V57() Element of K19(REAL)
(A (#) (g + (Z ^2))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((A (#) (g + (Z ^2))) ^) is V55() V56() V57() Element of K19(REAL)
dom (g + (Z ^2)) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
arccot . (x / A) is V28() V29() ext-real Element of REAL
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is V28() V29() ext-real 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() V29() ext-real set
K97(x,K99((A * (1 + ((x / A) ^2))))) is V28() V29() ext-real set
(arccot . (x / A)) - (x / (A * (1 + ((x / A) ^2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * (1 + ((x / A) ^2))))) is V28() V29() ext-real set
K96((arccot . (x / A)),K98((x / (A * (1 + ((x / A) ^2)))))) is V28() V29() ext-real set
((arccot * Z) - ((id f2) / (A (#) (g + (Z ^2))))) . x is V28() V29() ext-real Element of REAL
(arccot * Z) . x is V28() V29() ext-real Element of REAL
((id f2) / (A (#) (g + (Z ^2)))) . x is V28() V29() ext-real Element of REAL
((arccot * Z) . x) - (((id f2) / (A (#) (g + (Z ^2)))) . x) is V28() V29() ext-real Element of REAL
K98((((id f2) / (A (#) (g + (Z ^2)))) . x)) is V28() V29() ext-real set
K96(((arccot * Z) . x),K98((((id f2) / (A (#) (g + (Z ^2)))) . x))) is V28() V29() ext-real set
Z . x is V28() V29() ext-real Element of REAL
arccot . (Z . x) is V28() V29() ext-real Element of REAL
(arccot . (Z . x)) - (((id f2) / (A (#) (g + (Z ^2)))) . x) is V28() V29() ext-real Element of REAL
K96((arccot . (Z . x)),K98((((id f2) / (A (#) (g + (Z ^2)))) . x))) is V28() V29() ext-real set
(id f2) . x is V28() V29() ext-real Element of REAL
(A (#) (g + (Z ^2))) . x is V28() V29() ext-real Element of REAL
((id f2) . x) / ((A (#) (g + (Z ^2))) . x) is V28() V29() ext-real Element of REAL
K99(((A (#) (g + (Z ^2))) . x)) is V28() V29() ext-real set
K97(((id f2) . x),K99(((A (#) (g + (Z ^2))) . x))) is V28() V29() ext-real set
(arccot . (Z . x)) - (((id f2) . x) / ((A (#) (g + (Z ^2))) . x)) is V28() V29() ext-real Element of REAL
K98((((id f2) . x) / ((A (#) (g + (Z ^2))) . x))) is V28() V29() ext-real set
K96((arccot . (Z . x)),K98((((id f2) . x) / ((A (#) (g + (Z ^2))) . x)))) is V28() V29() ext-real set
x / ((A (#) (g + (Z ^2))) . x) is V28() V29() ext-real Element of REAL
K97(x,K99(((A (#) (g + (Z ^2))) . x))) is V28() V29() ext-real set
(arccot . (Z . x)) - (x / ((A (#) (g + (Z ^2))) . x)) is V28() V29() ext-real Element of REAL
K98((x / ((A (#) (g + (Z ^2))) . x))) is V28() V29() ext-real set
K96((arccot . (Z . x)),K98((x / ((A (#) (g + (Z ^2))) . x)))) is V28() V29() ext-real set
(g + (Z ^2)) . x is V28() V29() ext-real Element of REAL
A * ((g + (Z ^2)) . x) is V28() V29() ext-real Element of REAL
x / (A * ((g + (Z ^2)) . x)) is V28() V29() ext-real Element of REAL
K99((A * ((g + (Z ^2)) . x))) is V28() V29() ext-real set
K97(x,K99((A * ((g + (Z ^2)) . x)))) is V28() V29() ext-real set
(arccot . (Z . x)) - (x / (A * ((g + (Z ^2)) . x))) is V28() V29() ext-real Element of REAL
K98((x / (A * ((g + (Z ^2)) . x)))) is V28() V29() ext-real set
K96((arccot . (Z . x)),K98((x / (A * ((g + (Z ^2)) . x))))) is V28() V29() ext-real set
g . x is V28() V29() ext-real Element of REAL
(Z ^2) . x is V28() V29() ext-real Element of REAL
(g . x) + ((Z ^2) . x) is V28() V29() ext-real Element of REAL
A * ((g . x) + ((Z ^2) . x)) is V28() V29() ext-real Element of REAL
x / (A * ((g . x) + ((Z ^2) . x))) is V28() V29() ext-real Element of REAL
K99((A * ((g . x) + ((Z ^2) . x)))) is V28() V29() ext-real set
K97(x,K99((A * ((g . x) + ((Z ^2) . x))))) is V28() V29() ext-real set
(arccot . (Z . x)) - (x / (A * ((g . x) + ((Z ^2) . x)))) is V28() V29() ext-real Element of REAL
K98((x / (A * ((g . x) + ((Z ^2) . x))))) is V28() V29() ext-real set
K96((arccot . (Z . x)),K98((x / (A * ((g . x) + ((Z ^2) . x)))))) is V28() V29() ext-real set
(Z . x) ^2 is V28() V29() ext-real Element of REAL
K97((Z . x),(Z . x)) is V28() V29() ext-real set
(g . x) + ((Z . x) ^2) is V28() V29() ext-real Element of REAL
A * ((g . x) + ((Z . x) ^2)) is V28() V29() ext-real Element of REAL
x / (A * ((g . x) + ((Z . x) ^2))) is V28() V29() ext-real Element of REAL
K99((A * ((g . x) + ((Z . x) ^2)))) is V28() V29() ext-real set
K97(x,K99((A * ((g . x) + ((Z . x) ^2))))) is V28() V29() ext-real set
(arccot . (Z . x)) - (x / (A * ((g . x) + ((Z . x) ^2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * ((g . x) + ((Z . x) ^2))))) is V28() V29() ext-real set
K96((arccot . (Z . x)),K98((x / (A * ((g . x) + ((Z . x) ^2)))))) is V28() V29() ext-real set
(arccot . (x / A)) - (x / (A * ((g . x) + ((Z . x) ^2)))) is V28() V29() ext-real Element of REAL
K96((arccot . (x / A)),K98((x / (A * ((g . x) + ((Z . x) ^2)))))) is V28() V29() ext-real set
1 + ((Z . x) ^2) is V28() V29() ext-real Element of REAL
A * (1 + ((Z . x) ^2)) is V28() V29() ext-real Element of REAL
x / (A * (1 + ((Z . x) ^2))) is V28() V29() ext-real Element of REAL
K99((A * (1 + ((Z . x) ^2)))) is V28() V29() ext-real set
K97(x,K99((A * (1 + ((Z . x) ^2))))) is V28() V29() ext-real set
(arccot . (x / A)) - (x / (A * (1 + ((Z . x) ^2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * (1 + ((Z . x) ^2))))) is V28() V29() ext-real set
K96((arccot . (x / A)),K98((x / (A * (1 + ((Z . x) ^2)))))) is V28() V29() ext-real set
((id f2) (#) (arccot * Z)) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id f2) (#) (arccot * Z)) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((id f2) (#) (arccot * Z)) `| f2) . x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
arccot . (x / A) is V28() V29() ext-real Element of REAL
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is V28() V29() ext-real 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() V29() ext-real set
K97(x,K99((A * (1 + ((x / A) ^2))))) is V28() V29() ext-real set
(arccot . (x / A)) - (x / (A * (1 + ((x / A) ^2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * (1 + ((x / A) ^2))))) is V28() V29() ext-real set
K96((arccot . (x / A)),K98((x / (A * (1 + ((x / A) ^2)))))) is V28() V29() ext-real set
1 / 2 is V28() V29() ext-real non negative V67() Element of RAT
#R (1 / 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A is V21() V22() V23() V27() V28() V29() V30() ext-real non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
#Z A is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * arcsin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * arcsin) is V55() V56() V57() Element of K19(REAL)
A - 1 is V28() V29() V30() ext-real V67() Element of INT
K96(A,K98(1)) is V28() V29() V30() ext-real set
#Z (A - 1) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A - 1)) * arcsin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) ((#Z (A - 1)) * arcsin) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
((#Z A) * arcsin) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
((#Z A) * arcsin) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((#Z A) * arcsin) . (upper_bound f1)) - (((#Z A) * arcsin) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((#Z A) * arcsin) . (lower_bound f1))) is V28() V29() ext-real set
K96((((#Z A) * arcsin) . (upper_bound f1)),K98((((#Z A) * arcsin) . (lower_bound f1)))) is V28() V29() ext-real set
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f - (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
K98(1) (#) (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
f + (- (#Z 2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (f - (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(A (#) ((#Z (A - 1)) * arcsin)) / ((#R (1 / 2)) * (f - (#Z 2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V55() V56() V57() Element of K19(REAL)
integral (Z,f1) is V28() V29() ext-real Element of REAL
g is V55() V56() V57() open Element of K19(REAL)
dom (A (#) ((#Z (A - 1)) * arcsin)) is V55() V56() V57() Element of K19(REAL)
dom ((#R (1 / 2)) * (f - (#Z 2))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((#R (1 / 2)) * (f - (#Z 2))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((#R (1 / 2)) * (f - (#Z 2)))) \ (((#R (1 / 2)) * (f - (#Z 2))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (A (#) ((#Z (A - 1)) * arcsin))) /\ ((dom ((#R (1 / 2)) * (f - (#Z 2)))) \ (((#R (1 / 2)) * (f - (#Z 2))) " {0})) is V55() V56() V57() Element of K19(REAL)
dom ((#Z (A - 1)) * arcsin) is V55() V56() V57() Element of K19(REAL)
((#R (1 / 2)) * (f - (#Z 2))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#R (1 / 2)) * (f - (#Z 2))) ^) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
x is V21() V22() V23() V27() V28() V29() V30() ext-real non negative set
x + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
x is V28() V29() ext-real Element of REAL
(f - (#Z 2)) . x is V28() V29() ext-real Element of REAL
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
dom (f - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
y 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
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((f . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
1 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
x #Z (1 + 1) is V28() V29() ext-real Element of REAL
(f . x) - (x #Z (1 + 1)) is V28() V29() ext-real Element of REAL
K98((x #Z (1 + 1))) is V28() V29() ext-real set
K96((f . x),K98((x #Z (1 + 1)))) is V28() V29() ext-real set
x #Z 1 is V28() V29() ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V28() V29() ext-real Element of REAL
(f . x) - ((x #Z 1) * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98(((x #Z 1) * (x #Z 1))) is V28() V29() ext-real set
K96((f . x),K98(((x #Z 1) * (x #Z 1)))) is V28() V29() ext-real set
x * (x #Z 1) is V28() V29() ext-real Element of REAL
(f . x) - (x * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98((x * (x #Z 1))) is V28() V29() ext-real set
K96((f . x),K98((x * (x #Z 1)))) is V28() V29() ext-real set
x * x is V28() V29() ext-real Element of REAL
(f . x) - (x * x) is V28() V29() ext-real Element of REAL
K98((x * x)) is V28() V29() ext-real set
K96((f . x),K98((x * x))) is V28() V29() ext-real set
1 - (x * x) is V28() V29() ext-real Element of REAL
K96(1,K98((x * x))) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
(f - (#Z 2)) . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f - (#Z 2))) . x is V28() V29() ext-real Element of REAL
Z | g is Relation-like REAL -defined g -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
arcsin . x is V28() V29() ext-real Element of REAL
(arcsin . x) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arcsin . x) #Z (A - 1)) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
(A * ((arcsin . x) #Z (A - 1))) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97((A * ((arcsin . x) #Z (A - 1))),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
dom (f - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
(f - (#Z 2)) . x is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
((A (#) ((#Z (A - 1)) * arcsin)) / ((#R (1 / 2)) * (f - (#Z 2)))) . x is V28() V29() ext-real Element of REAL
(A (#) ((#Z (A - 1)) * arcsin)) . x is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f - (#Z 2))) . x is V28() V29() ext-real Element of REAL
((A (#) ((#Z (A - 1)) * arcsin)) . x) / (((#R (1 / 2)) * (f - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K99((((#R (1 / 2)) * (f - (#Z 2))) . x)) is V28() V29() ext-real set
K97(((A (#) ((#Z (A - 1)) * arcsin)) . x),K99((((#R (1 / 2)) * (f - (#Z 2))) . x))) is V28() V29() ext-real set
((#Z (A - 1)) * arcsin) . x is V28() V29() ext-real Element of REAL
A * (((#Z (A - 1)) * arcsin) . x) is V28() V29() ext-real Element of REAL
(A * (((#Z (A - 1)) * arcsin) . x)) / (((#R (1 / 2)) * (f - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K97((A * (((#Z (A - 1)) * arcsin) . x)),K99((((#R (1 / 2)) * (f - (#Z 2))) . x))) is V28() V29() ext-real set
(#Z (A - 1)) . (arcsin . x) is V28() V29() ext-real Element of REAL
A * ((#Z (A - 1)) . (arcsin . x)) is V28() V29() ext-real Element of REAL
(A * ((#Z (A - 1)) . (arcsin . x))) / (((#R (1 / 2)) * (f - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K97((A * ((#Z (A - 1)) . (arcsin . x))),K99((((#R (1 / 2)) * (f - (#Z 2))) . x))) is V28() V29() ext-real set
(A * ((arcsin . x) #Z (A - 1))) / (((#R (1 / 2)) * (f - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K97((A * ((arcsin . x) #Z (A - 1))),K99((((#R (1 / 2)) * (f - (#Z 2))) . x))) is V28() V29() ext-real set
(#R (1 / 2)) . ((f - (#Z 2)) . x) is V28() V29() ext-real Element of REAL
(A * ((arcsin . x) #Z (A - 1))) / ((#R (1 / 2)) . ((f - (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K99(((#R (1 / 2)) . ((f - (#Z 2)) . x))) is V28() V29() ext-real set
K97((A * ((arcsin . x) #Z (A - 1))),K99(((#R (1 / 2)) . ((f - (#Z 2)) . x)))) is V28() V29() ext-real set
((f - (#Z 2)) . x) #R (1 / 2) is V28() V29() ext-real set
(A * ((arcsin . x) #Z (A - 1))) / (((f - (#Z 2)) . x) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f - (#Z 2)) . x) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arcsin . x) #Z (A - 1))),K99((((f - (#Z 2)) . x) #R (1 / 2)))) is V28() V29() ext-real 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
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((f . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
((f . x) - ((#Z 2) . x)) #R (1 / 2) is V28() V29() ext-real set
(A * ((arcsin . x) #Z (A - 1))) / (((f . x) - ((#Z 2) . x)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f . x) - ((#Z 2) . x)) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arcsin . x) #Z (A - 1))),K99((((f . x) - ((#Z 2) . x)) #R (1 / 2)))) is V28() V29() ext-real set
x #Z 2 is V28() V29() ext-real Element of REAL
(f . x) - (x #Z 2) is V28() V29() ext-real Element of REAL
K98((x #Z 2)) is V28() V29() ext-real set
K96((f . x),K98((x #Z 2))) is V28() V29() ext-real set
((f . x) - (x #Z 2)) #R (1 / 2) is V28() V29() ext-real set
(A * ((arcsin . x) #Z (A - 1))) / (((f . x) - (x #Z 2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f . x) - (x #Z 2)) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arcsin . x) #Z (A - 1))),K99((((f . x) - (x #Z 2)) #R (1 / 2)))) is V28() V29() ext-real set
(f . x) - (x ^2) is V28() V29() ext-real Element of REAL
K96((f . x),K98((x ^2))) is V28() V29() ext-real set
((f . x) - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
(A * ((arcsin . x) #Z (A - 1))) / (((f . x) - (x ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f . x) - (x ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arcsin . x) #Z (A - 1))),K99((((f . x) - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(1 - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
(A * ((arcsin . x) #Z (A - 1))) / ((1 - (x ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99(((1 - (x ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arcsin . x) #Z (A - 1))),K99(((1 - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
((#Z A) * arcsin) `| g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#Z A) * arcsin) `| g) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((#Z A) * arcsin) `| g) . x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
arcsin . x is V28() V29() ext-real Element of REAL
(arcsin . x) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arcsin . x) #Z (A - 1)) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
(A * ((arcsin . x) #Z (A - 1))) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97((A * ((arcsin . x) #Z (A - 1))),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
A is V21() V22() V23() V27() V28() V29() V30() ext-real non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
#Z A is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * arccos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * arccos) is V55() V56() V57() Element of K19(REAL)
A - 1 is V28() V29() V30() ext-real V67() Element of INT
K96(A,K98(1)) is V28() V29() V30() ext-real set
#Z (A - 1) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A - 1)) * arccos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) ((#Z (A - 1)) * arccos) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((#Z A) * arccos) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((#Z A) * arccos) is Relation-like REAL -defined V6() V34() V35() V36() set
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
(- ((#Z A) * arccos)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
(- ((#Z A) * arccos)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
((- ((#Z A) * arccos)) . (upper_bound f1)) - ((- ((#Z A) * arccos)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98(((- ((#Z A) * arccos)) . (lower_bound f1))) is V28() V29() ext-real set
K96(((- ((#Z A) * arccos)) . (upper_bound f1)),K98(((- ((#Z A) * arccos)) . (lower_bound f1)))) is V28() V29() ext-real set
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f - (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
K98(1) (#) (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
f + (- (#Z 2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (f - (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(A (#) ((#Z (A - 1)) * arccos)) / ((#R (1 / 2)) * (f - (#Z 2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V55() V56() V57() Element of K19(REAL)
integral (Z,f1) is V28() V29() ext-real Element of REAL
g is V55() V56() V57() open Element of K19(REAL)
dom (A (#) ((#Z (A - 1)) * arccos)) is V55() V56() V57() Element of K19(REAL)
dom ((#R (1 / 2)) * (f - (#Z 2))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((#R (1 / 2)) * (f - (#Z 2))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((#R (1 / 2)) * (f - (#Z 2)))) \ (((#R (1 / 2)) * (f - (#Z 2))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (A (#) ((#Z (A - 1)) * arccos))) /\ ((dom ((#R (1 / 2)) * (f - (#Z 2)))) \ (((#R (1 / 2)) * (f - (#Z 2))) " {0})) is V55() V56() V57() Element of K19(REAL)
dom ((#Z (A - 1)) * arccos) is V55() V56() V57() Element of K19(REAL)
((#R (1 / 2)) * (f - (#Z 2))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#R (1 / 2)) * (f - (#Z 2))) ^) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
x is V21() V22() V23() V27() V28() V29() V30() ext-real non negative set
x + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
x is V28() V29() ext-real Element of REAL
(f - (#Z 2)) . x is V28() V29() ext-real Element of REAL
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
dom (f - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
y 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
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((f . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
1 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
x #Z (1 + 1) is V28() V29() ext-real Element of REAL
(f . x) - (x #Z (1 + 1)) is V28() V29() ext-real Element of REAL
K98((x #Z (1 + 1))) is V28() V29() ext-real set
K96((f . x),K98((x #Z (1 + 1)))) is V28() V29() ext-real set
x #Z 1 is V28() V29() ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V28() V29() ext-real Element of REAL
(f . x) - ((x #Z 1) * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98(((x #Z 1) * (x #Z 1))) is V28() V29() ext-real set
K96((f . x),K98(((x #Z 1) * (x #Z 1)))) is V28() V29() ext-real set
x * (x #Z 1) is V28() V29() ext-real Element of REAL
(f . x) - (x * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98((x * (x #Z 1))) is V28() V29() ext-real set
K96((f . x),K98((x * (x #Z 1)))) is V28() V29() ext-real set
x * x is V28() V29() ext-real Element of REAL
(f . x) - (x * x) is V28() V29() ext-real Element of REAL
K98((x * x)) is V28() V29() ext-real set
K96((f . x),K98((x * x))) is V28() V29() ext-real set
1 - (x * x) is V28() V29() ext-real Element of REAL
K96(1,K98((x * x))) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
(f - (#Z 2)) . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f - (#Z 2))) . x is V28() V29() ext-real Element of REAL
Z | g is Relation-like REAL -defined g -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- ((#Z A) * arccos)) is V55() V56() V57() Element of K19(REAL)
(- 1) (#) ((#Z A) * arccos) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
(arccos . x) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccos . x) #Z (A - 1)) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
(A * ((arccos . x) #Z (A - 1))) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
dom (f - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
(f - (#Z 2)) . x is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
((A (#) ((#Z (A - 1)) * arccos)) / ((#R (1 / 2)) * (f - (#Z 2)))) . x is V28() V29() ext-real Element of REAL
(A (#) ((#Z (A - 1)) * arccos)) . x is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f - (#Z 2))) . x is V28() V29() ext-real Element of REAL
((A (#) ((#Z (A - 1)) * arccos)) . x) / (((#R (1 / 2)) * (f - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K99((((#R (1 / 2)) * (f - (#Z 2))) . x)) is V28() V29() ext-real set
K97(((A (#) ((#Z (A - 1)) * arccos)) . x),K99((((#R (1 / 2)) * (f - (#Z 2))) . x))) is V28() V29() ext-real set
((#Z (A - 1)) * arccos) . x is V28() V29() ext-real Element of REAL
A * (((#Z (A - 1)) * arccos) . x) is V28() V29() ext-real Element of REAL
(A * (((#Z (A - 1)) * arccos) . x)) / (((#R (1 / 2)) * (f - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K97((A * (((#Z (A - 1)) * arccos) . x)),K99((((#R (1 / 2)) * (f - (#Z 2))) . x))) is V28() V29() ext-real set
(#Z (A - 1)) . (arccos . x) is V28() V29() ext-real Element of REAL
A * ((#Z (A - 1)) . (arccos . x)) is V28() V29() ext-real Element of REAL
(A * ((#Z (A - 1)) . (arccos . x))) / (((#R (1 / 2)) * (f - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K97((A * ((#Z (A - 1)) . (arccos . x))),K99((((#R (1 / 2)) * (f - (#Z 2))) . x))) is V28() V29() ext-real set
(A * ((arccos . x) #Z (A - 1))) / (((#R (1 / 2)) * (f - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K97((A * ((arccos . x) #Z (A - 1))),K99((((#R (1 / 2)) * (f - (#Z 2))) . x))) is V28() V29() ext-real set
(#R (1 / 2)) . ((f - (#Z 2)) . x) is V28() V29() ext-real Element of REAL
(A * ((arccos . x) #Z (A - 1))) / ((#R (1 / 2)) . ((f - (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K99(((#R (1 / 2)) . ((f - (#Z 2)) . x))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99(((#R (1 / 2)) . ((f - (#Z 2)) . x)))) is V28() V29() ext-real set
((f - (#Z 2)) . x) #R (1 / 2) is V28() V29() ext-real set
(A * ((arccos . x) #Z (A - 1))) / (((f - (#Z 2)) . x) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f - (#Z 2)) . x) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99((((f - (#Z 2)) . x) #R (1 / 2)))) is V28() V29() ext-real 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
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((f . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
((f . x) - ((#Z 2) . x)) #R (1 / 2) is V28() V29() ext-real set
(A * ((arccos . x) #Z (A - 1))) / (((f . x) - ((#Z 2) . x)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f . x) - ((#Z 2) . x)) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99((((f . x) - ((#Z 2) . x)) #R (1 / 2)))) is V28() V29() ext-real set
x #Z 2 is V28() V29() ext-real Element of REAL
(f . x) - (x #Z 2) is V28() V29() ext-real Element of REAL
K98((x #Z 2)) is V28() V29() ext-real set
K96((f . x),K98((x #Z 2))) is V28() V29() ext-real set
((f . x) - (x #Z 2)) #R (1 / 2) is V28() V29() ext-real set
(A * ((arccos . x) #Z (A - 1))) / (((f . x) - (x #Z 2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f . x) - (x #Z 2)) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99((((f . x) - (x #Z 2)) #R (1 / 2)))) is V28() V29() ext-real set
(f . x) - (x ^2) is V28() V29() ext-real Element of REAL
K96((f . x),K98((x ^2))) is V28() V29() ext-real set
((f . x) - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
(A * ((arccos . x) #Z (A - 1))) / (((f . x) - (x ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f . x) - (x ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99((((f . x) - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(1 - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
(A * ((arccos . x) #Z (A - 1))) / ((1 - (x ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99(((1 - (x ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99(((1 - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(- ((#Z A) * arccos)) `| g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
((- ((#Z A) * arccos)) `| g) . x is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
(arccos . x) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccos . x) #Z (A - 1)) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
(A * ((arccos . x) #Z (A - 1))) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
diff ((- ((#Z A) * arccos)),x) is V28() V29() ext-real Element of REAL
diff (((#Z A) * arccos),x) is V28() V29() ext-real Element of REAL
(- 1) * (diff (((#Z A) * arccos),x)) is V28() V29() ext-real Element of REAL
diff (arccos,x) is V28() V29() ext-real Element of REAL
(A * ((arccos . x) #Z (A - 1))) * (diff (arccos,x)) is V28() V29() ext-real Element of REAL
(- 1) * ((A * ((arccos . x) #Z (A - 1))) * (diff (arccos,x))) is V28() V29() ext-real Element of REAL
1 / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K97(1,K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
- (1 / (sqrt (1 - (x ^2)))) is V28() V29() ext-real Element of REAL
(A * ((arccos . x) #Z (A - 1))) * (- (1 / (sqrt (1 - (x ^2))))) is V28() V29() ext-real Element of REAL
(- 1) * ((A * ((arccos . x) #Z (A - 1))) * (- (1 / (sqrt (1 - (x ^2)))))) is V28() V29() ext-real Element of REAL
dom ((- ((#Z A) * arccos)) `| g) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- ((#Z A) * arccos)) `| g) . x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
(arccos . x) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((arccos . x) #Z (A - 1)) is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
(A * ((arccos . x) #Z (A - 1))) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97((A * ((arccos . x) #Z (A - 1))),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 - (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
K98(1) (#) (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
f1 + (- (#Z 2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (f1 - (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
id Z is Relation-like REAL -defined Z -defined REAL -valued Z -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) (#) arcsin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) (#) arcsin) . (upper_bound A) is V28() V29() ext-real Element of REAL
((id Z) (#) arcsin) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((id Z) (#) arcsin) . (upper_bound A)) - (((id Z) (#) arcsin) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((id Z) (#) arcsin) . (lower_bound A))) is V28() V29() ext-real set
K96((((id Z) (#) arcsin) . (upper_bound A)),K98((((id Z) (#) arcsin) . (lower_bound A)))) is V28() V29() ext-real set
dom arcsin is V55() V56() V57() Element of K19(REAL)
dom ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) is V55() V56() V57() Element of K19(REAL)
(dom arcsin) /\ (dom ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) is V55() V56() V57() Element of K19(REAL)
dom (id Z) is V55() V56() V57() Element of K19(Z)
K19(Z) is set
(dom (id Z)) /\ (dom arcsin) is V55() V56() V57() Element of K19(REAL)
dom ((id Z) (#) arcsin) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
(id Z) . g is V28() V29() ext-real Element of REAL
1 * g is V28() V29() ext-real Element of REAL
(1 * g) + 0 is V28() V29() ext-real Element of REAL
dom ((#R (1 / 2)) * (f1 - (#Z 2))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((#R (1 / 2)) * (f1 - (#Z 2))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (id Z)) /\ ((dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0})) is V55() V56() V57() Element of K19(REAL)
((#R (1 / 2)) * (f1 - (#Z 2))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
(f1 - (#Z 2)) . f2 is V28() V29() ext-real Element of REAL
1 + f2 is V28() V29() ext-real Element of REAL
1 - f2 is V28() V29() ext-real Element of REAL
K98(f2) is V28() V29() ext-real set
K96(1,K98(f2)) is V28() V29() ext-real set
(1 + f2) * (1 - f2) is V28() V29() ext-real Element of REAL
dom (f1 - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f1 . f2 is V28() V29() ext-real Element of REAL
(#Z 2) . f2 is V28() V29() ext-real Element of REAL
(f1 . f2) - ((#Z 2) . f2) is V28() V29() ext-real Element of REAL
K98(((#Z 2) . f2)) is V28() V29() ext-real set
K96((f1 . f2),K98(((#Z 2) . f2))) is V28() V29() ext-real set
1 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
f2 #Z (1 + 1) is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 #Z (1 + 1)) is V28() V29() ext-real Element of REAL
K98((f2 #Z (1 + 1))) is V28() V29() ext-real set
K96((f1 . f2),K98((f2 #Z (1 + 1)))) is V28() V29() ext-real set
f2 #Z 1 is V28() V29() ext-real Element of REAL
(f2 #Z 1) * (f2 #Z 1) is V28() V29() ext-real Element of REAL
(f1 . f2) - ((f2 #Z 1) * (f2 #Z 1)) is V28() V29() ext-real Element of REAL
K98(((f2 #Z 1) * (f2 #Z 1))) is V28() V29() ext-real set
K96((f1 . f2),K98(((f2 #Z 1) * (f2 #Z 1)))) is V28() V29() ext-real set
f2 * (f2 #Z 1) is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 * (f2 #Z 1)) is V28() V29() ext-real Element of REAL
K98((f2 * (f2 #Z 1))) is V28() V29() ext-real set
K96((f1 . f2),K98((f2 * (f2 #Z 1)))) is V28() V29() ext-real set
f2 * f2 is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 * f2) is V28() V29() ext-real Element of REAL
K98((f2 * f2)) is V28() V29() ext-real set
K96((f1 . f2),K98((f2 * f2))) is V28() V29() ext-real set
1 - (f2 * f2) is V28() V29() ext-real Element of REAL
K96(1,K98((f2 * f2))) is V28() V29() ext-real set
f2 is V28() V29() ext-real Element of REAL
f1 . f2 is V28() V29() ext-real Element of REAL
(f1 - (#Z 2)) . f2 is V28() V29() ext-real Element of REAL
f2 is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f1 - (#Z 2))) . f2 is V28() V29() ext-real Element of REAL
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is V28() V29() ext-real Element of REAL
f . f2 is V28() V29() ext-real Element of REAL
arcsin . f2 is V28() V29() ext-real Element of REAL
f2 ^2 is V28() V29() ext-real Element of REAL
K97(f2,f2) is V28() V29() ext-real set
1 - (f2 ^2) is V28() V29() ext-real Element of REAL
K98((f2 ^2)) is V28() V29() ext-real set
K96(1,K98((f2 ^2))) is V28() V29() ext-real set
sqrt (1 - (f2 ^2)) is V28() V29() ext-real Element of REAL
f2 / (sqrt (1 - (f2 ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (f2 ^2)))) is V28() V29() ext-real set
K97(f2,K99((sqrt (1 - (f2 ^2))))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / (sqrt (1 - (f2 ^2)))) is V28() V29() ext-real Element of REAL
dom (f1 - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
(f1 - (#Z 2)) . f2 is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
1 + f2 is V28() V29() ext-real Element of REAL
1 - f2 is V28() V29() ext-real Element of REAL
K98(f2) is V28() V29() ext-real set
K96(1,K98(f2)) is V28() V29() ext-real set
(1 + f2) * (1 - f2) is V28() V29() ext-real Element of REAL
(arcsin + ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) . f2 is V28() V29() ext-real Element of REAL
((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . f2 is V28() V29() ext-real Element of REAL
(arcsin . f2) + (((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . f2) is V28() V29() ext-real Element of REAL
(id Z) . f2 is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f1 - (#Z 2))) . f2 is V28() V29() ext-real Element of REAL
((id Z) . f2) / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2) is V28() V29() ext-real Element of REAL
K99((((#R (1 / 2)) * (f1 - (#Z 2))) . f2)) is V28() V29() ext-real set
K97(((id Z) . f2),K99((((#R (1 / 2)) * (f1 - (#Z 2))) . f2))) is V28() V29() ext-real set
(arcsin . f2) + (((id Z) . f2) / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2)) is V28() V29() ext-real Element of REAL
f2 / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2) is V28() V29() ext-real Element of REAL
K97(f2,K99((((#R (1 / 2)) * (f1 - (#Z 2))) . f2))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2)) is V28() V29() ext-real Element of REAL
(#R (1 / 2)) . ((f1 - (#Z 2)) . f2) is V28() V29() ext-real Element of REAL
f2 / ((#R (1 / 2)) . ((f1 - (#Z 2)) . f2)) is V28() V29() ext-real Element of REAL
K99(((#R (1 / 2)) . ((f1 - (#Z 2)) . f2))) is V28() V29() ext-real set
K97(f2,K99(((#R (1 / 2)) . ((f1 - (#Z 2)) . f2)))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / ((#R (1 / 2)) . ((f1 - (#Z 2)) . f2))) is V28() V29() ext-real Element of REAL
((f1 - (#Z 2)) . f2) #R (1 / 2) is V28() V29() ext-real set
f2 / (((f1 - (#Z 2)) . f2) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f1 - (#Z 2)) . f2) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99((((f1 - (#Z 2)) . f2) #R (1 / 2)))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / (((f1 - (#Z 2)) . f2) #R (1 / 2))) is V28() V29() ext-real Element of REAL
f1 . f2 is V28() V29() ext-real Element of REAL
(#Z 2) . f2 is V28() V29() ext-real Element of REAL
(f1 . f2) - ((#Z 2) . f2) is V28() V29() ext-real Element of REAL
K98(((#Z 2) . f2)) is V28() V29() ext-real set
K96((f1 . f2),K98(((#Z 2) . f2))) is V28() V29() ext-real set
((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2) is V28() V29() ext-real set
f2 / (((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99((((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2)))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / (((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
f2 #Z 2 is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 #Z 2) is V28() V29() ext-real Element of REAL
K98((f2 #Z 2)) is V28() V29() ext-real set
K96((f1 . f2),K98((f2 #Z 2))) is V28() V29() ext-real set
((f1 . f2) - (f2 #Z 2)) #R (1 / 2) is V28() V29() ext-real set
f2 / (((f1 . f2) - (f2 #Z 2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f1 . f2) - (f2 #Z 2)) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99((((f1 . f2) - (f2 #Z 2)) #R (1 / 2)))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / (((f1 . f2) - (f2 #Z 2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 ^2) is V28() V29() ext-real Element of REAL
K96((f1 . f2),K98((f2 ^2))) is V28() V29() ext-real set
((f1 . f2) - (f2 ^2)) #R (1 / 2) is V28() V29() ext-real set
f2 / (((f1 . f2) - (f2 ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f1 . f2) - (f2 ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99((((f1 . f2) - (f2 ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / (((f1 . f2) - (f2 ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
(1 - (f2 ^2)) #R (1 / 2) is V28() V29() ext-real set
f2 / ((1 - (f2 ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99(((1 - (f2 ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99(((1 - (f2 ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / ((1 - (f2 ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
((id Z) (#) arcsin) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id Z) (#) arcsin) `| Z) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
(((id Z) (#) arcsin) `| Z) . f2 is V28() V29() ext-real Element of REAL
f . f2 is V28() V29() ext-real Element of REAL
arcsin . f2 is V28() V29() ext-real Element of REAL
f2 ^2 is V28() V29() ext-real Element of REAL
K97(f2,f2) is V28() V29() ext-real set
1 - (f2 ^2) is V28() V29() ext-real Element of REAL
K98((f2 ^2)) is V28() V29() ext-real set
K96(1,K98((f2 ^2))) is V28() V29() ext-real set
sqrt (1 - (f2 ^2)) is V28() V29() ext-real Element of REAL
f2 / (sqrt (1 - (f2 ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (f2 ^2)))) is V28() V29() ext-real set
K97(f2,K99((sqrt (1 - (f2 ^2))))) is V28() V29() ext-real set
(arcsin . f2) + (f2 / (sqrt (1 - (f2 ^2)))) is V28() V29() ext-real Element of REAL
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 - (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
K98(1) (#) (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
f1 + (- (#Z 2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (f1 - (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
id Z is Relation-like REAL -defined Z -defined REAL -valued Z -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) is Relation-like REAL -defined V6() V34() V35() V36() set
arccos + (- ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) is Relation-like REAL -defined V6() V34() V35() V36() set
(id Z) (#) arccos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id Z) (#) arccos) . (upper_bound A) is V28() V29() ext-real Element of REAL
((id Z) (#) arccos) . (lower_bound A) is V28() V29() ext-real Element of REAL
(((id Z) (#) arccos) . (upper_bound A)) - (((id Z) (#) arccos) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98((((id Z) (#) arccos) . (lower_bound A))) is V28() V29() ext-real set
K96((((id Z) (#) arccos) . (upper_bound A)),K98((((id Z) (#) arccos) . (lower_bound A)))) is V28() V29() ext-real set
dom arccos is V55() V56() V57() Element of K19(REAL)
dom ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) is V55() V56() V57() Element of K19(REAL)
(dom arccos) /\ (dom ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) is V55() V56() V57() Element of K19(REAL)
dom (id Z) is V55() V56() V57() Element of K19(Z)
K19(Z) is set
(dom (id Z)) /\ (dom arccos) is V55() V56() V57() Element of K19(REAL)
dom ((id Z) (#) arccos) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
(id Z) . g is V28() V29() ext-real Element of REAL
1 * g is V28() V29() ext-real Element of REAL
(1 * g) + 0 is V28() V29() ext-real Element of REAL
dom ((#R (1 / 2)) * (f1 - (#Z 2))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((#R (1 / 2)) * (f1 - (#Z 2))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (id Z)) /\ ((dom ((#R (1 / 2)) * (f1 - (#Z 2)))) \ (((#R (1 / 2)) * (f1 - (#Z 2))) " {0})) is V55() V56() V57() Element of K19(REAL)
((#R (1 / 2)) * (f1 - (#Z 2))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#R (1 / 2)) * (f1 - (#Z 2))) ^) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
(f1 - (#Z 2)) . f2 is V28() V29() ext-real Element of REAL
1 + f2 is V28() V29() ext-real Element of REAL
1 - f2 is V28() V29() ext-real Element of REAL
K98(f2) is V28() V29() ext-real set
K96(1,K98(f2)) is V28() V29() ext-real set
(1 + f2) * (1 - f2) is V28() V29() ext-real Element of REAL
dom (f1 - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f1 . f2 is V28() V29() ext-real Element of REAL
(#Z 2) . f2 is V28() V29() ext-real Element of REAL
(f1 . f2) - ((#Z 2) . f2) is V28() V29() ext-real Element of REAL
K98(((#Z 2) . f2)) is V28() V29() ext-real set
K96((f1 . f2),K98(((#Z 2) . f2))) is V28() V29() ext-real set
1 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
f2 #Z (1 + 1) is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 #Z (1 + 1)) is V28() V29() ext-real Element of REAL
K98((f2 #Z (1 + 1))) is V28() V29() ext-real set
K96((f1 . f2),K98((f2 #Z (1 + 1)))) is V28() V29() ext-real set
f2 #Z 1 is V28() V29() ext-real Element of REAL
(f2 #Z 1) * (f2 #Z 1) is V28() V29() ext-real Element of REAL
(f1 . f2) - ((f2 #Z 1) * (f2 #Z 1)) is V28() V29() ext-real Element of REAL
K98(((f2 #Z 1) * (f2 #Z 1))) is V28() V29() ext-real set
K96((f1 . f2),K98(((f2 #Z 1) * (f2 #Z 1)))) is V28() V29() ext-real set
f2 * (f2 #Z 1) is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 * (f2 #Z 1)) is V28() V29() ext-real Element of REAL
K98((f2 * (f2 #Z 1))) is V28() V29() ext-real set
K96((f1 . f2),K98((f2 * (f2 #Z 1)))) is V28() V29() ext-real set
f2 * f2 is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 * f2) is V28() V29() ext-real Element of REAL
K98((f2 * f2)) is V28() V29() ext-real set
K96((f1 . f2),K98((f2 * f2))) is V28() V29() ext-real set
1 - (f2 * f2) is V28() V29() ext-real Element of REAL
K96(1,K98((f2 * f2))) is V28() V29() ext-real set
f2 is V28() V29() ext-real Element of REAL
f1 . f2 is V28() V29() ext-real Element of REAL
(f1 - (#Z 2)) . f2 is V28() V29() ext-real Element of REAL
f2 is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f1 - (#Z 2))) . f2 is V28() V29() ext-real Element of REAL
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is V28() V29() ext-real Element of REAL
f . f2 is V28() V29() ext-real Element of REAL
arccos . f2 is V28() V29() ext-real Element of REAL
f2 ^2 is V28() V29() ext-real Element of REAL
K97(f2,f2) is V28() V29() ext-real set
1 - (f2 ^2) is V28() V29() ext-real Element of REAL
K98((f2 ^2)) is V28() V29() ext-real set
K96(1,K98((f2 ^2))) is V28() V29() ext-real set
sqrt (1 - (f2 ^2)) is V28() V29() ext-real Element of REAL
f2 / (sqrt (1 - (f2 ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (f2 ^2)))) is V28() V29() ext-real set
K97(f2,K99((sqrt (1 - (f2 ^2))))) is V28() V29() ext-real set
(arccos . f2) - (f2 / (sqrt (1 - (f2 ^2)))) is V28() V29() ext-real Element of REAL
K98((f2 / (sqrt (1 - (f2 ^2))))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / (sqrt (1 - (f2 ^2)))))) is V28() V29() ext-real set
dom (f1 - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
(f1 - (#Z 2)) . f2 is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
1 + f2 is V28() V29() ext-real Element of REAL
1 - f2 is V28() V29() ext-real Element of REAL
K98(f2) is V28() V29() ext-real set
K96(1,K98(f2)) is V28() V29() ext-real set
(1 + f2) * (1 - f2) is V28() V29() ext-real Element of REAL
(arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2))))) . f2 is V28() V29() ext-real Element of REAL
((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . f2 is V28() V29() ext-real Element of REAL
(arccos . f2) - (((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . f2) is V28() V29() ext-real Element of REAL
K98((((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . f2)) is V28() V29() ext-real set
K96((arccos . f2),K98((((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) . f2))) is V28() V29() ext-real set
(id Z) . f2 is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f1 - (#Z 2))) . f2 is V28() V29() ext-real Element of REAL
((id Z) . f2) / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2) is V28() V29() ext-real Element of REAL
K99((((#R (1 / 2)) * (f1 - (#Z 2))) . f2)) is V28() V29() ext-real set
K97(((id Z) . f2),K99((((#R (1 / 2)) * (f1 - (#Z 2))) . f2))) is V28() V29() ext-real set
(arccos . f2) - (((id Z) . f2) / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2)) is V28() V29() ext-real Element of REAL
K98((((id Z) . f2) / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2))) is V28() V29() ext-real set
K96((arccos . f2),K98((((id Z) . f2) / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2)))) is V28() V29() ext-real set
f2 / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2) is V28() V29() ext-real Element of REAL
K97(f2,K99((((#R (1 / 2)) * (f1 - (#Z 2))) . f2))) is V28() V29() ext-real set
(arccos . f2) - (f2 / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2)) is V28() V29() ext-real Element of REAL
K98((f2 / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / (((#R (1 / 2)) * (f1 - (#Z 2))) . f2)))) is V28() V29() ext-real set
(#R (1 / 2)) . ((f1 - (#Z 2)) . f2) is V28() V29() ext-real Element of REAL
f2 / ((#R (1 / 2)) . ((f1 - (#Z 2)) . f2)) is V28() V29() ext-real Element of REAL
K99(((#R (1 / 2)) . ((f1 - (#Z 2)) . f2))) is V28() V29() ext-real set
K97(f2,K99(((#R (1 / 2)) . ((f1 - (#Z 2)) . f2)))) is V28() V29() ext-real set
(arccos . f2) - (f2 / ((#R (1 / 2)) . ((f1 - (#Z 2)) . f2))) is V28() V29() ext-real Element of REAL
K98((f2 / ((#R (1 / 2)) . ((f1 - (#Z 2)) . f2)))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / ((#R (1 / 2)) . ((f1 - (#Z 2)) . f2))))) is V28() V29() ext-real set
((f1 - (#Z 2)) . f2) #R (1 / 2) is V28() V29() ext-real set
f2 / (((f1 - (#Z 2)) . f2) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f1 - (#Z 2)) . f2) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99((((f1 - (#Z 2)) . f2) #R (1 / 2)))) is V28() V29() ext-real set
(arccos . f2) - (f2 / (((f1 - (#Z 2)) . f2) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((f2 / (((f1 - (#Z 2)) . f2) #R (1 / 2)))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / (((f1 - (#Z 2)) . f2) #R (1 / 2))))) is V28() V29() ext-real set
f1 . f2 is V28() V29() ext-real Element of REAL
(#Z 2) . f2 is V28() V29() ext-real Element of REAL
(f1 . f2) - ((#Z 2) . f2) is V28() V29() ext-real Element of REAL
K98(((#Z 2) . f2)) is V28() V29() ext-real set
K96((f1 . f2),K98(((#Z 2) . f2))) is V28() V29() ext-real set
((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2) is V28() V29() ext-real set
f2 / (((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99((((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2)))) is V28() V29() ext-real set
(arccos . f2) - (f2 / (((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((f2 / (((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2)))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / (((f1 . f2) - ((#Z 2) . f2)) #R (1 / 2))))) is V28() V29() ext-real set
f2 #Z 2 is V28() V29() ext-real Element of REAL
(f1 . f2) - (f2 #Z 2) is V28() V29() ext-real Element of REAL
K98((f2 #Z 2)) is V28() V29() ext-real set
K96((f1 . f2),K98((f2 #Z 2))) is V28() V29() ext-real set
((f1 . f2) - (f2 #Z 2)) #R (1 / 2) is V28() V29() ext-real set
f2 / (((f1 . f2) - (f2 #Z 2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f1 . f2) - (f2 #Z 2)) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99((((f1 . f2) - (f2 #Z 2)) #R (1 / 2)))) is V28() V29() ext-real set
(arccos . f2) - (f2 / (((f1 . f2) - (f2 #Z 2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((f2 / (((f1 . f2) - (f2 #Z 2)) #R (1 / 2)))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / (((f1 . f2) - (f2 #Z 2)) #R (1 / 2))))) is V28() V29() ext-real set
(f1 . f2) - (f2 ^2) is V28() V29() ext-real Element of REAL
K96((f1 . f2),K98((f2 ^2))) is V28() V29() ext-real set
((f1 . f2) - (f2 ^2)) #R (1 / 2) is V28() V29() ext-real set
f2 / (((f1 . f2) - (f2 ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((f1 . f2) - (f2 ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99((((f1 . f2) - (f2 ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(arccos . f2) - (f2 / (((f1 . f2) - (f2 ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((f2 / (((f1 . f2) - (f2 ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / (((f1 . f2) - (f2 ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(1 - (f2 ^2)) #R (1 / 2) is V28() V29() ext-real set
f2 / ((1 - (f2 ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99(((1 - (f2 ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97(f2,K99(((1 - (f2 ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(arccos . f2) - (f2 / ((1 - (f2 ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((f2 / ((1 - (f2 ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / ((1 - (f2 ^2)) #R (1 / 2))))) is V28() V29() ext-real set
((id Z) (#) arccos) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id Z) (#) arccos) `| Z) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
(((id Z) (#) arccos) `| Z) . f2 is V28() V29() ext-real Element of REAL
f . f2 is V28() V29() ext-real Element of REAL
arccos . f2 is V28() V29() ext-real Element of REAL
f2 ^2 is V28() V29() ext-real Element of REAL
K97(f2,f2) is V28() V29() ext-real set
1 - (f2 ^2) is V28() V29() ext-real Element of REAL
K98((f2 ^2)) is V28() V29() ext-real set
K96(1,K98((f2 ^2))) is V28() V29() ext-real set
sqrt (1 - (f2 ^2)) is V28() V29() ext-real Element of REAL
f2 / (sqrt (1 - (f2 ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (f2 ^2)))) is V28() V29() ext-real set
K97(f2,K99((sqrt (1 - (f2 ^2))))) is V28() V29() ext-real set
(arccos . f2) - (f2 / (sqrt (1 - (f2 ^2)))) is V28() V29() ext-real Element of REAL
K98((f2 / (sqrt (1 - (f2 ^2))))) is V28() V29() ext-real set
K96((arccos . f2),K98((f2 / (sqrt (1 - (f2 ^2)))))) is V28() V29() ext-real set
A is V28() V29() ext-real Element of REAL
A (#) arcsin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
f is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f is V28() V29() ext-real Element of REAL
lower_bound f is V28() V29() ext-real Element of REAL
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z (#) arcsin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(Z (#) arcsin) . (upper_bound f) is V28() V29() ext-real Element of REAL
(Z (#) arcsin) . (lower_bound f) is V28() V29() ext-real Element of REAL
((Z (#) arcsin) . (upper_bound f)) - ((Z (#) arcsin) . (lower_bound f)) is V28() V29() ext-real Element of REAL
K98(((Z (#) arcsin) . (lower_bound f))) is V28() V29() ext-real set
K96(((Z (#) arcsin) . (upper_bound f)),K98(((Z (#) arcsin) . (lower_bound f)))) is V28() V29() ext-real set
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g - (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
K98(1) (#) (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
g + (- (#Z 2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (g - (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z / ((#R (1 / 2)) * (g - (#Z 2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(A (#) arcsin) + (Z / ((#R (1 / 2)) * (g - (#Z 2)))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f2 is V55() V56() V57() Element of K19(REAL)
integral (f2,f) is V28() V29() ext-real Element of REAL
x is V55() V56() V57() open Element of K19(REAL)
dom (A (#) arcsin) is V55() V56() V57() Element of K19(REAL)
dom (Z / ((#R (1 / 2)) * (g - (#Z 2)))) is V55() V56() V57() Element of K19(REAL)
(dom (A (#) arcsin)) /\ (dom (Z / ((#R (1 / 2)) * (g - (#Z 2))))) is V55() V56() V57() Element of K19(REAL)
dom arcsin is V55() V56() V57() Element of K19(REAL)
dom Z is V55() V56() V57() Element of K19(REAL)
dom ((#R (1 / 2)) * (g - (#Z 2))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((#R (1 / 2)) * (g - (#Z 2))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((#R (1 / 2)) * (g - (#Z 2)))) \ (((#R (1 / 2)) * (g - (#Z 2))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom Z) /\ ((dom ((#R (1 / 2)) * (g - (#Z 2)))) \ (((#R (1 / 2)) * (g - (#Z 2))) " {0})) is V55() V56() V57() Element of K19(REAL)
(dom Z) /\ (dom arcsin) is V55() V56() V57() Element of K19(REAL)
dom (Z (#) arcsin) is V55() V56() V57() Element of K19(REAL)
((#R (1 / 2)) * (g - (#Z 2))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#R (1 / 2)) * (g - (#Z 2))) ^) is V55() V56() V57() Element of K19(REAL)
y is V28() V29() ext-real Element of REAL
Z . y is V28() V29() ext-real Element of REAL
A * y is V28() V29() ext-real Element of REAL
(A * y) + f1 is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
(g - (#Z 2)) . x is V28() V29() ext-real Element of REAL
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
dom (g - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
c₁₀ is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(g . x) - ((#Z 2) . x) is V28() V29() ext-real Element of REAL
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((g . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
1 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
x #Z (1 + 1) is V28() V29() ext-real Element of REAL
(g . x) - (x #Z (1 + 1)) is V28() V29() ext-real Element of REAL
K98((x #Z (1 + 1))) is V28() V29() ext-real set
K96((g . x),K98((x #Z (1 + 1)))) is V28() V29() ext-real set
x #Z 1 is V28() V29() ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V28() V29() ext-real Element of REAL
(g . x) - ((x #Z 1) * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98(((x #Z 1) * (x #Z 1))) is V28() V29() ext-real set
K96((g . x),K98(((x #Z 1) * (x #Z 1)))) is V28() V29() ext-real set
x * (x #Z 1) is V28() V29() ext-real Element of REAL
(g . x) - (x * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98((x * (x #Z 1))) is V28() V29() ext-real set
K96((g . x),K98((x * (x #Z 1)))) is V28() V29() ext-real set
x * x is V28() V29() ext-real Element of REAL
(g . x) - (x * x) is V28() V29() ext-real Element of REAL
K98((x * x)) is V28() V29() ext-real set
K96((g . x),K98((x * x))) is V28() V29() ext-real set
1 - (x * x) is V28() V29() ext-real Element of REAL
K96(1,K98((x * x))) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
(g - (#Z 2)) . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (g - (#Z 2))) . x is V28() V29() ext-real Element of REAL
f2 | x is Relation-like REAL -defined x -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
f2 . x is V28() V29() ext-real Element of REAL
arcsin . x is V28() V29() ext-real Element of REAL
A * (arcsin . x) is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
((A * x) + f1) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / (sqrt (1 - (x ^2)))) is V28() V29() ext-real Element of REAL
dom (g - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
(g - (#Z 2)) . x is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
((A (#) arcsin) + (Z / ((#R (1 / 2)) * (g - (#Z 2))))) . x is V28() V29() ext-real Element of REAL
(A (#) arcsin) . x is V28() V29() ext-real Element of REAL
(Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x is V28() V29() ext-real Element of REAL
((A (#) arcsin) . x) + ((Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x) is V28() V29() ext-real Element of REAL
(A * (arcsin . x)) + ((Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x) is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (g - (#Z 2))) . x is V28() V29() ext-real Element of REAL
(Z . x) / (((#R (1 / 2)) * (g - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K99((((#R (1 / 2)) * (g - (#Z 2))) . x)) is V28() V29() ext-real set
K97((Z . x),K99((((#R (1 / 2)) * (g - (#Z 2))) . x))) is V28() V29() ext-real set
(A * (arcsin . x)) + ((Z . x) / (((#R (1 / 2)) * (g - (#Z 2))) . x)) is V28() V29() ext-real Element of REAL
((A * x) + f1) / (((#R (1 / 2)) * (g - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K97(((A * x) + f1),K99((((#R (1 / 2)) * (g - (#Z 2))) . x))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / (((#R (1 / 2)) * (g - (#Z 2))) . x)) is V28() V29() ext-real Element of REAL
(#R (1 / 2)) . ((g - (#Z 2)) . x) is V28() V29() ext-real Element of REAL
((A * x) + f1) / ((#R (1 / 2)) . ((g - (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K99(((#R (1 / 2)) . ((g - (#Z 2)) . x))) is V28() V29() ext-real set
K97(((A * x) + f1),K99(((#R (1 / 2)) . ((g - (#Z 2)) . x)))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / ((#R (1 / 2)) . ((g - (#Z 2)) . x))) is V28() V29() ext-real Element of REAL
((g - (#Z 2)) . x) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / (((g - (#Z 2)) . x) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((g - (#Z 2)) . x) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((((g - (#Z 2)) . x) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / (((g - (#Z 2)) . x) #R (1 / 2))) is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(g . x) - ((#Z 2) . x) is V28() V29() ext-real Element of REAL
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((g . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
((g . x) - ((#Z 2) . x)) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / (((g . x) - ((#Z 2) . x)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((g . x) - ((#Z 2) . x)) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((((g . x) - ((#Z 2) . x)) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / (((g . x) - ((#Z 2) . x)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
x #Z 2 is V28() V29() ext-real Element of REAL
(g . x) - (x #Z 2) is V28() V29() ext-real Element of REAL
K98((x #Z 2)) is V28() V29() ext-real set
K96((g . x),K98((x #Z 2))) is V28() V29() ext-real set
((g . x) - (x #Z 2)) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / (((g . x) - (x #Z 2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((g . x) - (x #Z 2)) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((((g . x) - (x #Z 2)) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / (((g . x) - (x #Z 2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
(g . x) - (x ^2) is V28() V29() ext-real Element of REAL
K96((g . x),K98((x ^2))) is V28() V29() ext-real set
((g . x) - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / (((g . x) - (x ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((g . x) - (x ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((((g . x) - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / (((g . x) - (x ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
(1 - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / ((1 - (x ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99(((1 - (x ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99(((1 - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / ((1 - (x ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
(Z (#) arcsin) `| x is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((Z (#) arcsin) `| x) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((Z (#) arcsin) `| x) . x is V28() V29() ext-real Element of REAL
f2 . x is V28() V29() ext-real Element of REAL
arcsin . x is V28() V29() ext-real Element of REAL
A * (arcsin . x) is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
((A * x) + f1) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
(A * (arcsin . x)) + (((A * x) + f1) / (sqrt (1 - (x ^2)))) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
A (#) arccos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
f is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f is V28() V29() ext-real Element of REAL
lower_bound f is V28() V29() ext-real Element of REAL
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z (#) arccos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(Z (#) arccos) . (upper_bound f) is V28() V29() ext-real Element of REAL
(Z (#) arccos) . (lower_bound f) is V28() V29() ext-real Element of REAL
((Z (#) arccos) . (upper_bound f)) - ((Z (#) arccos) . (lower_bound f)) is V28() V29() ext-real Element of REAL
K98(((Z (#) arccos) . (lower_bound f))) is V28() V29() ext-real set
K96(((Z (#) arccos) . (upper_bound f)),K98(((Z (#) arccos) . (lower_bound f)))) is V28() V29() ext-real set
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g - (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
K98(1) (#) (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
g + (- (#Z 2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (g - (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z / ((#R (1 / 2)) * (g - (#Z 2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(A (#) arccos) - (Z / ((#R (1 / 2)) * (g - (#Z 2)))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (Z / ((#R (1 / 2)) * (g - (#Z 2)))) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (Z / ((#R (1 / 2)) * (g - (#Z 2)))) is Relation-like REAL -defined V6() V34() V35() V36() set
(A (#) arccos) + (- (Z / ((#R (1 / 2)) * (g - (#Z 2))))) is Relation-like REAL -defined V6() V34() V35() V36() set
f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f2 is V55() V56() V57() Element of K19(REAL)
integral (f2,f) is V28() V29() ext-real Element of REAL
x is V55() V56() V57() open Element of K19(REAL)
dom (A (#) arccos) is V55() V56() V57() Element of K19(REAL)
dom (Z / ((#R (1 / 2)) * (g - (#Z 2)))) is V55() V56() V57() Element of K19(REAL)
(dom (A (#) arccos)) /\ (dom (Z / ((#R (1 / 2)) * (g - (#Z 2))))) is V55() V56() V57() Element of K19(REAL)
dom arccos is V55() V56() V57() Element of K19(REAL)
dom Z is V55() V56() V57() Element of K19(REAL)
dom ((#R (1 / 2)) * (g - (#Z 2))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((#R (1 / 2)) * (g - (#Z 2))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((#R (1 / 2)) * (g - (#Z 2)))) \ (((#R (1 / 2)) * (g - (#Z 2))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom Z) /\ ((dom ((#R (1 / 2)) * (g - (#Z 2)))) \ (((#R (1 / 2)) * (g - (#Z 2))) " {0})) is V55() V56() V57() Element of K19(REAL)
(dom Z) /\ (dom arccos) is V55() V56() V57() Element of K19(REAL)
dom (Z (#) arccos) is V55() V56() V57() Element of K19(REAL)
((#R (1 / 2)) * (g - (#Z 2))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#R (1 / 2)) * (g - (#Z 2))) ^) is V55() V56() V57() Element of K19(REAL)
y is V28() V29() ext-real Element of REAL
Z . y is V28() V29() ext-real Element of REAL
A * y is V28() V29() ext-real Element of REAL
(A * y) + f1 is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
(g - (#Z 2)) . x is V28() V29() ext-real Element of REAL
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
dom (g - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
c₁₀ is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(g . x) - ((#Z 2) . x) is V28() V29() ext-real Element of REAL
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((g . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
1 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
x #Z (1 + 1) is V28() V29() ext-real Element of REAL
(g . x) - (x #Z (1 + 1)) is V28() V29() ext-real Element of REAL
K98((x #Z (1 + 1))) is V28() V29() ext-real set
K96((g . x),K98((x #Z (1 + 1)))) is V28() V29() ext-real set
x #Z 1 is V28() V29() ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V28() V29() ext-real Element of REAL
(g . x) - ((x #Z 1) * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98(((x #Z 1) * (x #Z 1))) is V28() V29() ext-real set
K96((g . x),K98(((x #Z 1) * (x #Z 1)))) is V28() V29() ext-real set
x * (x #Z 1) is V28() V29() ext-real Element of REAL
(g . x) - (x * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98((x * (x #Z 1))) is V28() V29() ext-real set
K96((g . x),K98((x * (x #Z 1)))) is V28() V29() ext-real set
x * x is V28() V29() ext-real Element of REAL
(g . x) - (x * x) is V28() V29() ext-real Element of REAL
K98((x * x)) is V28() V29() ext-real set
K96((g . x),K98((x * x))) is V28() V29() ext-real set
1 - (x * x) is V28() V29() ext-real Element of REAL
K96(1,K98((x * x))) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
(g - (#Z 2)) . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (g - (#Z 2))) . x is V28() V29() ext-real Element of REAL
f2 | x is Relation-like REAL -defined x -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
f2 . x is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
A * (arccos . x) is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
((A * x) + f1) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / (sqrt (1 - (x ^2)))) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / (sqrt (1 - (x ^2))))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / (sqrt (1 - (x ^2)))))) is V28() V29() ext-real set
dom (g - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
(g - (#Z 2)) . x is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
((A (#) arccos) - (Z / ((#R (1 / 2)) * (g - (#Z 2))))) . x is V28() V29() ext-real Element of REAL
(A (#) arccos) . x is V28() V29() ext-real Element of REAL
(Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x is V28() V29() ext-real Element of REAL
((A (#) arccos) . x) - ((Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x) is V28() V29() ext-real Element of REAL
K98(((Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x)) is V28() V29() ext-real set
K96(((A (#) arccos) . x),K98(((Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x))) is V28() V29() ext-real set
(A * (arccos . x)) - ((Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x) is V28() V29() ext-real Element of REAL
K96((A * (arccos . x)),K98(((Z / ((#R (1 / 2)) * (g - (#Z 2)))) . x))) is V28() V29() ext-real set
Z . x is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (g - (#Z 2))) . x is V28() V29() ext-real Element of REAL
(Z . x) / (((#R (1 / 2)) * (g - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K99((((#R (1 / 2)) * (g - (#Z 2))) . x)) is V28() V29() ext-real set
K97((Z . x),K99((((#R (1 / 2)) * (g - (#Z 2))) . x))) is V28() V29() ext-real set
(A * (arccos . x)) - ((Z . x) / (((#R (1 / 2)) * (g - (#Z 2))) . x)) is V28() V29() ext-real Element of REAL
K98(((Z . x) / (((#R (1 / 2)) * (g - (#Z 2))) . x))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98(((Z . x) / (((#R (1 / 2)) * (g - (#Z 2))) . x)))) is V28() V29() ext-real set
((A * x) + f1) / (((#R (1 / 2)) * (g - (#Z 2))) . x) is V28() V29() ext-real Element of REAL
K97(((A * x) + f1),K99((((#R (1 / 2)) * (g - (#Z 2))) . x))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / (((#R (1 / 2)) * (g - (#Z 2))) . x)) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / (((#R (1 / 2)) * (g - (#Z 2))) . x))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / (((#R (1 / 2)) * (g - (#Z 2))) . x)))) is V28() V29() ext-real set
(#R (1 / 2)) . ((g - (#Z 2)) . x) is V28() V29() ext-real Element of REAL
((A * x) + f1) / ((#R (1 / 2)) . ((g - (#Z 2)) . x)) is V28() V29() ext-real Element of REAL
K99(((#R (1 / 2)) . ((g - (#Z 2)) . x))) is V28() V29() ext-real set
K97(((A * x) + f1),K99(((#R (1 / 2)) . ((g - (#Z 2)) . x)))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / ((#R (1 / 2)) . ((g - (#Z 2)) . x))) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / ((#R (1 / 2)) . ((g - (#Z 2)) . x)))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / ((#R (1 / 2)) . ((g - (#Z 2)) . x))))) is V28() V29() ext-real set
((g - (#Z 2)) . x) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / (((g - (#Z 2)) . x) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((g - (#Z 2)) . x) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((((g - (#Z 2)) . x) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / (((g - (#Z 2)) . x) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / (((g - (#Z 2)) . x) #R (1 / 2)))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / (((g - (#Z 2)) . x) #R (1 / 2))))) is V28() V29() ext-real set
g . x is V28() V29() ext-real Element of REAL
(#Z 2) . x is V28() V29() ext-real Element of REAL
(g . x) - ((#Z 2) . x) is V28() V29() ext-real Element of REAL
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((g . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
((g . x) - ((#Z 2) . x)) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / (((g . x) - ((#Z 2) . x)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((g . x) - ((#Z 2) . x)) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((((g . x) - ((#Z 2) . x)) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / (((g . x) - ((#Z 2) . x)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / (((g . x) - ((#Z 2) . x)) #R (1 / 2)))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / (((g . x) - ((#Z 2) . x)) #R (1 / 2))))) is V28() V29() ext-real set
x #Z 2 is V28() V29() ext-real Element of REAL
(g . x) - (x #Z 2) is V28() V29() ext-real Element of REAL
K98((x #Z 2)) is V28() V29() ext-real set
K96((g . x),K98((x #Z 2))) is V28() V29() ext-real set
((g . x) - (x #Z 2)) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / (((g . x) - (x #Z 2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((g . x) - (x #Z 2)) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((((g . x) - (x #Z 2)) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / (((g . x) - (x #Z 2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / (((g . x) - (x #Z 2)) #R (1 / 2)))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / (((g . x) - (x #Z 2)) #R (1 / 2))))) is V28() V29() ext-real set
(g . x) - (x ^2) is V28() V29() ext-real Element of REAL
K96((g . x),K98((x ^2))) is V28() V29() ext-real set
((g . x) - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / (((g . x) - (x ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99((((g . x) - (x ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((((g . x) - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / (((g . x) - (x ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / (((g . x) - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / (((g . x) - (x ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(1 - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
((A * x) + f1) / ((1 - (x ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
K99(((1 - (x ^2)) #R (1 / 2))) is V28() V29() ext-real set
K97(((A * x) + f1),K99(((1 - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / ((1 - (x ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / ((1 - (x ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / ((1 - (x ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(Z (#) arccos) `| x is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((Z (#) arccos) `| x) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((Z (#) arccos) `| x) . x is V28() V29() ext-real Element of REAL
f2 . x is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
A * (arccos . x) is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
((A * x) + f1) / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97(((A * x) + f1),K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
(A * (arccos . x)) - (((A * x) + f1) / (sqrt (1 - (x ^2)))) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / (sqrt (1 - (x ^2))))) is V28() V29() ext-real set
K96((A * (arccos . x)),K98((((A * x) + f1) / (sqrt (1 - (x ^2)))))) is V28() V29() ext-real set
A is V28() V29() ext-real Element of REAL
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arcsin * Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z ^2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z (#) Z is Relation-like REAL -defined V6() V34() V35() V36() set
f - (Z ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (Z ^2) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (Z ^2) is Relation-like REAL -defined V6() V34() V35() V36() set
f + (- (Z ^2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (f - (Z ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) ((#R (1 / 2)) * (f - (Z ^2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom g is V55() V56() V57() Element of K19(REAL)
g | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (g,f1) is V28() V29() ext-real Element of REAL
f2 is V55() V56() V57() open Element of K19(REAL)
id f2 is Relation-like REAL -defined f2 -defined REAL -valued f2 -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2)))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arcsin * Z) + ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f2) (#) (arcsin * Z) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id f2) (#) (arcsin * Z)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
((id f2) (#) (arcsin * Z)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((id f2) (#) (arcsin * Z)) . (upper_bound f1)) - (((id f2) (#) (arcsin * Z)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((id f2) (#) (arcsin * Z)) . (lower_bound f1))) is V28() V29() ext-real set
K96((((id f2) (#) (arcsin * Z)) . (upper_bound f1)),K98((((id f2) (#) (arcsin * Z)) . (lower_bound f1)))) is V28() V29() ext-real set
dom (arcsin * Z) is V55() V56() V57() Element of K19(REAL)
dom ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) is V55() V56() V57() Element of K19(REAL)
(dom (arcsin * Z)) /\ (dom ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2)))))) is V55() V56() V57() Element of K19(REAL)
dom (id f2) is V55() V56() V57() Element of K19(f2)
K19(f2) is set
(dom (id f2)) /\ (dom (arcsin * Z)) is V55() V56() V57() Element of K19(REAL)
dom ((id f2) (#) (arcsin * Z)) is V55() V56() V57() Element of K19(REAL)
dom (A (#) ((#R (1 / 2)) * (f - (Z ^2)))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(A (#) ((#R (1 / 2)) * (f - (Z ^2)))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) \ ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (id f2)) /\ ((dom (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) \ ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) " {0})) is V55() V56() V57() Element of K19(REAL)
(A (#) ((#R (1 / 2)) * (f - (Z ^2)))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) ^) is V55() V56() V57() Element of K19(REAL)
dom ((#R (1 / 2)) * (f - (Z ^2))) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
arcsin . (x / A) is V28() V29() ext-real Element of REAL
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is V28() V29() ext-real set
1 - ((x / A) ^2) is V28() V29() ext-real Element of REAL
K98(((x / A) ^2)) is V28() V29() ext-real set
K96(1,K98(((x / A) ^2))) is V28() V29() ext-real set
sqrt (1 - ((x / A) ^2)) is V28() V29() ext-real Element of REAL
A * (sqrt (1 - ((x / A) ^2))) is V28() V29() ext-real Element of REAL
x / (A * (sqrt (1 - ((x / A) ^2)))) is V28() V29() ext-real Element of REAL
K99((A * (sqrt (1 - ((x / A) ^2))))) is V28() V29() ext-real set
K97(x,K99((A * (sqrt (1 - ((x / A) ^2)))))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * (sqrt (1 - ((x / A) ^2))))) is V28() V29() ext-real Element of REAL
dom (f - (Z ^2)) is V55() V56() V57() Element of K19(REAL)
(f - (Z ^2)) . x is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
Z . x is V28() V29() ext-real Element of REAL
1 + (Z . x) is V28() V29() ext-real Element of REAL
1 - (Z . x) is V28() V29() ext-real Element of REAL
K98((Z . x)) is V28() V29() ext-real set
K96(1,K98((Z . x))) is V28() V29() ext-real set
(1 + (Z . x)) * (1 - (Z . x)) is V28() V29() ext-real Element of REAL
((arcsin * Z) + ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2)))))) . x is V28() V29() ext-real Element of REAL
(arcsin * Z) . x is V28() V29() ext-real Element of REAL
((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x is V28() V29() ext-real Element of REAL
((arcsin * Z) . x) + (((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x) is V28() V29() ext-real Element of REAL
arcsin . (Z . x) is V28() V29() ext-real Element of REAL
(arcsin . (Z . x)) + (((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x) is V28() V29() ext-real Element of REAL
(arcsin . (x / A)) + (((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x) is V28() V29() ext-real Element of REAL
(id f2) . x is V28() V29() ext-real Element of REAL
(A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x is V28() V29() ext-real Element of REAL
((id f2) . x) / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x) is V28() V29() ext-real Element of REAL
K99(((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x)) is V28() V29() ext-real set
K97(((id f2) . x),K99(((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x))) is V28() V29() ext-real set
(arcsin . (x / A)) + (((id f2) . x) / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x)) is V28() V29() ext-real Element of REAL
x / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x) is V28() V29() ext-real Element of REAL
K97(x,K99(((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x)) is V28() V29() ext-real Element of REAL
((#R (1 / 2)) * (f - (Z ^2))) . x is V28() V29() ext-real Element of REAL
A * (((#R (1 / 2)) * (f - (Z ^2))) . x) is V28() V29() ext-real Element of REAL
x / (A * (((#R (1 / 2)) * (f - (Z ^2))) . x)) is V28() V29() ext-real Element of REAL
K99((A * (((#R (1 / 2)) * (f - (Z ^2))) . x))) is V28() V29() ext-real set
K97(x,K99((A * (((#R (1 / 2)) * (f - (Z ^2))) . x)))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * (((#R (1 / 2)) * (f - (Z ^2))) . x))) is V28() V29() ext-real Element of REAL
(#R (1 / 2)) . ((f - (Z ^2)) . x) is V28() V29() ext-real Element of REAL
A * ((#R (1 / 2)) . ((f - (Z ^2)) . x)) is V28() V29() ext-real Element of REAL
x / (A * ((#R (1 / 2)) . ((f - (Z ^2)) . x))) is V28() V29() ext-real Element of REAL
K99((A * ((#R (1 / 2)) . ((f - (Z ^2)) . x)))) is V28() V29() ext-real set
K97(x,K99((A * ((#R (1 / 2)) . ((f - (Z ^2)) . x))))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * ((#R (1 / 2)) . ((f - (Z ^2)) . x)))) is V28() V29() ext-real Element of REAL
((f - (Z ^2)) . x) #R (1 / 2) is V28() V29() ext-real set
A * (((f - (Z ^2)) . x) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * (((f - (Z ^2)) . x) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * (((f - (Z ^2)) . x) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * (((f - (Z ^2)) . x) #R (1 / 2))))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * (((f - (Z ^2)) . x) #R (1 / 2)))) 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
K98(((Z ^2) . x)) is V28() V29() ext-real set
K96((f . x),K98(((Z ^2) . x))) is V28() V29() ext-real set
((f . x) - ((Z ^2) . x)) #R (1 / 2) is V28() V29() ext-real set
A * (((f . x) - ((Z ^2) . x)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * (((f . x) - ((Z ^2) . x)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * (((f . x) - ((Z ^2) . x)) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * (((f . x) - ((Z ^2) . x)) #R (1 / 2))))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * (((f . x) - ((Z ^2) . x)) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
(Z . x) ^2 is V28() V29() ext-real Element of REAL
K97((Z . x),(Z . x)) is V28() V29() ext-real set
(f . x) - ((Z . x) ^2) is V28() V29() ext-real Element of REAL
K98(((Z . x) ^2)) is V28() V29() ext-real set
K96((f . x),K98(((Z . x) ^2))) is V28() V29() ext-real set
((f . x) - ((Z . x) ^2)) #R (1 / 2) is V28() V29() ext-real set
A * (((f . x) - ((Z . x) ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * (((f . x) - ((Z . x) ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * (((f . x) - ((Z . x) ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * (((f . x) - ((Z . x) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * (((f . x) - ((Z . x) ^2)) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
1 - ((Z . x) ^2) is V28() V29() ext-real Element of REAL
K96(1,K98(((Z . x) ^2))) is V28() V29() ext-real set
(1 - ((Z . x) ^2)) #R (1 / 2) is V28() V29() ext-real set
A * ((1 - ((Z . x) ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * ((1 - ((Z . x) ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * ((1 - ((Z . x) ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * ((1 - ((Z . x) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * ((1 - ((Z . x) ^2)) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
(1 - ((x / A) ^2)) #R (1 / 2) is V28() V29() ext-real set
A * ((1 - ((x / A) ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * ((1 - ((x / A) ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * ((1 - ((x / A) ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * ((1 - ((x / A) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * ((1 - ((x / A) ^2)) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
((id f2) (#) (arcsin * Z)) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id f2) (#) (arcsin * Z)) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((id f2) (#) (arcsin * Z)) `| f2) . x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
arcsin . (x / A) is V28() V29() ext-real Element of REAL
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is V28() V29() ext-real set
1 - ((x / A) ^2) is V28() V29() ext-real Element of REAL
K98(((x / A) ^2)) is V28() V29() ext-real set
K96(1,K98(((x / A) ^2))) is V28() V29() ext-real set
sqrt (1 - ((x / A) ^2)) is V28() V29() ext-real Element of REAL
A * (sqrt (1 - ((x / A) ^2))) is V28() V29() ext-real Element of REAL
x / (A * (sqrt (1 - ((x / A) ^2)))) is V28() V29() ext-real Element of REAL
K99((A * (sqrt (1 - ((x / A) ^2))))) is V28() V29() ext-real set
K97(x,K99((A * (sqrt (1 - ((x / A) ^2)))))) is V28() V29() ext-real set
(arcsin . (x / A)) + (x / (A * (sqrt (1 - ((x / A) ^2))))) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccos * Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z ^2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z (#) Z is Relation-like REAL -defined V6() V34() V35() V36() set
f - (Z ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (Z ^2) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (Z ^2) is Relation-like REAL -defined V6() V34() V35() V36() set
f + (- (Z ^2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (f - (Z ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) ((#R (1 / 2)) * (f - (Z ^2))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom g is V55() V56() V57() Element of K19(REAL)
g | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (g,f1) is V28() V29() ext-real Element of REAL
f2 is V55() V56() V57() open Element of K19(REAL)
id f2 is Relation-like REAL -defined f2 -defined REAL -valued f2 -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2)))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(arccos * Z) - ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) is Relation-like REAL -defined V6() V34() V35() V36() set
(arccos * Z) + (- ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2)))))) is Relation-like REAL -defined V6() V34() V35() V36() set
(id f2) (#) (arccos * Z) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((id f2) (#) (arccos * Z)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
((id f2) (#) (arccos * Z)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((id f2) (#) (arccos * Z)) . (upper_bound f1)) - (((id f2) (#) (arccos * Z)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((id f2) (#) (arccos * Z)) . (lower_bound f1))) is V28() V29() ext-real set
K96((((id f2) (#) (arccos * Z)) . (upper_bound f1)),K98((((id f2) (#) (arccos * Z)) . (lower_bound f1)))) is V28() V29() ext-real set
dom (arccos * Z) is V55() V56() V57() Element of K19(REAL)
dom ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) is V55() V56() V57() Element of K19(REAL)
(dom (arccos * Z)) /\ (dom ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2)))))) is V55() V56() V57() Element of K19(REAL)
dom (id f2) is V55() V56() V57() Element of K19(f2)
K19(f2) is set
(dom (id f2)) /\ (dom (arccos * Z)) is V55() V56() V57() Element of K19(REAL)
dom ((id f2) (#) (arccos * Z)) is V55() V56() V57() Element of K19(REAL)
dom (A (#) ((#R (1 / 2)) * (f - (Z ^2)))) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(A (#) ((#R (1 / 2)) * (f - (Z ^2)))) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) \ ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (id f2)) /\ ((dom (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) \ ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) " {0})) is V55() V56() V57() Element of K19(REAL)
(A (#) ((#R (1 / 2)) * (f - (Z ^2)))) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) ^) is V55() V56() V57() Element of K19(REAL)
dom ((#R (1 / 2)) * (f - (Z ^2))) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
arccos . (x / A) is V28() V29() ext-real Element of REAL
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is V28() V29() ext-real set
1 - ((x / A) ^2) is V28() V29() ext-real Element of REAL
K98(((x / A) ^2)) is V28() V29() ext-real set
K96(1,K98(((x / A) ^2))) is V28() V29() ext-real set
sqrt (1 - ((x / A) ^2)) is V28() V29() ext-real Element of REAL
A * (sqrt (1 - ((x / A) ^2))) is V28() V29() ext-real Element of REAL
x / (A * (sqrt (1 - ((x / A) ^2)))) is V28() V29() ext-real Element of REAL
K99((A * (sqrt (1 - ((x / A) ^2))))) is V28() V29() ext-real set
K97(x,K99((A * (sqrt (1 - ((x / A) ^2)))))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * (sqrt (1 - ((x / A) ^2))))) is V28() V29() ext-real Element of REAL
K98((x / (A * (sqrt (1 - ((x / A) ^2)))))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * (sqrt (1 - ((x / A) ^2))))))) is V28() V29() ext-real set
dom (f - (Z ^2)) is V55() V56() V57() Element of K19(REAL)
(f - (Z ^2)) . x is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
Z . x is V28() V29() ext-real Element of REAL
1 + (Z . x) is V28() V29() ext-real Element of REAL
1 - (Z . x) is V28() V29() ext-real Element of REAL
K98((Z . x)) is V28() V29() ext-real set
K96(1,K98((Z . x))) is V28() V29() ext-real set
(1 + (Z . x)) * (1 - (Z . x)) is V28() V29() ext-real Element of REAL
((arccos * Z) - ((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2)))))) . x is V28() V29() ext-real Element of REAL
(arccos * Z) . x is V28() V29() ext-real Element of REAL
((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x is V28() V29() ext-real Element of REAL
((arccos * Z) . x) - (((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x) is V28() V29() ext-real Element of REAL
K98((((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x)) is V28() V29() ext-real set
K96(((arccos * Z) . x),K98((((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x))) is V28() V29() ext-real set
arccos . (Z . x) is V28() V29() ext-real Element of REAL
(arccos . (Z . x)) - (((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x) is V28() V29() ext-real Element of REAL
K96((arccos . (Z . x)),K98((((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x))) is V28() V29() ext-real set
(arccos . (x / A)) - (((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x) is V28() V29() ext-real Element of REAL
K96((arccos . (x / A)),K98((((id f2) / (A (#) ((#R (1 / 2)) * (f - (Z ^2))))) . x))) is V28() V29() ext-real set
(id f2) . x is V28() V29() ext-real Element of REAL
(A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x is V28() V29() ext-real Element of REAL
((id f2) . x) / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x) is V28() V29() ext-real Element of REAL
K99(((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x)) is V28() V29() ext-real set
K97(((id f2) . x),K99(((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x))) is V28() V29() ext-real set
(arccos . (x / A)) - (((id f2) . x) / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x)) is V28() V29() ext-real Element of REAL
K98((((id f2) . x) / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((((id f2) . x) / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x)))) is V28() V29() ext-real set
x / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x) is V28() V29() ext-real Element of REAL
K97(x,K99(((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x)) is V28() V29() ext-real Element of REAL
K98((x / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / ((A (#) ((#R (1 / 2)) * (f - (Z ^2)))) . x)))) is V28() V29() ext-real set
((#R (1 / 2)) * (f - (Z ^2))) . x is V28() V29() ext-real Element of REAL
A * (((#R (1 / 2)) * (f - (Z ^2))) . x) is V28() V29() ext-real Element of REAL
x / (A * (((#R (1 / 2)) * (f - (Z ^2))) . x)) is V28() V29() ext-real Element of REAL
K99((A * (((#R (1 / 2)) * (f - (Z ^2))) . x))) is V28() V29() ext-real set
K97(x,K99((A * (((#R (1 / 2)) * (f - (Z ^2))) . x)))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * (((#R (1 / 2)) * (f - (Z ^2))) . x))) is V28() V29() ext-real Element of REAL
K98((x / (A * (((#R (1 / 2)) * (f - (Z ^2))) . x)))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * (((#R (1 / 2)) * (f - (Z ^2))) . x))))) is V28() V29() ext-real set
(#R (1 / 2)) . ((f - (Z ^2)) . x) is V28() V29() ext-real Element of REAL
A * ((#R (1 / 2)) . ((f - (Z ^2)) . x)) is V28() V29() ext-real Element of REAL
x / (A * ((#R (1 / 2)) . ((f - (Z ^2)) . x))) is V28() V29() ext-real Element of REAL
K99((A * ((#R (1 / 2)) . ((f - (Z ^2)) . x)))) is V28() V29() ext-real set
K97(x,K99((A * ((#R (1 / 2)) . ((f - (Z ^2)) . x))))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * ((#R (1 / 2)) . ((f - (Z ^2)) . x)))) is V28() V29() ext-real Element of REAL
K98((x / (A * ((#R (1 / 2)) . ((f - (Z ^2)) . x))))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * ((#R (1 / 2)) . ((f - (Z ^2)) . x)))))) is V28() V29() ext-real set
((f - (Z ^2)) . x) #R (1 / 2) is V28() V29() ext-real set
A * (((f - (Z ^2)) . x) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * (((f - (Z ^2)) . x) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * (((f - (Z ^2)) . x) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * (((f - (Z ^2)) . x) #R (1 / 2))))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * (((f - (Z ^2)) . x) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * (((f - (Z ^2)) . x) #R (1 / 2))))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * (((f - (Z ^2)) . x) #R (1 / 2)))))) is V28() V29() ext-real 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
K98(((Z ^2) . x)) is V28() V29() ext-real set
K96((f . x),K98(((Z ^2) . x))) is V28() V29() ext-real set
((f . x) - ((Z ^2) . x)) #R (1 / 2) is V28() V29() ext-real set
A * (((f . x) - ((Z ^2) . x)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * (((f . x) - ((Z ^2) . x)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * (((f . x) - ((Z ^2) . x)) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * (((f . x) - ((Z ^2) . x)) #R (1 / 2))))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * (((f . x) - ((Z ^2) . x)) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * (((f . x) - ((Z ^2) . x)) #R (1 / 2))))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * (((f . x) - ((Z ^2) . x)) #R (1 / 2)))))) is V28() V29() ext-real set
(Z . x) ^2 is V28() V29() ext-real Element of REAL
K97((Z . x),(Z . x)) is V28() V29() ext-real set
(f . x) - ((Z . x) ^2) is V28() V29() ext-real Element of REAL
K98(((Z . x) ^2)) is V28() V29() ext-real set
K96((f . x),K98(((Z . x) ^2))) is V28() V29() ext-real set
((f . x) - ((Z . x) ^2)) #R (1 / 2) is V28() V29() ext-real set
A * (((f . x) - ((Z . x) ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * (((f . x) - ((Z . x) ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * (((f . x) - ((Z . x) ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * (((f . x) - ((Z . x) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * (((f . x) - ((Z . x) ^2)) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * (((f . x) - ((Z . x) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * (((f . x) - ((Z . x) ^2)) #R (1 / 2)))))) is V28() V29() ext-real set
1 - ((Z . x) ^2) is V28() V29() ext-real Element of REAL
K96(1,K98(((Z . x) ^2))) is V28() V29() ext-real set
(1 - ((Z . x) ^2)) #R (1 / 2) is V28() V29() ext-real set
A * ((1 - ((Z . x) ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * ((1 - ((Z . x) ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * ((1 - ((Z . x) ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * ((1 - ((Z . x) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * ((1 - ((Z . x) ^2)) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * ((1 - ((Z . x) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * ((1 - ((Z . x) ^2)) #R (1 / 2)))))) is V28() V29() ext-real set
(1 - ((x / A) ^2)) #R (1 / 2) is V28() V29() ext-real set
A * ((1 - ((x / A) ^2)) #R (1 / 2)) is V28() V29() ext-real Element of REAL
x / (A * ((1 - ((x / A) ^2)) #R (1 / 2))) is V28() V29() ext-real Element of REAL
K99((A * ((1 - ((x / A) ^2)) #R (1 / 2)))) is V28() V29() ext-real set
K97(x,K99((A * ((1 - ((x / A) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * ((1 - ((x / A) ^2)) #R (1 / 2)))) is V28() V29() ext-real Element of REAL
K98((x / (A * ((1 - ((x / A) ^2)) #R (1 / 2))))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * ((1 - ((x / A) ^2)) #R (1 / 2)))))) is V28() V29() ext-real set
((id f2) (#) (arccos * Z)) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((id f2) (#) (arccos * Z)) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(((id f2) (#) (arccos * Z)) `| f2) . x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
x / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(x,K99(A)) is V28() V29() ext-real set
arccos . (x / A) is V28() V29() ext-real Element of REAL
(x / A) ^2 is V28() V29() ext-real Element of REAL
K97((x / A),(x / A)) is V28() V29() ext-real set
1 - ((x / A) ^2) is V28() V29() ext-real Element of REAL
K98(((x / A) ^2)) is V28() V29() ext-real set
K96(1,K98(((x / A) ^2))) is V28() V29() ext-real set
sqrt (1 - ((x / A) ^2)) is V28() V29() ext-real Element of REAL
A * (sqrt (1 - ((x / A) ^2))) is V28() V29() ext-real Element of REAL
x / (A * (sqrt (1 - ((x / A) ^2)))) is V28() V29() ext-real Element of REAL
K99((A * (sqrt (1 - ((x / A) ^2))))) is V28() V29() ext-real set
K97(x,K99((A * (sqrt (1 - ((x / A) ^2)))))) is V28() V29() ext-real set
(arccos . (x / A)) - (x / (A * (sqrt (1 - ((x / A) ^2))))) is V28() V29() ext-real Element of REAL
K98((x / (A * (sqrt (1 - ((x / A) ^2)))))) is V28() V29() ext-real set
K96((arccos . (x / A)),K98((x / (A * (sqrt (1 - ((x / A) ^2))))))) is V28() V29() ext-real set
A is V21() V22() V23() V27() V28() V29() V30() ext-real non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
A - 1 is V28() V29() V30() ext-real V67() Element of INT
K96(A,K98(1)) is V28() V29() V30() ext-real set
#Z (A - 1) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A - 1)) * sin is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) ((#Z (A - 1)) * sin) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
A + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
#Z (A + 1) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A + 1)) * cos is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(A (#) ((#Z (A - 1)) * sin)) / ((#Z (A + 1)) * cos) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
#Z A is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * tan is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * tan) is V55() V56() V57() Element of K19(REAL)
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
((#Z A) * tan) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
((#Z A) * tan) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((#Z A) * tan) . (upper_bound f1)) - (((#Z A) * tan) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((#Z A) * tan) . (lower_bound f1))) is V28() V29() ext-real set
K96((((#Z A) * tan) . (upper_bound f1)),K98((((#Z A) * tan) . (lower_bound f1)))) is V28() V29() ext-real set
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,f1) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom (A (#) ((#Z (A - 1)) * sin)) is non empty V55() V56() V57() Element of K19(REAL)
dom ((#Z (A + 1)) * cos) is non empty V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((#Z (A + 1)) * cos) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((#Z (A + 1)) * cos)) \ (((#Z (A + 1)) * cos) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (A (#) ((#Z (A - 1)) * sin))) /\ ((dom ((#Z (A + 1)) * cos)) \ (((#Z (A + 1)) * cos) " {0})) is V55() V56() V57() Element of K19(REAL)
((#Z (A + 1)) * cos) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#Z (A + 1)) * cos) ^) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
((#Z (A + 1)) * cos) . g is V28() V29() ext-real Element of REAL
dom ((#Z (A - 1)) * sin) is non empty V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
f2 is V21() V22() V23() V27() V28() V29() V30() ext-real non negative set
f2 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
g is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
(sin . g) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((sin . g) #Z (A - 1)) is V28() V29() ext-real Element of REAL
cos . g is V28() V29() ext-real Element of REAL
(cos . g) #Z (A + 1) is V28() V29() ext-real Element of REAL
(A * ((sin . g) #Z (A - 1))) / ((cos . g) #Z (A + 1)) is V28() V29() ext-real Element of REAL
K99(((cos . g) #Z (A + 1))) is V28() V29() ext-real set
K97((A * ((sin . g) #Z (A - 1))),K99(((cos . g) #Z (A + 1)))) is V28() V29() ext-real set
((A (#) ((#Z (A - 1)) * sin)) / ((#Z (A + 1)) * cos)) . g is V28() V29() ext-real Element of REAL
(A (#) ((#Z (A - 1)) * sin)) . g is V28() V29() ext-real Element of REAL
((#Z (A + 1)) * cos) . g is V28() V29() ext-real Element of REAL
((A (#) ((#Z (A - 1)) * sin)) . g) / (((#Z (A + 1)) * cos) . g) is V28() V29() ext-real Element of REAL
K99((((#Z (A + 1)) * cos) . g)) is V28() V29() ext-real set
K97(((A (#) ((#Z (A - 1)) * sin)) . g),K99((((#Z (A + 1)) * cos) . g))) is V28() V29() ext-real set
((#Z (A - 1)) * sin) . g is V28() V29() ext-real Element of REAL
A * (((#Z (A - 1)) * sin) . g) is V28() V29() ext-real Element of REAL
(A * (((#Z (A - 1)) * sin) . g)) / (((#Z (A + 1)) * cos) . g) is V28() V29() ext-real Element of REAL
K97((A * (((#Z (A - 1)) * sin) . g)),K99((((#Z (A + 1)) * cos) . g))) is V28() V29() ext-real set
(#Z (A - 1)) . (sin . g) is V28() V29() ext-real Element of REAL
A * ((#Z (A - 1)) . (sin . g)) is V28() V29() ext-real Element of REAL
(A * ((#Z (A - 1)) . (sin . g))) / (((#Z (A + 1)) * cos) . g) is V28() V29() ext-real Element of REAL
K97((A * ((#Z (A - 1)) . (sin . g))),K99((((#Z (A + 1)) * cos) . g))) is V28() V29() ext-real set
(A * ((sin . g) #Z (A - 1))) / (((#Z (A + 1)) * cos) . g) is V28() V29() ext-real Element of REAL
K97((A * ((sin . g) #Z (A - 1))),K99((((#Z (A + 1)) * cos) . g))) is V28() V29() ext-real set
(#Z (A + 1)) . (cos . g) is V28() V29() ext-real Element of REAL
(A * ((sin . g) #Z (A - 1))) / ((#Z (A + 1)) . (cos . g)) is V28() V29() ext-real Element of REAL
K99(((#Z (A + 1)) . (cos . g))) is V28() V29() ext-real set
K97((A * ((sin . g) #Z (A - 1))),K99(((#Z (A + 1)) . (cos . g)))) is V28() V29() ext-real set
((#Z A) * tan) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#Z A) * tan) `| Z) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
(((#Z A) * tan) `| Z) . g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
(sin . g) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((sin . g) #Z (A - 1)) is V28() V29() ext-real Element of REAL
cos . g is V28() V29() ext-real Element of REAL
(cos . g) #Z (A + 1) is V28() V29() ext-real Element of REAL
(A * ((sin . g) #Z (A - 1))) / ((cos . g) #Z (A + 1)) is V28() V29() ext-real Element of REAL
K99(((cos . g) #Z (A + 1))) is V28() V29() ext-real set
K97((A * ((sin . g) #Z (A - 1))),K99(((cos . g) #Z (A + 1)))) is V28() V29() ext-real set
A is V21() V22() V23() V27() V28() V29() V30() ext-real non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
A - 1 is V28() V29() V30() ext-real V67() Element of INT
K96(A,K98(1)) is V28() V29() V30() ext-real set
#Z (A - 1) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A - 1)) * cos is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
A (#) ((#Z (A - 1)) * cos) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
A + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
#Z (A + 1) is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z (A + 1)) * sin is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(A (#) ((#Z (A - 1)) * cos)) / ((#Z (A + 1)) * sin) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
#Z A is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * cot is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * cot) is V55() V56() V57() Element of K19(REAL)
- ((#Z A) * cot) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((#Z A) * cot) is Relation-like REAL -defined V6() V34() V35() V36() set
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
(- ((#Z A) * cot)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
(- ((#Z A) * cot)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
((- ((#Z A) * cot)) . (upper_bound f1)) - ((- ((#Z A) * cot)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98(((- ((#Z A) * cot)) . (lower_bound f1))) is V28() V29() ext-real set
K96(((- ((#Z A) * cot)) . (upper_bound f1)),K98(((- ((#Z A) * cot)) . (lower_bound f1)))) is V28() V29() ext-real set
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,f1) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom (A (#) ((#Z (A - 1)) * cos)) is non empty V55() V56() V57() Element of K19(REAL)
dom ((#Z (A + 1)) * sin) is non empty V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((#Z (A + 1)) * sin) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((#Z (A + 1)) * sin)) \ (((#Z (A + 1)) * sin) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom (A (#) ((#Z (A - 1)) * cos))) /\ ((dom ((#Z (A + 1)) * sin)) \ (((#Z (A + 1)) * sin) " {0})) is V55() V56() V57() Element of K19(REAL)
((#Z (A + 1)) * sin) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((#Z (A + 1)) * sin) ^) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
((#Z (A + 1)) * sin) . g is V28() V29() ext-real Element of REAL
dom ((#Z (A - 1)) * cos) is non empty V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
f2 is V21() V22() V23() V27() V28() V29() V30() ext-real non negative set
f2 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
g is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- ((#Z A) * cot)) is V55() V56() V57() Element of K19(REAL)
(- 1) (#) ((#Z A) * cot) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
dom (cos / sin) is V55() V56() V57() Element of K19(REAL)
(- ((#Z A) * cot)) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is V28() V29() ext-real Element of REAL
((- ((#Z A) * cot)) `| Z) . g is V28() V29() ext-real Element of REAL
cos . g is V28() V29() ext-real Element of REAL
(cos . g) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((cos . g) #Z (A - 1)) is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
(sin . g) #Z (A + 1) is V28() V29() ext-real Element of REAL
(A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z (A + 1)) is V28() V29() ext-real Element of REAL
K99(((sin . g) #Z (A + 1))) is V28() V29() ext-real set
K97((A * ((cos . g) #Z (A - 1))),K99(((sin . g) #Z (A + 1)))) is V28() V29() ext-real set
f2 is V21() V22() V23() V27() V28() V29() V30() ext-real non negative set
f2 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
diff ((- ((#Z A) * cot)),g) is V28() V29() ext-real Element of REAL
diff (((#Z A) * cot),g) is V28() V29() ext-real Element of REAL
(- 1) * (diff (((#Z A) * cot),g)) is V28() V29() ext-real Element of REAL
cot . g is V28() V29() ext-real Element of REAL
(cot . g) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((cot . g) #Z (A - 1)) is V28() V29() ext-real Element of REAL
diff (cot,g) is V28() V29() ext-real Element of REAL
(A * ((cot . g) #Z (A - 1))) * (diff (cot,g)) is V28() V29() ext-real Element of REAL
(- 1) * ((A * ((cot . g) #Z (A - 1))) * (diff (cot,g))) is V28() V29() ext-real Element of REAL
(sin . g) ^2 is V28() V29() ext-real Element of REAL
K97((sin . g),(sin . g)) is V28() V29() ext-real set
1 / ((sin . g) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . g) ^2)) is V28() V29() ext-real set
K97(1,K99(((sin . g) ^2))) is V28() V29() ext-real set
- (1 / ((sin . g) ^2)) is V28() V29() ext-real Element of REAL
(A * ((cot . g) #Z (A - 1))) * (- (1 / ((sin . g) ^2))) is V28() V29() ext-real Element of REAL
(- 1) * ((A * ((cot . g) #Z (A - 1))) * (- (1 / ((sin . g) ^2)))) is V28() V29() ext-real Element of REAL
(A * ((cot . g) #Z (A - 1))) / ((sin . g) ^2) is V28() V29() ext-real Element of REAL
K97((A * ((cot . g) #Z (A - 1))),K99(((sin . g) ^2))) is V28() V29() ext-real set
- ((A * ((cot . g) #Z (A - 1))) / ((sin . g) ^2)) is V28() V29() ext-real Element of REAL
(- 1) * (- ((A * ((cot . g) #Z (A - 1))) / ((sin . g) ^2))) is V28() V29() ext-real Element of REAL
(sin . g) #Z (A - 1) is V28() V29() ext-real Element of REAL
((cos . g) #Z (A - 1)) / ((sin . g) #Z (A - 1)) is V28() V29() ext-real Element of REAL
K99(((sin . g) #Z (A - 1))) is V28() V29() ext-real set
K97(((cos . g) #Z (A - 1)),K99(((sin . g) #Z (A - 1)))) is V28() V29() ext-real set
A * (((cos . g) #Z (A - 1)) / ((sin . g) #Z (A - 1))) is V28() V29() ext-real Element of REAL
(A * (((cos . g) #Z (A - 1)) / ((sin . g) #Z (A - 1)))) / ((sin . g) ^2) is V28() V29() ext-real Element of REAL
K97((A * (((cos . g) #Z (A - 1)) / ((sin . g) #Z (A - 1)))),K99(((sin . g) ^2))) is V28() V29() ext-real set
- ((A * (((cos . g) #Z (A - 1)) / ((sin . g) #Z (A - 1)))) / ((sin . g) ^2)) is V28() V29() ext-real Element of REAL
(- 1) * (- ((A * (((cos . g) #Z (A - 1)) / ((sin . g) #Z (A - 1)))) / ((sin . g) ^2))) is V28() V29() ext-real Element of REAL
(A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z (A - 1)) is V28() V29() ext-real Element of REAL
K97((A * ((cos . g) #Z (A - 1))),K99(((sin . g) #Z (A - 1)))) is V28() V29() ext-real set
((A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z (A - 1))) / ((sin . g) ^2) is V28() V29() ext-real Element of REAL
K97(((A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z (A - 1))),K99(((sin . g) ^2))) is V28() V29() ext-real set
- (((A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z (A - 1))) / ((sin . g) ^2)) is V28() V29() ext-real Element of REAL
(- 1) * (- (((A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z (A - 1))) / ((sin . g) ^2))) is V28() V29() ext-real Element of REAL
((sin . g) #Z (A - 1)) * ((sin . g) ^2) is V28() V29() ext-real Element of REAL
(A * ((cos . g) #Z (A - 1))) / (((sin . g) #Z (A - 1)) * ((sin . g) ^2)) is V28() V29() ext-real Element of REAL
K99((((sin . g) #Z (A - 1)) * ((sin . g) ^2))) is V28() V29() ext-real set
K97((A * ((cos . g) #Z (A - 1))),K99((((sin . g) #Z (A - 1)) * ((sin . g) ^2)))) is V28() V29() ext-real set
- ((A * ((cos . g) #Z (A - 1))) / (((sin . g) #Z (A - 1)) * ((sin . g) ^2))) is V28() V29() ext-real Element of REAL
(- 1) * (- ((A * ((cos . g) #Z (A - 1))) / (((sin . g) #Z (A - 1)) * ((sin . g) ^2)))) is V28() V29() ext-real Element of REAL
(sin . g) #Z 2 is V28() V29() ext-real Element of REAL
((sin . g) #Z (A - 1)) * ((sin . g) #Z 2) is V28() V29() ext-real Element of REAL
(A * ((cos . g) #Z (A - 1))) / (((sin . g) #Z (A - 1)) * ((sin . g) #Z 2)) is V28() V29() ext-real Element of REAL
K99((((sin . g) #Z (A - 1)) * ((sin . g) #Z 2))) is V28() V29() ext-real set
K97((A * ((cos . g) #Z (A - 1))),K99((((sin . g) #Z (A - 1)) * ((sin . g) #Z 2)))) is V28() V29() ext-real set
- ((A * ((cos . g) #Z (A - 1))) / (((sin . g) #Z (A - 1)) * ((sin . g) #Z 2))) is V28() V29() ext-real Element of REAL
(- 1) * (- ((A * ((cos . g) #Z (A - 1))) / (((sin . g) #Z (A - 1)) * ((sin . g) #Z 2)))) is V28() V29() ext-real Element of REAL
(A - 1) + 2 is V28() V29() V30() ext-real V67() Element of INT
(sin . g) #Z ((A - 1) + 2) is V28() V29() ext-real Element of REAL
(A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z ((A - 1) + 2)) is V28() V29() ext-real Element of REAL
K99(((sin . g) #Z ((A - 1) + 2))) is V28() V29() ext-real set
K97((A * ((cos . g) #Z (A - 1))),K99(((sin . g) #Z ((A - 1) + 2)))) is V28() V29() ext-real set
- ((A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z ((A - 1) + 2))) is V28() V29() ext-real Element of REAL
(- 1) * (- ((A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z ((A - 1) + 2)))) is V28() V29() ext-real Element of REAL
g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
cos . g is V28() V29() ext-real Element of REAL
(cos . g) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((cos . g) #Z (A - 1)) is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
(sin . g) #Z (A + 1) is V28() V29() ext-real Element of REAL
(A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z (A + 1)) is V28() V29() ext-real Element of REAL
K99(((sin . g) #Z (A + 1))) is V28() V29() ext-real set
K97((A * ((cos . g) #Z (A - 1))),K99(((sin . g) #Z (A + 1)))) is V28() V29() ext-real set
((A (#) ((#Z (A - 1)) * cos)) / ((#Z (A + 1)) * sin)) . g is V28() V29() ext-real Element of REAL
(A (#) ((#Z (A - 1)) * cos)) . g is V28() V29() ext-real Element of REAL
((#Z (A + 1)) * sin) . g is V28() V29() ext-real Element of REAL
((A (#) ((#Z (A - 1)) * cos)) . g) / (((#Z (A + 1)) * sin) . g) is V28() V29() ext-real Element of REAL
K99((((#Z (A + 1)) * sin) . g)) is V28() V29() ext-real set
K97(((A (#) ((#Z (A - 1)) * cos)) . g),K99((((#Z (A + 1)) * sin) . g))) is V28() V29() ext-real set
((#Z (A - 1)) * cos) . g is V28() V29() ext-real Element of REAL
A * (((#Z (A - 1)) * cos) . g) is V28() V29() ext-real Element of REAL
(A * (((#Z (A - 1)) * cos) . g)) / (((#Z (A + 1)) * sin) . g) is V28() V29() ext-real Element of REAL
K97((A * (((#Z (A - 1)) * cos) . g)),K99((((#Z (A + 1)) * sin) . g))) is V28() V29() ext-real set
(#Z (A - 1)) . (cos . g) is V28() V29() ext-real Element of REAL
A * ((#Z (A - 1)) . (cos . g)) is V28() V29() ext-real Element of REAL
(A * ((#Z (A - 1)) . (cos . g))) / (((#Z (A + 1)) * sin) . g) is V28() V29() ext-real Element of REAL
K97((A * ((#Z (A - 1)) . (cos . g))),K99((((#Z (A + 1)) * sin) . g))) is V28() V29() ext-real set
(A * ((cos . g) #Z (A - 1))) / (((#Z (A + 1)) * sin) . g) is V28() V29() ext-real Element of REAL
K97((A * ((cos . g) #Z (A - 1))),K99((((#Z (A + 1)) * sin) . g))) is V28() V29() ext-real set
(#Z (A + 1)) . (sin . g) is V28() V29() ext-real Element of REAL
(A * ((cos . g) #Z (A - 1))) / ((#Z (A + 1)) . (sin . g)) is V28() V29() ext-real Element of REAL
K99(((#Z (A + 1)) . (sin . g))) is V28() V29() ext-real set
K97((A * ((cos . g) #Z (A - 1))),K99(((#Z (A + 1)) . (sin . g)))) is V28() V29() ext-real set
dom ((- ((#Z A) * cot)) `| Z) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
((- ((#Z A) * cot)) `| Z) . g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
cos . g is V28() V29() ext-real Element of REAL
(cos . g) #Z (A - 1) is V28() V29() ext-real Element of REAL
A * ((cos . g) #Z (A - 1)) is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
(sin . g) #Z (A + 1) is V28() V29() ext-real Element of REAL
(A * ((cos . g) #Z (A - 1))) / ((sin . g) #Z (A + 1)) is V28() V29() ext-real Element of REAL
K99(((sin . g) #Z (A + 1))) is V28() V29() ext-real set
K97((A * ((cos . g) #Z (A - 1))),K99(((sin . g) #Z (A + 1)))) is V28() V29() ext-real set
A is V28() V29() ext-real Element of REAL
1 / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(1,K99(A)) is V28() V29() ext-real set
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
tan * f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (tan * f) is V55() V56() V57() Element of K19(REAL)
sin * f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * f) ^2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * f) (#) (sin * f) is Relation-like REAL -defined V6() V34() V35() V36() set
cos * f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * f) ^2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * f) (#) (cos * f) is Relation-like REAL -defined V6() V34() V35() V36() set
((sin * f) ^2) / ((cos * f) ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(1 / A) (#) (tan * f) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V55() V56() V57() Element of K19(REAL)
integral (Z,f1) is V28() V29() ext-real Element of REAL
g is V55() V56() V57() open Element of K19(REAL)
id g is Relation-like REAL -defined g -defined REAL -valued g -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
((1 / A) (#) (tan * f)) - (id g) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (id g) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (id g) is Relation-like REAL -defined V6() V34() V35() V36() set
((1 / A) (#) (tan * f)) + (- (id g)) is Relation-like REAL -defined V6() V34() V35() V36() set
(((1 / A) (#) (tan * f)) - (id g)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
(((1 / A) (#) (tan * f)) - (id g)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
((((1 / A) (#) (tan * f)) - (id g)) . (upper_bound f1)) - ((((1 / A) (#) (tan * f)) - (id g)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98(((((1 / A) (#) (tan * f)) - (id g)) . (lower_bound f1))) is V28() V29() ext-real set
K96(((((1 / A) (#) (tan * f)) - (id g)) . (upper_bound f1)),K98(((((1 / A) (#) (tan * f)) - (id g)) . (lower_bound f1)))) is V28() V29() ext-real set
dom ((1 / A) (#) (tan * f)) is V55() V56() V57() Element of K19(REAL)
dom (id g) is V55() V56() V57() Element of K19(g)
K19(g) is set
(dom ((1 / A) (#) (tan * f))) /\ (dom (id g)) is V55() V56() V57() Element of K19(g)
dom (((1 / A) (#) (tan * f)) - (id g)) is V55() V56() V57() Element of K19(g)
f2 is V28() V29() ext-real Element of REAL
f . f2 is V28() V29() ext-real Element of REAL
A * f2 is V28() V29() ext-real Element of REAL
(A * f2) + 0 is V28() V29() ext-real Element of REAL
dom ((sin * f) ^2) is V55() V56() V57() Element of K19(REAL)
dom ((cos * f) ^2) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((cos * f) ^2) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((cos * f) ^2)) \ (((cos * f) ^2) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom ((sin * f) ^2)) /\ ((dom ((cos * f) ^2)) \ (((cos * f) ^2) " {0})) is V55() V56() V57() Element of K19(REAL)
dom (sin * f) is V55() V56() V57() Element of K19(REAL)
((cos * f) ^2) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((cos * f) ^2) ^) is V55() V56() V57() Element of K19(REAL)
dom (cos * f) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
((cos * f) ^2) . f2 is V28() V29() ext-real Element of REAL
Z | g is Relation-like REAL -defined g -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is V28() V29() ext-real Element of REAL
Z . f2 is V28() V29() ext-real Element of REAL
A * f2 is V28() V29() ext-real Element of REAL
sin . (A * f2) is V28() V29() ext-real Element of REAL
(sin . (A * f2)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (A * f2)),(sin . (A * f2))) is V28() V29() ext-real set
cos . (A * f2) is V28() V29() ext-real Element of REAL
(cos . (A * f2)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (A * f2)),(cos . (A * f2))) is V28() V29() ext-real set
((sin . (A * f2)) ^2) / ((cos . (A * f2)) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (A * f2)) ^2)) is V28() V29() ext-real set
K97(((sin . (A * f2)) ^2),K99(((cos . (A * f2)) ^2))) is V28() V29() ext-real set
(((sin * f) ^2) / ((cos * f) ^2)) . f2 is V28() V29() ext-real Element of REAL
((sin * f) ^2) . f2 is V28() V29() ext-real Element of REAL
((cos * f) ^2) . f2 is V28() V29() ext-real Element of REAL
(((sin * f) ^2) . f2) / (((cos * f) ^2) . f2) is V28() V29() ext-real Element of REAL
K99((((cos * f) ^2) . f2)) is V28() V29() ext-real set
K97((((sin * f) ^2) . f2),K99((((cos * f) ^2) . f2))) is V28() V29() ext-real set
(sin * f) . f2 is V28() V29() ext-real Element of REAL
((sin * f) . f2) ^2 is V28() V29() ext-real Element of REAL
K97(((sin * f) . f2),((sin * f) . f2)) is V28() V29() ext-real set
(((sin * f) . f2) ^2) / (((cos * f) ^2) . f2) is V28() V29() ext-real Element of REAL
K97((((sin * f) . f2) ^2),K99((((cos * f) ^2) . f2))) is V28() V29() ext-real set
(cos * f) . f2 is V28() V29() ext-real Element of REAL
((cos * f) . f2) ^2 is V28() V29() ext-real Element of REAL
K97(((cos * f) . f2),((cos * f) . f2)) is V28() V29() ext-real set
(((sin * f) . f2) ^2) / (((cos * f) . f2) ^2) is V28() V29() ext-real Element of REAL
K99((((cos * f) . f2) ^2)) is V28() V29() ext-real set
K97((((sin * f) . f2) ^2),K99((((cos * f) . f2) ^2))) is V28() V29() ext-real set
f . f2 is V28() V29() ext-real Element of REAL
sin . (f . f2) is V28() V29() ext-real Element of REAL
(sin . (f . f2)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (f . f2)),(sin . (f . f2))) is V28() V29() ext-real set
((sin . (f . f2)) ^2) / (((cos * f) . f2) ^2) is V28() V29() ext-real Element of REAL
K97(((sin . (f . f2)) ^2),K99((((cos * f) . f2) ^2))) is V28() V29() ext-real set
cos . (f . f2) is V28() V29() ext-real Element of REAL
(cos . (f . f2)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (f . f2)),(cos . (f . f2))) is V28() V29() ext-real set
((sin . (f . f2)) ^2) / ((cos . (f . f2)) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (f . f2)) ^2)) is V28() V29() ext-real set
K97(((sin . (f . f2)) ^2),K99(((cos . (f . f2)) ^2))) is V28() V29() ext-real set
((sin . (A * f2)) ^2) / ((cos . (f . f2)) ^2) is V28() V29() ext-real Element of REAL
K97(((sin . (A * f2)) ^2),K99(((cos . (f . f2)) ^2))) is V28() V29() ext-real set
(((1 / A) (#) (tan * f)) - (id g)) `| g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((((1 / A) (#) (tan * f)) - (id g)) `| g) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
((((1 / A) (#) (tan * f)) - (id g)) `| g) . f2 is V28() V29() ext-real Element of REAL
Z . f2 is V28() V29() ext-real Element of REAL
A * f2 is V28() V29() ext-real Element of REAL
sin . (A * f2) is V28() V29() ext-real Element of REAL
(sin . (A * f2)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (A * f2)),(sin . (A * f2))) is V28() V29() ext-real set
cos . (A * f2) is V28() V29() ext-real Element of REAL
(cos . (A * f2)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (A * f2)),(cos . (A * f2))) is V28() V29() ext-real set
((sin . (A * f2)) ^2) / ((cos . (A * f2)) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . (A * f2)) ^2)) is V28() V29() ext-real set
K97(((sin . (A * f2)) ^2),K99(((cos . (A * f2)) ^2))) is V28() V29() ext-real set
A is V28() V29() ext-real Element of REAL
1 / A is V28() V29() ext-real Element of REAL
K99(A) is V28() V29() ext-real set
K97(1,K99(A)) is V28() V29() ext-real set
- (1 / A) is V28() V29() ext-real Element of REAL
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
cot * f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (cot * f) is V55() V56() V57() Element of K19(REAL)
cos * f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * f) ^2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(cos * f) (#) (cos * f) is Relation-like REAL -defined V6() V34() V35() V36() set
sin * f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * f) ^2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(sin * f) (#) (sin * f) is Relation-like REAL -defined V6() V34() V35() V36() set
((cos * f) ^2) / ((sin * f) ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(- (1 / A)) (#) (cot * f) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V55() V56() V57() Element of K19(REAL)
integral (Z,f1) is V28() V29() ext-real Element of REAL
g is V55() V56() V57() open Element of K19(REAL)
id g is Relation-like REAL -defined g -defined REAL -valued g -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
((- (1 / A)) (#) (cot * f)) - (id g) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (id g) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (id g) is Relation-like REAL -defined V6() V34() V35() V36() set
((- (1 / A)) (#) (cot * f)) + (- (id g)) is Relation-like REAL -defined V6() V34() V35() V36() set
(((- (1 / A)) (#) (cot * f)) - (id g)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
(((- (1 / A)) (#) (cot * f)) - (id g)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
((((- (1 / A)) (#) (cot * f)) - (id g)) . (upper_bound f1)) - ((((- (1 / A)) (#) (cot * f)) - (id g)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98(((((- (1 / A)) (#) (cot * f)) - (id g)) . (lower_bound f1))) is V28() V29() ext-real set
K96(((((- (1 / A)) (#) (cot * f)) - (id g)) . (upper_bound f1)),K98(((((- (1 / A)) (#) (cot * f)) - (id g)) . (lower_bound f1)))) is V28() V29() ext-real set
dom ((- (1 / A)) (#) (cot * f)) is V55() V56() V57() Element of K19(REAL)
dom (id g) is V55() V56() V57() Element of K19(g)
K19(g) is set
(dom ((- (1 / A)) (#) (cot * f))) /\ (dom (id g)) is V55() V56() V57() Element of K19(g)
dom (((- (1 / A)) (#) (cot * f)) - (id g)) is V55() V56() V57() Element of K19(g)
f2 is V28() V29() ext-real Element of REAL
f . f2 is V28() V29() ext-real Element of REAL
A * f2 is V28() V29() ext-real Element of REAL
(A * f2) + 0 is V28() V29() ext-real Element of REAL
dom ((cos * f) ^2) is V55() V56() V57() Element of K19(REAL)
dom ((sin * f) ^2) is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
((sin * f) ^2) " {0} is V55() V56() V57() Element of K19(REAL)
(dom ((sin * f) ^2)) \ (((sin * f) ^2) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom ((cos * f) ^2)) /\ ((dom ((sin * f) ^2)) \ (((sin * f) ^2) " {0})) is V55() V56() V57() Element of K19(REAL)
dom (cos * f) is V55() V56() V57() Element of K19(REAL)
((sin * f) ^2) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((sin * f) ^2) ^) is V55() V56() V57() Element of K19(REAL)
dom (sin * f) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
((sin * f) ^2) . f2 is V28() V29() ext-real Element of REAL
Z | g is Relation-like REAL -defined g -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is V28() V29() ext-real Element of REAL
Z . f2 is V28() V29() ext-real Element of REAL
A * f2 is V28() V29() ext-real Element of REAL
cos . (A * f2) is V28() V29() ext-real Element of REAL
(cos . (A * f2)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (A * f2)),(cos . (A * f2))) is V28() V29() ext-real set
sin . (A * f2) is V28() V29() ext-real Element of REAL
(sin . (A * f2)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (A * f2)),(sin . (A * f2))) is V28() V29() ext-real set
((cos . (A * f2)) ^2) / ((sin . (A * f2)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (A * f2)) ^2)) is V28() V29() ext-real set
K97(((cos . (A * f2)) ^2),K99(((sin . (A * f2)) ^2))) is V28() V29() ext-real set
(((cos * f) ^2) / ((sin * f) ^2)) . f2 is V28() V29() ext-real Element of REAL
((cos * f) ^2) . f2 is V28() V29() ext-real Element of REAL
((sin * f) ^2) . f2 is V28() V29() ext-real Element of REAL
(((cos * f) ^2) . f2) / (((sin * f) ^2) . f2) is V28() V29() ext-real Element of REAL
K99((((sin * f) ^2) . f2)) is V28() V29() ext-real set
K97((((cos * f) ^2) . f2),K99((((sin * f) ^2) . f2))) is V28() V29() ext-real set
(cos * f) . f2 is V28() V29() ext-real Element of REAL
((cos * f) . f2) ^2 is V28() V29() ext-real Element of REAL
K97(((cos * f) . f2),((cos * f) . f2)) is V28() V29() ext-real set
(((cos * f) . f2) ^2) / (((sin * f) ^2) . f2) is V28() V29() ext-real Element of REAL
K97((((cos * f) . f2) ^2),K99((((sin * f) ^2) . f2))) is V28() V29() ext-real set
(sin * f) . f2 is V28() V29() ext-real Element of REAL
((sin * f) . f2) ^2 is V28() V29() ext-real Element of REAL
K97(((sin * f) . f2),((sin * f) . f2)) is V28() V29() ext-real set
(((cos * f) . f2) ^2) / (((sin * f) . f2) ^2) is V28() V29() ext-real Element of REAL
K99((((sin * f) . f2) ^2)) is V28() V29() ext-real set
K97((((cos * f) . f2) ^2),K99((((sin * f) . f2) ^2))) is V28() V29() ext-real set
f . f2 is V28() V29() ext-real Element of REAL
cos . (f . f2) is V28() V29() ext-real Element of REAL
(cos . (f . f2)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (f . f2)),(cos . (f . f2))) is V28() V29() ext-real set
((cos . (f . f2)) ^2) / (((sin * f) . f2) ^2) is V28() V29() ext-real Element of REAL
K97(((cos . (f . f2)) ^2),K99((((sin * f) . f2) ^2))) is V28() V29() ext-real set
sin . (f . f2) is V28() V29() ext-real Element of REAL
(sin . (f . f2)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (f . f2)),(sin . (f . f2))) is V28() V29() ext-real set
((cos . (f . f2)) ^2) / ((sin . (f . f2)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (f . f2)) ^2)) is V28() V29() ext-real set
K97(((cos . (f . f2)) ^2),K99(((sin . (f . f2)) ^2))) is V28() V29() ext-real set
((cos . (A * f2)) ^2) / ((sin . (f . f2)) ^2) is V28() V29() ext-real Element of REAL
K97(((cos . (A * f2)) ^2),K99(((sin . (f . f2)) ^2))) is V28() V29() ext-real set
(((- (1 / A)) (#) (cot * f)) - (id g)) `| g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((((- (1 / A)) (#) (cot * f)) - (id g)) `| g) is V55() V56() V57() Element of K19(REAL)
f2 is V28() V29() ext-real Element of REAL
((((- (1 / A)) (#) (cot * f)) - (id g)) `| g) . f2 is V28() V29() ext-real Element of REAL
Z . f2 is V28() V29() ext-real Element of REAL
A * f2 is V28() V29() ext-real Element of REAL
cos . (A * f2) is V28() V29() ext-real Element of REAL
(cos . (A * f2)) ^2 is V28() V29() ext-real Element of REAL
K97((cos . (A * f2)),(cos . (A * f2))) is V28() V29() ext-real set
sin . (A * f2) is V28() V29() ext-real Element of REAL
(sin . (A * f2)) ^2 is V28() V29() ext-real Element of REAL
K97((sin . (A * f2)),(sin . (A * f2))) is V28() V29() ext-real set
((cos . (A * f2)) ^2) / ((sin . (A * f2)) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . (A * f2)) ^2)) is V28() V29() ext-real set
K97(((cos . (A * f2)) ^2),K99(((sin . (A * f2)) ^2))) is V28() V29() ext-real set
A is V28() V29() ext-real Element of REAL
A (#) (sin / cos) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
f is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f is V28() V29() ext-real Element of REAL
lower_bound f is V28() V29() ext-real Element of REAL
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z / (cos ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(A (#) (sin / cos)) + (Z / (cos ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z (#) tan is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(Z (#) tan) . (upper_bound f) is V28() V29() ext-real Element of REAL
(Z (#) tan) . (lower_bound f) is V28() V29() ext-real Element of REAL
((Z (#) tan) . (upper_bound f)) - ((Z (#) tan) . (lower_bound f)) is V28() V29() ext-real Element of REAL
K98(((Z (#) tan) . (lower_bound f))) is V28() V29() ext-real set
K96(((Z (#) tan) . (upper_bound f)),K98(((Z (#) tan) . (lower_bound f)))) is V28() V29() ext-real set
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom g is V55() V56() V57() Element of K19(REAL)
integral (g,f) is V28() V29() ext-real Element of REAL
f2 is V55() V56() V57() open Element of K19(REAL)
dom (A (#) (sin / cos)) is V55() V56() V57() Element of K19(REAL)
dom (Z / (cos ^2)) is V55() V56() V57() Element of K19(REAL)
(dom (A (#) (sin / cos))) /\ (dom (Z / (cos ^2))) is V55() V56() V57() Element of K19(REAL)
dom (sin / cos) is V55() V56() V57() Element of K19(REAL)
dom Z is V55() V56() V57() Element of K19(REAL)
dom (cos ^2) is non empty V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(cos ^2) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (cos ^2)) \ ((cos ^2) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom Z) /\ ((dom (cos ^2)) \ ((cos ^2) " {0})) is V55() V56() V57() Element of K19(REAL)
(dom Z) /\ (dom tan) is V55() V56() V57() Element of K19(REAL)
dom (Z (#) tan) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
cos . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
(cos ^2) . x is V28() V29() ext-real Element of REAL
(cos ^2) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((cos ^2) ^) is V55() V56() V57() Element of K19(REAL)
g | f2 is Relation-like REAL -defined f2 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
sin . x is V28() V29() ext-real Element of REAL
A * (sin . x) is V28() V29() ext-real Element of REAL
cos . x is V28() V29() ext-real Element of REAL
(A * (sin . x)) / (cos . x) is V28() V29() ext-real Element of REAL
K99((cos . x)) is V28() V29() ext-real set
K97((A * (sin . x)),K99((cos . x))) is V28() V29() ext-real set
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
(cos . x) ^2 is V28() V29() ext-real Element of REAL
K97((cos . x),(cos . x)) is V28() V29() ext-real set
((A * x) + f1) / ((cos . x) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . x) ^2)) is V28() V29() ext-real set
K97(((A * x) + f1),K99(((cos . x) ^2))) is V28() V29() ext-real set
((A * (sin . x)) / (cos . x)) + (((A * x) + f1) / ((cos . x) ^2)) is V28() V29() ext-real Element of REAL
((A (#) (sin / cos)) + (Z / (cos ^2))) . x is V28() V29() ext-real Element of REAL
(A (#) (sin / cos)) . x is V28() V29() ext-real Element of REAL
(Z / (cos ^2)) . x is V28() V29() ext-real Element of REAL
((A (#) (sin / cos)) . x) + ((Z / (cos ^2)) . x) is V28() V29() ext-real Element of REAL
(sin / cos) . x is V28() V29() ext-real Element of REAL
A * ((sin / cos) . x) is V28() V29() ext-real Element of REAL
(A * ((sin / cos) . x)) + ((Z / (cos ^2)) . x) is V28() V29() ext-real Element of REAL
(sin . x) / (cos . x) is V28() V29() ext-real Element of REAL
K97((sin . x),K99((cos . x))) is V28() V29() ext-real set
A * ((sin . x) / (cos . x)) is V28() V29() ext-real Element of REAL
(A * ((sin . x) / (cos . x))) + ((Z / (cos ^2)) . x) is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
(cos ^2) . x is V28() V29() ext-real Element of REAL
(Z . x) / ((cos ^2) . x) is V28() V29() ext-real Element of REAL
K99(((cos ^2) . x)) is V28() V29() ext-real set
K97((Z . x),K99(((cos ^2) . x))) is V28() V29() ext-real set
((A * (sin . x)) / (cos . x)) + ((Z . x) / ((cos ^2) . x)) is V28() V29() ext-real Element of REAL
(Z . x) / ((cos . x) ^2) is V28() V29() ext-real Element of REAL
K97((Z . x),K99(((cos . x) ^2))) is V28() V29() ext-real set
((A * (sin . x)) / (cos . x)) + ((Z . x) / ((cos . x) ^2)) is V28() V29() ext-real Element of REAL
(Z (#) tan) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((Z (#) tan) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((Z (#) tan) `| f2) . x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
sin . x is V28() V29() ext-real Element of REAL
A * (sin . x) is V28() V29() ext-real Element of REAL
cos . x is V28() V29() ext-real Element of REAL
(A * (sin . x)) / (cos . x) is V28() V29() ext-real Element of REAL
K99((cos . x)) is V28() V29() ext-real set
K97((A * (sin . x)),K99((cos . x))) is V28() V29() ext-real set
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
(cos . x) ^2 is V28() V29() ext-real Element of REAL
K97((cos . x),(cos . x)) is V28() V29() ext-real set
((A * x) + f1) / ((cos . x) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . x) ^2)) is V28() V29() ext-real set
K97(((A * x) + f1),K99(((cos . x) ^2))) is V28() V29() ext-real set
((A * (sin . x)) / (cos . x)) + (((A * x) + f1) / ((cos . x) ^2)) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
A (#) (cos / sin) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is V28() V29() ext-real Element of REAL
f is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f is V28() V29() ext-real Element of REAL
lower_bound f is V28() V29() ext-real Element of REAL
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z / (sin ^2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(A (#) (cos / sin)) - (Z / (sin ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (Z / (sin ^2)) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (Z / (sin ^2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(A (#) (cos / sin)) + (- (Z / (sin ^2))) is Relation-like REAL -defined V6() V34() V35() V36() set
Z (#) cot is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(Z (#) cot) . (upper_bound f) is V28() V29() ext-real Element of REAL
(Z (#) cot) . (lower_bound f) is V28() V29() ext-real Element of REAL
((Z (#) cot) . (upper_bound f)) - ((Z (#) cot) . (lower_bound f)) is V28() V29() ext-real Element of REAL
K98(((Z (#) cot) . (lower_bound f))) is V28() V29() ext-real set
K96(((Z (#) cot) . (upper_bound f)),K98(((Z (#) cot) . (lower_bound f)))) is V28() V29() ext-real set
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom g is V55() V56() V57() Element of K19(REAL)
integral (g,f) is V28() V29() ext-real Element of REAL
f2 is V55() V56() V57() open Element of K19(REAL)
dom (A (#) (cos / sin)) is V55() V56() V57() Element of K19(REAL)
- (Z / (sin ^2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (- (Z / (sin ^2))) is V55() V56() V57() Element of K19(REAL)
(dom (A (#) (cos / sin))) /\ (dom (- (Z / (sin ^2)))) is V55() V56() V57() Element of K19(REAL)
dom (cos / sin) is V55() V56() V57() Element of K19(REAL)
dom (Z / (sin ^2)) is V55() V56() V57() Element of K19(REAL)
dom Z is V55() V56() V57() Element of K19(REAL)
dom (sin ^2) is non empty V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
(sin ^2) " {0} is V55() V56() V57() Element of K19(REAL)
(dom (sin ^2)) \ ((sin ^2) " {0}) is V55() V56() V57() Element of K19(REAL)
(dom Z) /\ ((dom (sin ^2)) \ ((sin ^2) " {0})) is V55() V56() V57() Element of K19(REAL)
(dom Z) /\ (dom cot) is V55() V56() V57() Element of K19(REAL)
dom (Z (#) cot) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
sin . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
(sin ^2) . x is V28() V29() ext-real Element of REAL
(sin ^2) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((sin ^2) ^) is V55() V56() V57() Element of K19(REAL)
g | f2 is Relation-like REAL -defined f2 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
cos . x is V28() V29() ext-real Element of REAL
A * (cos . x) is V28() V29() ext-real Element of REAL
sin . x is V28() V29() ext-real Element of REAL
(A * (cos . x)) / (sin . x) is V28() V29() ext-real Element of REAL
K99((sin . x)) is V28() V29() ext-real set
K97((A * (cos . x)),K99((sin . x))) is V28() V29() ext-real set
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
(sin . x) ^2 is V28() V29() ext-real Element of REAL
K97((sin . x),(sin . x)) is V28() V29() ext-real set
((A * x) + f1) / ((sin . x) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . x) ^2)) is V28() V29() ext-real set
K97(((A * x) + f1),K99(((sin . x) ^2))) is V28() V29() ext-real set
((A * (cos . x)) / (sin . x)) - (((A * x) + f1) / ((sin . x) ^2)) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / ((sin . x) ^2))) is V28() V29() ext-real set
K96(((A * (cos . x)) / (sin . x)),K98((((A * x) + f1) / ((sin . x) ^2)))) is V28() V29() ext-real set
((A (#) (cos / sin)) - (Z / (sin ^2))) . x is V28() V29() ext-real Element of REAL
(A (#) (cos / sin)) . x is V28() V29() ext-real Element of REAL
(Z / (sin ^2)) . x is V28() V29() ext-real Element of REAL
((A (#) (cos / sin)) . x) - ((Z / (sin ^2)) . x) is V28() V29() ext-real Element of REAL
K98(((Z / (sin ^2)) . x)) is V28() V29() ext-real set
K96(((A (#) (cos / sin)) . x),K98(((Z / (sin ^2)) . x))) is V28() V29() ext-real set
(cos / sin) . x is V28() V29() ext-real Element of REAL
A * ((cos / sin) . x) is V28() V29() ext-real Element of REAL
(A * ((cos / sin) . x)) - ((Z / (sin ^2)) . x) is V28() V29() ext-real Element of REAL
K96((A * ((cos / sin) . x)),K98(((Z / (sin ^2)) . x))) is V28() V29() ext-real set
(cos . x) / (sin . x) is V28() V29() ext-real Element of REAL
K97((cos . x),K99((sin . x))) is V28() V29() ext-real set
A * ((cos . x) / (sin . x)) is V28() V29() ext-real Element of REAL
(A * ((cos . x) / (sin . x))) - ((Z / (sin ^2)) . x) is V28() V29() ext-real Element of REAL
K96((A * ((cos . x) / (sin . x))),K98(((Z / (sin ^2)) . x))) is V28() V29() ext-real set
Z . x is V28() V29() ext-real Element of REAL
(sin ^2) . x is V28() V29() ext-real Element of REAL
(Z . x) / ((sin ^2) . x) is V28() V29() ext-real Element of REAL
K99(((sin ^2) . x)) is V28() V29() ext-real set
K97((Z . x),K99(((sin ^2) . x))) is V28() V29() ext-real set
((A * (cos . x)) / (sin . x)) - ((Z . x) / ((sin ^2) . x)) is V28() V29() ext-real Element of REAL
K98(((Z . x) / ((sin ^2) . x))) is V28() V29() ext-real set
K96(((A * (cos . x)) / (sin . x)),K98(((Z . x) / ((sin ^2) . x)))) is V28() V29() ext-real set
(Z . x) / ((sin . x) ^2) is V28() V29() ext-real Element of REAL
K97((Z . x),K99(((sin . x) ^2))) is V28() V29() ext-real set
((A * (cos . x)) / (sin . x)) - ((Z . x) / ((sin . x) ^2)) is V28() V29() ext-real Element of REAL
K98(((Z . x) / ((sin . x) ^2))) is V28() V29() ext-real set
K96(((A * (cos . x)) / (sin . x)),K98(((Z . x) / ((sin . x) ^2)))) is V28() V29() ext-real set
(Z (#) cot) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((Z (#) cot) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((Z (#) cot) `| f2) . x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
cos . x is V28() V29() ext-real Element of REAL
A * (cos . x) is V28() V29() ext-real Element of REAL
sin . x is V28() V29() ext-real Element of REAL
(A * (cos . x)) / (sin . x) is V28() V29() ext-real Element of REAL
K99((sin . x)) is V28() V29() ext-real set
K97((A * (cos . x)),K99((sin . x))) is V28() V29() ext-real set
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
(sin . x) ^2 is V28() V29() ext-real Element of REAL
K97((sin . x),(sin . x)) is V28() V29() ext-real set
((A * x) + f1) / ((sin . x) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . x) ^2)) is V28() V29() ext-real set
K97(((A * x) + f1),K99(((sin . x) ^2))) is V28() V29() ext-real set
((A * (cos . x)) / (sin . x)) - (((A * x) + f1) / ((sin . x) ^2)) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) / ((sin . x) ^2))) is V28() V29() ext-real set
K96(((A * (cos . x)) / (sin . x)),K98((((A * x) + f1) / ((sin . x) ^2)))) is V28() V29() ext-real set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
id f is Relation-like REAL -defined f -defined REAL -valued f -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
tan - (id f) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (id f) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (id f) is Relation-like REAL -defined V6() V34() V35() V36() set
tan + (- (id f)) is Relation-like REAL -defined V6() V34() V35() V36() set
dom (tan - (id f)) is V55() V56() V57() Element of K19(f)
K19(f) is 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() V29() ext-real set
K96(((tan - (id f)) . (upper_bound A)),K98(((tan - (id f)) . (lower_bound A)))) is V28() V29() ext-real set
(tan - (id f)) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((tan - (id f)) `| f) is V55() V56() V57() 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
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 V28() V29() ext-real set
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 V28() V29() ext-real set
((sin . Z) ^2) / ((cos . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((cos . Z) ^2)) is V28() V29() ext-real set
K97(((sin . Z) ^2),K99(((cos . Z) ^2))) is V28() V29() ext-real set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
id f is Relation-like REAL -defined f -defined REAL -valued f -valued V6() V7() total V34() V35() V36() Element of K19(K20(REAL,REAL))
(- cot) - (id f) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (id f) is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) (id f) is Relation-like REAL -defined V6() V34() V35() V36() set
(- cot) + (- (id f)) is Relation-like REAL -defined V6() V34() V35() V36() set
dom ((- cot) - (id f)) is V55() V56() V57() Element of K19(f)
K19(f) is set
((- 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() V29() ext-real set
K96((((- cot) - (id f)) . (upper_bound A)),K98((((- cot) - (id f)) . (lower_bound A)))) is V28() V29() ext-real set
((- cot) - (id f)) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((- cot) - (id f)) `| f) is V55() V56() V57() 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
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 V28() V29() ext-real set
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 V28() V29() ext-real set
((cos . Z) ^2) / ((sin . Z) ^2) is V28() V29() ext-real Element of REAL
K99(((sin . Z) ^2)) is V28() V29() ext-real set
K97(((cos . Z) ^2),K99(((sin . Z) ^2))) is V28() V29() ext-real set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(arctan * ln) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(arctan * ln) . (lower_bound A) is V28() V29() ext-real Element of REAL
((arctan * ln) . (upper_bound A)) - ((arctan * ln) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((arctan * ln) . (lower_bound A))) is V28() V29() ext-real set
K96(((arctan * ln) . (upper_bound A)),K98(((arctan * ln) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
ln . Z is V28() V29() ext-real Element of REAL
(arctan * ln) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((arctan * ln) `| f) is V55() V56() V57() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((arctan * 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
(ln . Z) ^2 is V28() V29() ext-real Element of REAL
K97((ln . Z),(ln . Z)) is V28() V29() ext-real set
1 + ((ln . Z) ^2) is V28() V29() ext-real Element of REAL
Z * (1 + ((ln . Z) ^2)) is V28() V29() ext-real Element of REAL
1 / (Z * (1 + ((ln . Z) ^2))) is V28() V29() ext-real Element of REAL
K99((Z * (1 + ((ln . Z) ^2)))) is V28() V29() ext-real set
K97(1,K99((Z * (1 + ((ln . Z) ^2))))) is V28() V29() ext-real set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(arccot * ln) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(arccot * ln) . (lower_bound A) is V28() V29() ext-real Element of REAL
((arccot * ln) . (upper_bound A)) - ((arccot * ln) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((arccot * ln) . (lower_bound A))) is V28() V29() ext-real set
K96(((arccot * ln) . (upper_bound A)),K98(((arccot * ln) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f1,A) is V28() V29() ext-real Element of REAL
f is V55() V56() V57() open Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
ln . Z is V28() V29() ext-real Element of REAL
(arccot * ln) `| f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((arccot * ln) `| f) is V55() V56() V57() Element of K19(REAL)
Z is V28() V29() ext-real Element of REAL
((arccot * 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
(ln . Z) ^2 is V28() V29() ext-real Element of REAL
K97((ln . Z),(ln . Z)) is V28() V29() ext-real set
1 + ((ln . Z) ^2) is V28() V29() ext-real Element of REAL
Z * (1 + ((ln . Z) ^2)) is V28() V29() ext-real Element of REAL
1 / (Z * (1 + ((ln . Z) ^2))) is V28() V29() ext-real Element of REAL
K99((Z * (1 + ((ln . Z) ^2)))) is V28() V29() ext-real set
K97(1,K99((Z * (1 + ((ln . Z) ^2))))) is V28() V29() ext-real set
- (1 / (Z * (1 + ((ln . Z) ^2)))) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
f is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f is V28() V29() ext-real Element of REAL
lower_bound f is V28() V29() ext-real Element of REAL
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V55() V56() V57() Element of K19(REAL)
Z | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (Z,f) is V28() V29() ext-real Element of REAL
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arcsin * g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (arcsin * g) is V55() V56() V57() Element of K19(REAL)
(arcsin * g) . (upper_bound f) is V28() V29() ext-real Element of REAL
(arcsin * g) . (lower_bound f) is V28() V29() ext-real Element of REAL
((arcsin * g) . (upper_bound f)) - ((arcsin * g) . (lower_bound f)) is V28() V29() ext-real Element of REAL
K98(((arcsin * g) . (lower_bound f))) is V28() V29() ext-real set
K96(((arcsin * g) . (upper_bound f)),K98(((arcsin * g) . (lower_bound f)))) is V28() V29() ext-real set
f2 is V55() V56() V57() open Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
(arcsin * g) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((arcsin * g) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((arcsin * g) `| f2) . x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
((A * x) + f1) ^2 is V28() V29() ext-real Element of REAL
K97(((A * x) + f1),((A * x) + f1)) is V28() V29() ext-real set
1 - (((A * x) + f1) ^2) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) ^2)) is V28() V29() ext-real set
K96(1,K98((((A * x) + f1) ^2))) is V28() V29() ext-real set
sqrt (1 - (((A * x) + f1) ^2)) is V28() V29() ext-real Element of REAL
A / (sqrt (1 - (((A * x) + f1) ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (((A * x) + f1) ^2)))) is V28() V29() ext-real set
K97(A,K99((sqrt (1 - (((A * x) + f1) ^2))))) is V28() V29() ext-real set
A is V28() V29() ext-real Element of REAL
f1 is V28() V29() ext-real Element of REAL
f is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f is V28() V29() ext-real Element of REAL
lower_bound f is V28() V29() ext-real Element of REAL
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom Z is V55() V56() V57() Element of K19(REAL)
Z | f is Relation-like REAL -defined f -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (Z,f) is V28() V29() ext-real Element of REAL
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
arccos * g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (arccos * g) is V55() V56() V57() Element of K19(REAL)
- (arccos * g) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (arccos * g) is Relation-like REAL -defined V6() V34() V35() V36() set
(- (arccos * g)) . (upper_bound f) is V28() V29() ext-real Element of REAL
(- (arccos * g)) . (lower_bound f) is V28() V29() ext-real Element of REAL
((- (arccos * g)) . (upper_bound f)) - ((- (arccos * g)) . (lower_bound f)) is V28() V29() ext-real Element of REAL
K98(((- (arccos * g)) . (lower_bound f))) is V28() V29() ext-real set
K96(((- (arccos * g)) . (upper_bound f)),K98(((- (arccos * g)) . (lower_bound f)))) is V28() V29() ext-real set
f2 is V55() V56() V57() open Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
dom (- (arccos * g)) is V55() V56() V57() Element of K19(REAL)
(- 1) (#) (arccos * g) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom g is V55() V56() V57() Element of K19(REAL)
x is set
g `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
(g `| f2) . x is V28() V29() ext-real Element of REAL
(- (arccos * g)) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
((- (arccos * g)) `| f2) . x is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
((A * x) + f1) ^2 is V28() V29() ext-real Element of REAL
K97(((A * x) + f1),((A * x) + f1)) is V28() V29() ext-real set
1 - (((A * x) + f1) ^2) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) ^2)) is V28() V29() ext-real set
K96(1,K98((((A * x) + f1) ^2))) is V28() V29() ext-real set
sqrt (1 - (((A * x) + f1) ^2)) is V28() V29() ext-real Element of REAL
A / (sqrt (1 - (((A * x) + f1) ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (((A * x) + f1) ^2)))) is V28() V29() ext-real set
K97(A,K99((sqrt (1 - (((A * x) + f1) ^2))))) is V28() V29() ext-real set
g . x is V28() V29() ext-real Element of REAL
diff ((- (arccos * g)),x) is V28() V29() ext-real Element of REAL
diff ((arccos * g),x) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((arccos * g),x)) is V28() V29() ext-real Element of REAL
diff (g,x) is V28() V29() ext-real Element of REAL
(g . x) ^2 is V28() V29() ext-real Element of REAL
K97((g . x),(g . x)) is V28() V29() ext-real set
1 - ((g . x) ^2) is V28() V29() ext-real Element of REAL
K98(((g . x) ^2)) is V28() V29() ext-real set
K96(1,K98(((g . x) ^2))) is V28() V29() ext-real set
sqrt (1 - ((g . x) ^2)) is V28() V29() ext-real Element of REAL
(diff (g,x)) / (sqrt (1 - ((g . x) ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - ((g . x) ^2)))) is V28() V29() ext-real set
K97((diff (g,x)),K99((sqrt (1 - ((g . x) ^2))))) is V28() V29() ext-real set
- ((diff (g,x)) / (sqrt (1 - ((g . x) ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * (- ((diff (g,x)) / (sqrt (1 - ((g . x) ^2))))) is V28() V29() ext-real Element of REAL
(g `| f2) . x is V28() V29() ext-real Element of REAL
((g `| f2) . x) / (sqrt (1 - ((g . x) ^2))) is V28() V29() ext-real Element of REAL
K97(((g `| f2) . x),K99((sqrt (1 - ((g . x) ^2))))) is V28() V29() ext-real set
- (((g `| f2) . x) / (sqrt (1 - ((g . x) ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * (- (((g `| f2) . x) / (sqrt (1 - ((g . x) ^2))))) is V28() V29() ext-real Element of REAL
A / (sqrt (1 - ((g . x) ^2))) is V28() V29() ext-real Element of REAL
K97(A,K99((sqrt (1 - ((g . x) ^2))))) is V28() V29() ext-real set
- (A / (sqrt (1 - ((g . x) ^2)))) is V28() V29() ext-real Element of REAL
(- 1) * (- (A / (sqrt (1 - ((g . x) ^2))))) is V28() V29() ext-real Element of REAL
dom ((- (arccos * g)) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- (arccos * g)) `| f2) . x is V28() V29() ext-real Element of REAL
Z . x is V28() V29() ext-real Element of REAL
A * x is V28() V29() ext-real Element of REAL
(A * x) + f1 is V28() V29() ext-real Element of REAL
((A * x) + f1) ^2 is V28() V29() ext-real Element of REAL
K97(((A * x) + f1),((A * x) + f1)) is V28() V29() ext-real set
1 - (((A * x) + f1) ^2) is V28() V29() ext-real Element of REAL
K98((((A * x) + f1) ^2)) is V28() V29() ext-real set
K96(1,K98((((A * x) + f1) ^2))) is V28() V29() ext-real set
sqrt (1 - (((A * x) + f1) ^2)) is V28() V29() ext-real Element of REAL
A / (sqrt (1 - (((A * x) + f1) ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (((A * x) + f1) ^2)))) is V28() V29() ext-real set
K97(A,K99((sqrt (1 - (((A * x) + f1) ^2))))) is V28() V29() ext-real set
- (1 / 2) is V28() V29() ext-real non positive V67() Element of RAT
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(#R (1 / 2)) * f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#R (1 / 2)) * f1) is V55() V56() V57() Element of K19(REAL)
- ((#R (1 / 2)) * f1) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((#R (1 / 2)) * f1) is Relation-like REAL -defined V6() V34() V35() V36() set
(- ((#R (1 / 2)) * f1)) . (upper_bound A) is V28() V29() ext-real Element of REAL
(- ((#R (1 / 2)) * f1)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- ((#R (1 / 2)) * f1)) . (lower_bound A))) is V28() V29() ext-real set
K96(((- ((#R (1 / 2)) * f1)) . (upper_bound A)),K98(((- ((#R (1 / 2)) * f1)) . (lower_bound A)))) is V28() V29() ext-real set
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f - Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- Z is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) Z is Relation-like REAL -defined V6() V34() V35() V36() set
f + (- Z) is Relation-like REAL -defined V6() V34() V35() V36() set
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom g is V55() V56() V57() Element of K19(REAL)
g | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (g,A) is V28() V29() ext-real Element of REAL
f2 is V55() V56() V57() open Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
dom (- ((#R (1 / 2)) * f1)) is V55() V56() V57() Element of K19(REAL)
dom f1 is V55() V56() V57() Element of K19(REAL)
x is set
(- 1) (#) Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + ((- 1) (#) Z) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (f + ((- 1) (#) Z)) is V55() V56() V57() Element of K19(REAL)
(- 1) (#) ((#R (1 / 2)) * f1) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
f . x is V28() V29() ext-real Element of REAL
0 * x is V28() V29() ext-real Element of REAL
1 + (0 * x) is V28() V29() ext-real Element of REAL
f1 `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
2 * (- 1) is V28() V29() V30() ext-real non positive V67() Element of INT
x is V28() V29() ext-real Element of REAL
(f1 `| f2) . x is V28() V29() ext-real Element of REAL
(2 * (- 1)) * x is V28() V29() ext-real Element of REAL
0 + ((2 * (- 1)) * x) is V28() V29() ext-real Element of REAL
(- ((#R (1 / 2)) * f1)) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
((- ((#R (1 / 2)) * f1)) `| f2) . 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
K98((x #Z 2)) is V28() V29() ext-real set
K96(1,K98((x #Z 2))) is V28() V29() ext-real set
(1 - (x #Z 2)) #R (- (1 / 2)) is V28() V29() ext-real set
x * ((1 - (x #Z 2)) #R (- (1 / 2))) is V28() V29() ext-real Element of REAL
dom (f - Z) is V55() V56() V57() Element of K19(REAL)
(f - Z) . x is V28() V29() ext-real Element of REAL
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
K98((Z . x)) is V28() V29() ext-real set
K96((f . x),K98((Z . x))) is V28() V29() ext-real set
1 - (Z . x) is V28() V29() ext-real Element of REAL
K96(1,K98((Z . x))) is V28() V29() ext-real set
f1 . x is V28() V29() ext-real Element of REAL
diff ((- ((#R (1 / 2)) * f1)),x) is V28() V29() ext-real Element of REAL
diff (((#R (1 / 2)) * f1),x) is V28() V29() ext-real Element of REAL
(- 1) * (diff (((#R (1 / 2)) * f1),x)) is V28() V29() ext-real Element of REAL
(1 / 2) - 1 is V28() V29() ext-real V67() Element of RAT
K96((1 / 2),K98(1)) is V28() V29() ext-real set
(f1 . x) #R ((1 / 2) - 1) is V28() V29() ext-real set
(1 / 2) * ((f1 . x) #R ((1 / 2) - 1)) is V28() V29() ext-real Element of REAL
diff (f1,x) is V28() V29() ext-real Element of REAL
((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * (diff (f1,x)) is V28() V29() ext-real Element of REAL
(- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * (diff (f1,x))) is V28() V29() ext-real Element of REAL
(f1 `| f2) . x is V28() V29() ext-real Element of REAL
((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * ((f1 `| f2) . x) is V28() V29() ext-real Element of REAL
(- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * ((f1 `| f2) . x)) is V28() V29() ext-real Element of REAL
(2 * (- 1)) * x is V28() V29() ext-real Element of REAL
0 + ((2 * (- 1)) * x) is V28() V29() ext-real Element of REAL
((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * x)) is V28() V29() ext-real Element of REAL
(- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * x))) is V28() V29() ext-real Element of REAL
dom ((- ((#R (1 / 2)) * f1)) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- ((#R (1 / 2)) * f1)) `| f2) . x is V28() V29() ext-real Element of REAL
g . 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
K98((x #Z 2)) is V28() V29() ext-real set
K96(1,K98((x #Z 2))) is V28() V29() ext-real set
(1 - (x #Z 2)) #R (- (1 / 2)) is V28() V29() ext-real set
x * ((1 - (x #Z 2)) #R (- (1 / 2))) is V28() V29() ext-real Element of REAL
A is V28() V29() ext-real Element of REAL
A ^2 is V28() V29() ext-real Element of REAL
K97(A,A) is V28() V29() ext-real set
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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 Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(#R (1 / 2)) * f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#R (1 / 2)) * f) is V55() V56() V57() Element of K19(REAL)
- ((#R (1 / 2)) * f) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) ((#R (1 / 2)) * f) is Relation-like REAL -defined V6() V34() V35() V36() set
(- ((#R (1 / 2)) * f)) . (upper_bound f1) is V28() V29() ext-real Element of REAL
(- ((#R (1 / 2)) * f)) . (lower_bound f1) is V28() V29() ext-real Element of REAL
((- ((#R (1 / 2)) * f)) . (upper_bound f1)) - ((- ((#R (1 / 2)) * f)) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98(((- ((#R (1 / 2)) * f)) . (lower_bound f1))) is V28() V29() ext-real set
K96(((- ((#R (1 / 2)) * f)) . (upper_bound f1)),K98(((- ((#R (1 / 2)) * f)) . (lower_bound f1)))) is V28() V29() ext-real set
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z - g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- g is Relation-like REAL -defined V6() V34() V35() V36() set
K98(1) (#) g is Relation-like REAL -defined V6() V34() V35() V36() set
Z + (- g) is Relation-like REAL -defined V6() V34() V35() V36() set
f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f2 is V55() V56() V57() Element of K19(REAL)
f2 | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f2,f1) is V28() V29() ext-real Element of REAL
x is V55() V56() V57() open Element of K19(REAL)
y is V28() V29() ext-real Element of REAL
Z . y is V28() V29() ext-real Element of REAL
f . y is V28() V29() ext-real Element of REAL
dom (- ((#R (1 / 2)) * f)) is V55() V56() V57() Element of K19(REAL)
dom f is V55() V56() V57() Element of K19(REAL)
y is set
(- 1) (#) g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z + ((- 1) (#) g) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (Z + ((- 1) (#) g)) is V55() V56() V57() Element of K19(REAL)
(- 1) (#) ((#R (1 / 2)) * f) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
y is V28() V29() ext-real Element of REAL
Z . y is V28() V29() ext-real Element of REAL
0 * y is V28() V29() ext-real Element of REAL
(A ^2) + (0 * y) is V28() V29() ext-real Element of REAL
f `| x is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
2 * (- 1) is V28() V29() V30() ext-real non positive V67() Element of INT
y is V28() V29() ext-real Element of REAL
(f `| x) . y is V28() V29() ext-real Element of REAL
(2 * (- 1)) * y is V28() V29() ext-real Element of REAL
0 + ((2 * (- 1)) * y) is V28() V29() ext-real Element of REAL
(- ((#R (1 / 2)) * f)) `| x is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
y is V28() V29() ext-real Element of REAL
((- ((#R (1 / 2)) * f)) `| x) . y is V28() V29() ext-real Element of REAL
y #Z 2 is V28() V29() ext-real Element of REAL
(A ^2) - (y #Z 2) is V28() V29() ext-real Element of REAL
K98((y #Z 2)) is V28() V29() ext-real set
K96((A ^2),K98((y #Z 2))) is V28() V29() ext-real set
((A ^2) - (y #Z 2)) #R (- (1 / 2)) is V28() V29() ext-real set
y * (((A ^2) - (y #Z 2)) #R (- (1 / 2))) is V28() V29() ext-real Element of REAL
dom (Z - g) is V55() V56() V57() Element of K19(REAL)
(Z - g) . y is V28() V29() ext-real Element of REAL
Z . y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
(Z . y) - (g . y) is V28() V29() ext-real Element of REAL
K98((g . y)) is V28() V29() ext-real set
K96((Z . y),K98((g . y))) is V28() V29() ext-real set
(A ^2) - (g . y) is V28() V29() ext-real Element of REAL
K96((A ^2),K98((g . y))) is V28() V29() ext-real set
f . y is V28() V29() ext-real Element of REAL
diff ((- ((#R (1 / 2)) * f)),y) is V28() V29() ext-real Element of REAL
diff (((#R (1 / 2)) * f),y) is V28() V29() ext-real Element of REAL
(- 1) * (diff (((#R (1 / 2)) * f),y)) is V28() V29() ext-real Element of REAL
(1 / 2) - 1 is V28() V29() ext-real V67() Element of RAT
K96((1 / 2),K98(1)) is V28() V29() ext-real set
(f . y) #R ((1 / 2) - 1) is V28() V29() ext-real set
(1 / 2) * ((f . y) #R ((1 / 2) - 1)) is V28() V29() ext-real Element of REAL
diff (f,y) is V28() V29() ext-real Element of REAL
((1 / 2) * ((f . y) #R ((1 / 2) - 1))) * (diff (f,y)) is V28() V29() ext-real Element of REAL
(- 1) * (((1 / 2) * ((f . y) #R ((1 / 2) - 1))) * (diff (f,y))) is V28() V29() ext-real Element of REAL
(f `| x) . y is V28() V29() ext-real Element of REAL
((1 / 2) * ((f . y) #R ((1 / 2) - 1))) * ((f `| x) . y) is V28() V29() ext-real Element of REAL
(- 1) * (((1 / 2) * ((f . y) #R ((1 / 2) - 1))) * ((f `| x) . y)) is V28() V29() ext-real Element of REAL
(2 * (- 1)) * y is V28() V29() ext-real Element of REAL
0 + ((2 * (- 1)) * y) is V28() V29() ext-real Element of REAL
((1 / 2) * ((f . y) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * y)) is V28() V29() ext-real Element of REAL
(- 1) * (((1 / 2) * ((f . y) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * y))) is V28() V29() ext-real Element of REAL
dom ((- ((#R (1 / 2)) * f)) `| x) is V55() V56() V57() Element of K19(REAL)
y is V28() V29() ext-real Element of REAL
((- ((#R (1 / 2)) * f)) `| x) . y is V28() V29() ext-real Element of REAL
f2 . y is V28() V29() ext-real Element of REAL
y #Z 2 is V28() V29() ext-real Element of REAL
(A ^2) - (y #Z 2) is V28() V29() ext-real Element of REAL
K98((y #Z 2)) is V28() V29() ext-real set
K96((A ^2),K98((y #Z 2))) is V28() V29() ext-real set
((A ^2) - (y #Z 2)) #R (- (1 / 2)) is V28() V29() ext-real set
y * (((A ^2) - (y #Z 2)) #R (- (1 / 2))) is V28() V29() ext-real Element of REAL
A is V21() V22() V23() V27() V28() V29() V30() ext-real non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
A + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
#Z A is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * (sin ^) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * (sin ^)) is V55() V56() V57() Element of K19(REAL)
1 / A is V28() V29() ext-real non negative V67() Element of RAT
K99(A) is V28() V29() ext-real non negative set
K97(1,K99(A)) is V28() V29() ext-real non negative set
- (1 / A) is V28() V29() ext-real non positive V67() Element of RAT
(- (1 / A)) (#) ((#Z A) * (sin ^)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
((- (1 / A)) (#) ((#Z A) * (sin ^))) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
((- (1 / A)) (#) ((#Z A) * (sin ^))) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((- (1 / A)) (#) ((#Z A) * (sin ^))) . (upper_bound f1)) - (((- (1 / A)) (#) ((#Z A) * (sin ^))) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((- (1 / A)) (#) ((#Z A) * (sin ^))) . (lower_bound f1))) is V28() V29() ext-real set
K96((((- (1 / A)) (#) ((#Z A) * (sin ^))) . (upper_bound f1)),K98((((- (1 / A)) (#) ((#Z A) * (sin ^))) . (lower_bound f1)))) is V28() V29() ext-real set
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
f | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,f1) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom ((- (1 / A)) (#) ((#Z A) * (sin ^))) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
((- (1 / A)) (#) ((#Z A) * (sin ^))) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((- (1 / A)) (#) ((#Z A) * (sin ^))) `| Z) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
(((- (1 / A)) (#) ((#Z A) * (sin ^))) `| Z) . g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
cos . g is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
(sin . g) #Z (A + 1) is V28() V29() ext-real Element of REAL
(cos . g) / ((sin . g) #Z (A + 1)) is V28() V29() ext-real Element of REAL
K99(((sin . g) #Z (A + 1))) is V28() V29() ext-real set
K97((cos . g),K99(((sin . g) #Z (A + 1)))) is V28() V29() ext-real set
A is V21() V22() V23() V27() V28() V29() V30() ext-real non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
A + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
#Z A is Relation-like REAL -defined REAL -valued V6() non empty total V18( REAL , REAL ) V34() V35() V36() Element of K19(K20(REAL,REAL))
(#Z A) * (cos ^) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((#Z A) * (cos ^)) is V55() V56() V57() Element of K19(REAL)
1 / A is V28() V29() ext-real non negative V67() Element of RAT
K99(A) is V28() V29() ext-real non negative set
K97(1,K99(A)) is V28() V29() ext-real non negative set
(1 / A) (#) ((#Z A) * (cos ^)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound f1 is V28() V29() ext-real Element of REAL
((1 / A) (#) ((#Z A) * (cos ^))) . (upper_bound f1) is V28() V29() ext-real Element of REAL
lower_bound f1 is V28() V29() ext-real Element of REAL
((1 / A) (#) ((#Z A) * (cos ^))) . (lower_bound f1) is V28() V29() ext-real Element of REAL
(((1 / A) (#) ((#Z A) * (cos ^))) . (upper_bound f1)) - (((1 / A) (#) ((#Z A) * (cos ^))) . (lower_bound f1)) is V28() V29() ext-real Element of REAL
K98((((1 / A) (#) ((#Z A) * (cos ^))) . (lower_bound f1))) is V28() V29() ext-real set
K96((((1 / A) (#) ((#Z A) * (cos ^))) . (upper_bound f1)),K98((((1 / A) (#) ((#Z A) * (cos ^))) . (lower_bound f1)))) is V28() V29() ext-real set
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
f | f1 is Relation-like REAL -defined f1 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
integral (f,f1) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom ((1 / A) (#) ((#Z A) * (cos ^))) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
cos . g is V28() V29() ext-real Element of REAL
((1 / A) (#) ((#Z A) * (cos ^))) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (((1 / A) (#) ((#Z A) * (cos ^))) `| Z) is V55() V56() V57() Element of K19(REAL)
g is V28() V29() ext-real Element of REAL
(((1 / A) (#) ((#Z A) * (cos ^))) `| Z) . g is V28() V29() ext-real Element of REAL
f . g is V28() V29() ext-real Element of REAL
sin . g is V28() V29() ext-real Element of REAL
cos . g is V28() V29() ext-real Element of REAL
(cos . g) #Z (A + 1) is V28() V29() ext-real Element of REAL
(sin . g) / ((cos . g) #Z (A + 1)) is V28() V29() ext-real Element of REAL
K99(((cos . g) #Z (A + 1))) is V28() V29() ext-real set
K97((sin . g),K99(((cos . g) #Z (A + 1)))) is V28() V29() ext-real set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed 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() V29() ext-real set
K96(((- (ln * arccot)) . (upper_bound A)),K98(((- (ln * arccot)) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f1 is V55() V56() V57() Element of K19(REAL)
integral (f1,A) is V28() V29() ext-real Element of REAL
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f + Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(f + Z) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((f + Z) ^) / g is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f2 is V55() V56() V57() open Element of K19(REAL)
dom ((f + Z) ^) is V55() V56() V57() Element of K19(REAL)
dom g is V55() V56() V57() Element of K19(REAL)
{0} is V9() non empty V55() V56() V57() V58() V59() V60() Element of K19(REAL)
g " {0} is V55() V56() V57() Element of K19(REAL)
(dom g) \ (g " {0}) is V55() V56() V57() Element of K19(REAL)
(dom ((f + Z) ^)) /\ ((dom g) \ (g " {0})) is V55() V56() V57() 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
g . x is V28() V29() ext-real Element of REAL
f1 | f2 is Relation-like REAL -defined f2 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
g ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom (g ^) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
g . x is V28() V29() ext-real Element of REAL
g | f2 is Relation-like REAL -defined f2 -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
rng (g | f2) is V55() V56() V57() Element of K19(REAL)
x is set
dom (g | f2) is V55() V56() V57() Element of K19(f2)
K19(f2) is set
y is set
(g | f2) . y is V28() V29() ext-real Element of REAL
g . y is V28() V29() ext-real Element of REAL
g .: f2 is V55() V56() V57() Element of K19(REAL)
dom (- (ln * arccot)) is V55() V56() V57() Element of K19(REAL)
(- (ln * arccot)) `| f2 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
((- (ln * arccot)) `| f2) . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 + (x ^2) is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
(1 + (x ^2)) * (arccot . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x ^2)) * (arccot . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x ^2)) * (arccot . x))) is V28() V29() ext-real set
K97(1,K99(((1 + (x ^2)) * (arccot . x)))) is V28() V29() ext-real set
diff ((- (ln * arccot)),x) is V28() V29() ext-real Element of REAL
diff ((ln * arccot),x) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((ln * arccot),x)) is V28() V29() ext-real Element of REAL
diff (arccot,x) is V28() V29() ext-real Element of REAL
(diff (arccot,x)) / (arccot . x) is V28() V29() ext-real Element of REAL
K99((arccot . x)) is V28() V29() ext-real set
K97((diff (arccot,x)),K99((arccot . x))) is V28() V29() ext-real set
(- 1) * ((diff (arccot,x)) / (arccot . x)) is V28() V29() ext-real Element of REAL
1 / (1 + (x ^2)) is V28() V29() ext-real Element of REAL
K99((1 + (x ^2))) is V28() V29() ext-real set
K97(1,K99((1 + (x ^2)))) is V28() V29() ext-real set
- (1 / (1 + (x ^2))) is V28() V29() ext-real Element of REAL
(- (1 / (1 + (x ^2)))) / (arccot . x) is V28() V29() ext-real Element of REAL
K97((- (1 / (1 + (x ^2)))),K99((arccot . x))) is V28() V29() ext-real set
(- 1) * ((- (1 / (1 + (x ^2)))) / (arccot . x)) is V28() V29() ext-real Element of REAL
(1 / (1 + (x ^2))) / (arccot . x) is V28() V29() ext-real Element of REAL
K97((1 / (1 + (x ^2))),K99((arccot . x))) is V28() V29() ext-real set
dom ((- (ln * arccot)) `| f2) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- (ln * arccot)) `| f2) . x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 + (x ^2) is V28() V29() ext-real Element of REAL
arccot . x is V28() V29() ext-real Element of REAL
(1 + (x ^2)) * (arccot . x) is V28() V29() ext-real Element of REAL
1 / ((1 + (x ^2)) * (arccot . x)) is V28() V29() ext-real Element of REAL
K99(((1 + (x ^2)) * (arccot . x))) is V28() V29() ext-real set
K97(1,K99(((1 + (x ^2)) * (arccot . x)))) is V28() V29() ext-real set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(ln * arcsin) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(ln * arcsin) . (lower_bound A) is V28() V29() ext-real Element of REAL
((ln * arcsin) . (upper_bound A)) - ((ln * arcsin) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((ln * arcsin) . (lower_bound A))) is V28() V29() ext-real set
K96(((ln * arcsin) . (upper_bound A)),K98(((ln * arcsin) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 - (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
K98(1) (#) (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
f1 + (- (#Z 2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (f1 - (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) is V55() V56() V57() Element of K19(REAL)
dom ((#R (1 / 2)) * (f1 - (#Z 2))) is V55() V56() V57() Element of K19(REAL)
dom arcsin is V55() V56() V57() Element of K19(REAL)
(dom ((#R (1 / 2)) * (f1 - (#Z 2)))) /\ (dom arcsin) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(f1 - (#Z 2)) . x is V28() V29() ext-real Element of REAL
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
dom (f1 - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
y 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
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((f1 . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
1 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
x #Z (1 + 1) is V28() V29() ext-real Element of REAL
(f1 . x) - (x #Z (1 + 1)) is V28() V29() ext-real Element of REAL
K98((x #Z (1 + 1))) is V28() V29() ext-real set
K96((f1 . x),K98((x #Z (1 + 1)))) is V28() V29() ext-real set
x #Z 1 is V28() V29() ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V28() V29() ext-real Element of REAL
(f1 . x) - ((x #Z 1) * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98(((x #Z 1) * (x #Z 1))) is V28() V29() ext-real set
K96((f1 . x),K98(((x #Z 1) * (x #Z 1)))) is V28() V29() ext-real set
x * (x #Z 1) is V28() V29() ext-real Element of REAL
(f1 . x) - (x * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98((x * (x #Z 1))) is V28() V29() ext-real set
K96((f1 . x),K98((x * (x #Z 1)))) is V28() V29() ext-real set
x * x is V28() V29() ext-real Element of REAL
(f1 . x) - (x * x) is V28() V29() ext-real Element of REAL
K98((x * x)) is V28() V29() ext-real set
K96((f1 . x),K98((x * x))) is V28() V29() ext-real set
1 - (x * x) is V28() V29() ext-real Element of REAL
K96(1,K98((x * x))) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
(f1 - (#Z 2)) . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x is V28() V29() ext-real Element of REAL
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
arcsin . x is V28() V29() ext-real Element of 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 V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
arcsin . x is V28() V29() ext-real Element of REAL
(sqrt (1 - (x ^2))) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99(((sqrt (1 - (x ^2))) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99(((sqrt (1 - (x ^2))) * (arcsin . x)))) is V28() V29() ext-real set
dom (f1 - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
(f1 - (#Z 2)) . x is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) ^) . x is V28() V29() ext-real Element of REAL
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x is V28() V29() ext-real Element of REAL
1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x) is V28() V29() ext-real Element of REAL
K99(((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x)) is V28() V29() ext-real set
K97(1,K99(((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arcsin) . x))) is V28() V29() ext-real set
((#R (1 / 2)) * (f1 - (#Z 2))) . x is V28() V29() ext-real Element of REAL
(((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99(((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99(((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arcsin . x)))) is V28() V29() ext-real set
(#R (1 / 2)) . ((f1 - (#Z 2)) . x) is V28() V29() ext-real Element of REAL
((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / (((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99((((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99((((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arcsin . x)))) is V28() V29() ext-real set
((f1 - (#Z 2)) . x) #R (1 / 2) is V28() V29() ext-real set
(((f1 - (#Z 2)) . x) #R (1 / 2)) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / ((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99(((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99(((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arcsin . x)))) is V28() V29() ext-real 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
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((f1 . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
((f1 . x) - ((#Z 2) . x)) #R (1 / 2) is V28() V29() ext-real set
(((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / ((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99(((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99(((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arcsin . x)))) is V28() V29() ext-real 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
K98((x #Z 2)) is V28() V29() ext-real set
K96((f1 . x),K98((x #Z 2))) is V28() V29() ext-real set
((f1 . x) - (x #Z 2)) #R (1 / 2) is V28() V29() ext-real set
(((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / ((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99(((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99(((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arcsin . x)))) is V28() V29() ext-real set
(f1 . x) - (x ^2) is V28() V29() ext-real Element of REAL
K96((f1 . x),K98((x ^2))) is V28() V29() ext-real set
((f1 . x) - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
(((f1 . x) - (x ^2)) #R (1 / 2)) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / ((((f1 . x) - (x ^2)) #R (1 / 2)) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99(((((f1 . x) - (x ^2)) #R (1 / 2)) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99(((((f1 . x) - (x ^2)) #R (1 / 2)) * (arcsin . x)))) is V28() V29() ext-real set
(1 - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
((1 - (x ^2)) #R (1 / 2)) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / (((1 - (x ^2)) #R (1 / 2)) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99((((1 - (x ^2)) #R (1 / 2)) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99((((1 - (x ^2)) #R (1 / 2)) * (arcsin . x)))) is V28() V29() ext-real set
(ln * arcsin) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom ((ln * arcsin) `| Z) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((ln * arcsin) `| 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 V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
arcsin . x is V28() V29() ext-real Element of REAL
(sqrt (1 - (x ^2))) * (arcsin . x) is V28() V29() ext-real Element of REAL
1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) is V28() V29() ext-real Element of REAL
K99(((sqrt (1 - (x ^2))) * (arcsin . x))) is V28() V29() ext-real set
K97(1,K99(((sqrt (1 - (x ^2))) * (arcsin . x)))) is V28() V29() ext-real set
- (ln * arccos) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
K98(1) (#) (ln * arccos) is Relation-like REAL -defined V6() V34() V35() V36() set
A is non empty V55() V56() V57() closed_interval V86() V87() compact closed Element of K19(REAL)
upper_bound A is V28() V29() ext-real Element of REAL
(- (ln * arccos)) . (upper_bound A) is V28() V29() ext-real Element of REAL
lower_bound A is V28() V29() ext-real Element of REAL
(- (ln * arccos)) . (lower_bound A) is V28() V29() ext-real Element of REAL
((- (ln * arccos)) . (upper_bound A)) - ((- (ln * arccos)) . (lower_bound A)) is V28() V29() ext-real Element of REAL
K98(((- (ln * arccos)) . (lower_bound A))) is V28() V29() ext-real set
K96(((- (ln * arccos)) . (upper_bound A)),K98(((- (ln * arccos)) . (lower_bound A)))) is V28() V29() ext-real set
f1 is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f1 - (#Z 2) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
- (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
K98(1) (#) (#Z 2) is Relation-like REAL -defined V6() total V34() V35() V36() set
f1 + (- (#Z 2)) is Relation-like REAL -defined V6() V34() V35() V36() set
(#R (1 / 2)) * (f1 - (#Z 2)) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^ is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
dom f is V55() V56() V57() Element of K19(REAL)
integral (f,A) is V28() V29() ext-real Element of REAL
Z is V55() V56() V57() open Element of K19(REAL)
dom (((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) is V55() V56() V57() Element of K19(REAL)
dom ((#R (1 / 2)) * (f1 - (#Z 2))) is V55() V56() V57() Element of K19(REAL)
dom arccos is V55() V56() V57() Element of K19(REAL)
(dom ((#R (1 / 2)) * (f1 - (#Z 2)))) /\ (dom arccos) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
(f1 - (#Z 2)) . x is V28() V29() ext-real Element of REAL
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
dom (f1 - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
y 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
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((f1 . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
1 + 1 is non empty V21() V22() V23() V27() V28() V29() V30() ext-real positive non negative V55() V56() V57() V58() V59() V60() V67() Element of NAT
x #Z (1 + 1) is V28() V29() ext-real Element of REAL
(f1 . x) - (x #Z (1 + 1)) is V28() V29() ext-real Element of REAL
K98((x #Z (1 + 1))) is V28() V29() ext-real set
K96((f1 . x),K98((x #Z (1 + 1)))) is V28() V29() ext-real set
x #Z 1 is V28() V29() ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V28() V29() ext-real Element of REAL
(f1 . x) - ((x #Z 1) * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98(((x #Z 1) * (x #Z 1))) is V28() V29() ext-real set
K96((f1 . x),K98(((x #Z 1) * (x #Z 1)))) is V28() V29() ext-real set
x * (x #Z 1) is V28() V29() ext-real Element of REAL
(f1 . x) - (x * (x #Z 1)) is V28() V29() ext-real Element of REAL
K98((x * (x #Z 1))) is V28() V29() ext-real set
K96((f1 . x),K98((x * (x #Z 1)))) is V28() V29() ext-real set
x * x is V28() V29() ext-real Element of REAL
(f1 . x) - (x * x) is V28() V29() ext-real Element of REAL
K98((x * x)) is V28() V29() ext-real set
K96((f1 . x),K98((x * x))) is V28() V29() ext-real set
1 - (x * x) is V28() V29() ext-real Element of REAL
K96(1,K98((x * x))) is V28() V29() ext-real set
x is V28() V29() ext-real Element of REAL
f1 . x is V28() V29() ext-real Element of REAL
(f1 - (#Z 2)) . x is V28() V29() ext-real Element of REAL
x is V28() V29() ext-real Element of REAL
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) . x is V28() V29() ext-real Element of REAL
f | Z is Relation-like REAL -defined Z -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
f | A is Relation-like REAL -defined A -defined REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
dom (- (ln * arccos)) is V55() V56() V57() Element of K19(REAL)
(- 1) (#) (ln * arccos) is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
(- (ln * arccos)) `| Z is Relation-like REAL -defined REAL -valued V6() V34() V35() V36() Element of K19(K20(REAL,REAL))
x is V28() V29() ext-real Element of REAL
((- (ln * arccos)) `| Z) . x is V28() V29() ext-real Element of REAL
x ^2 is V28() V29() ext-real Element of REAL
K97(x,x) is V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
(sqrt (1 - (x ^2))) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / ((sqrt (1 - (x ^2))) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99(((sqrt (1 - (x ^2))) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99(((sqrt (1 - (x ^2))) * (arccos . x)))) is V28() V29() ext-real set
diff ((- (ln * arccos)),x) is V28() V29() ext-real Element of REAL
diff ((ln * arccos),x) is V28() V29() ext-real Element of REAL
(- 1) * (diff ((ln * arccos),x)) is V28() V29() ext-real Element of REAL
diff (arccos,x) is V28() V29() ext-real Element of REAL
(diff (arccos,x)) / (arccos . x) is V28() V29() ext-real Element of REAL
K99((arccos . x)) is V28() V29() ext-real set
K97((diff (arccos,x)),K99((arccos . x))) is V28() V29() ext-real set
(- 1) * ((diff (arccos,x)) / (arccos . x)) is V28() V29() ext-real Element of REAL
1 / (sqrt (1 - (x ^2))) is V28() V29() ext-real Element of REAL
K99((sqrt (1 - (x ^2)))) is V28() V29() ext-real set
K97(1,K99((sqrt (1 - (x ^2))))) is V28() V29() ext-real set
- (1 / (sqrt (1 - (x ^2)))) is V28() V29() ext-real Element of REAL
(- (1 / (sqrt (1 - (x ^2))))) / (arccos . x) is V28() V29() ext-real Element of REAL
K97((- (1 / (sqrt (1 - (x ^2))))),K99((arccos . x))) is V28() V29() ext-real set
(- 1) * ((- (1 / (sqrt (1 - (x ^2))))) / (arccos . x)) is V28() V29() ext-real Element of REAL
(1 / (sqrt (1 - (x ^2)))) / (arccos . x) is V28() V29() ext-real Element of REAL
K97((1 / (sqrt (1 - (x ^2)))),K99((arccos . x))) is V28() V29() ext-real set
- ((1 / (sqrt (1 - (x ^2)))) / (arccos . x)) is V28() V29() ext-real Element of REAL
(- 1) * (- ((1 / (sqrt (1 - (x ^2)))) / (arccos . x))) is V28() V29() ext-real Element of 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 V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
(sqrt (1 - (x ^2))) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / ((sqrt (1 - (x ^2))) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99(((sqrt (1 - (x ^2))) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99(((sqrt (1 - (x ^2))) * (arccos . x)))) is V28() V29() ext-real set
dom (f1 - (#Z 2)) is V55() V56() V57() Element of K19(REAL)
(f1 - (#Z 2)) . x is V28() V29() ext-real Element of REAL
dom (#R (1 / 2)) is V55() V56() V57() Element of K19(REAL)
1 + x is V28() V29() ext-real Element of REAL
1 - x is V28() V29() ext-real Element of REAL
K98(x) is V28() V29() ext-real set
K96(1,K98(x)) is V28() V29() ext-real set
(1 + x) * (1 - x) is V28() V29() ext-real Element of REAL
((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) ^) . x is V28() V29() ext-real Element of REAL
(((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) . x is V28() V29() ext-real Element of REAL
1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) . x) is V28() V29() ext-real Element of REAL
K99(((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) . x)) is V28() V29() ext-real set
K97(1,K99(((((#R (1 / 2)) * (f1 - (#Z 2))) (#) arccos) . x))) is V28() V29() ext-real set
((#R (1 / 2)) * (f1 - (#Z 2))) . x is V28() V29() ext-real Element of REAL
(((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / ((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99(((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99(((((#R (1 / 2)) * (f1 - (#Z 2))) . x) * (arccos . x)))) is V28() V29() ext-real set
(#R (1 / 2)) . ((f1 - (#Z 2)) . x) is V28() V29() ext-real Element of REAL
((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / (((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99((((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99((((#R (1 / 2)) . ((f1 - (#Z 2)) . x)) * (arccos . x)))) is V28() V29() ext-real set
((f1 - (#Z 2)) . x) #R (1 / 2) is V28() V29() ext-real set
(((f1 - (#Z 2)) . x) #R (1 / 2)) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / ((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99(((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99(((((f1 - (#Z 2)) . x) #R (1 / 2)) * (arccos . x)))) is V28() V29() ext-real 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
K98(((#Z 2) . x)) is V28() V29() ext-real set
K96((f1 . x),K98(((#Z 2) . x))) is V28() V29() ext-real set
((f1 . x) - ((#Z 2) . x)) #R (1 / 2) is V28() V29() ext-real set
(((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / ((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99(((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99(((((f1 . x) - ((#Z 2) . x)) #R (1 / 2)) * (arccos . x)))) is V28() V29() ext-real 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
K98((x #Z 2)) is V28() V29() ext-real set
K96((f1 . x),K98((x #Z 2))) is V28() V29() ext-real set
((f1 . x) - (x #Z 2)) #R (1 / 2) is V28() V29() ext-real set
(((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / ((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99(((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99(((((f1 . x) - (x #Z 2)) #R (1 / 2)) * (arccos . x)))) is V28() V29() ext-real set
(f1 . x) - (x ^2) is V28() V29() ext-real Element of REAL
K96((f1 . x),K98((x ^2))) is V28() V29() ext-real set
((f1 . x) - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
(((f1 . x) - (x ^2)) #R (1 / 2)) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / ((((f1 . x) - (x ^2)) #R (1 / 2)) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99(((((f1 . x) - (x ^2)) #R (1 / 2)) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99(((((f1 . x) - (x ^2)) #R (1 / 2)) * (arccos . x)))) is V28() V29() ext-real set
(1 - (x ^2)) #R (1 / 2) is V28() V29() ext-real set
((1 - (x ^2)) #R (1 / 2)) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / (((1 - (x ^2)) #R (1 / 2)) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99((((1 - (x ^2)) #R (1 / 2)) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99((((1 - (x ^2)) #R (1 / 2)) * (arccos . x)))) is V28() V29() ext-real set
dom ((- (ln * arccos)) `| Z) is V55() V56() V57() Element of K19(REAL)
x is V28() V29() ext-real Element of REAL
((- (ln * arccos)) `| 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 V28() V29() ext-real set
1 - (x ^2) is V28() V29() ext-real Element of REAL
K98((x ^2)) is V28() V29() ext-real set
K96(1,K98((x ^2))) is V28() V29() ext-real set
sqrt (1 - (x ^2)) is V28() V29() ext-real Element of REAL
arccos . x is V28() V29() ext-real Element of REAL
(sqrt (1 - (x ^2))) * (arccos . x) is V28() V29() ext-real Element of REAL
1 / ((sqrt (1 - (x ^2))) * (arccos . x)) is V28() V29() ext-real Element of REAL
K99(((sqrt (1 - (x ^2))) * (arccos . x))) is V28() V29() ext-real set
K97(1,K99(((sqrt (1 - (x ^2))) * (arccos . x)))) is V28() V29() ext-real set