:: SIN_COS9 semantic presentation

REAL is non empty V46() V47() V48() V52() V73() V84() V85() V87() set
NAT is V46() V47() V48() V49() V50() V51() V52() V84() Element of K6(REAL)
K6(REAL) is set
COMPLEX is non empty V46() V52() V73() set
0 is set
1 is non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() Element of NAT
{0,1} is set
K7(REAL,REAL) is V36() V37() V38() set
K6(K7(REAL,REAL)) is set
K7(NAT,REAL) is V36() V37() V38() set
K6(K7(NAT,REAL)) is set
K7(NAT,COMPLEX) is V36() set
K6(K7(NAT,COMPLEX)) is set
K7(COMPLEX,COMPLEX) is V36() set
K6(K7(COMPLEX,COMPLEX)) is set
PFuncs (REAL,REAL) is set
K7(NAT,(PFuncs (REAL,REAL))) is set
K6(K7(NAT,(PFuncs (REAL,REAL)))) is set
RAT is non empty V46() V47() V48() V49() V52() V73() set
INT is non empty V46() V47() V48() V49() V50() V52() V73() set
0 is natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() Element of NAT
sin is V19() V22( REAL ) V23( REAL ) Function-like V33( REAL , REAL ) V36() V37() V38() continuous Element of K6(K7(REAL,REAL))
dom sin is V46() V47() V48() Element of K6(REAL)
cos is V19() V22( REAL ) V23( REAL ) Function-like V33( REAL , REAL ) V36() V37() V38() continuous Element of K6(K7(REAL,REAL))
dom cos is V46() V47() V48() Element of K6(REAL)
cos . 0 is V11() real ext-real Element of REAL
sin . 0 is V11() real ext-real Element of REAL
cos 0 is V11() real ext-real Element of REAL
sin 0 is V11() real ext-real Element of REAL
exp_R is V19() V22( REAL ) V23( REAL ) Function-like V33( REAL , REAL ) V36() V37() V38() continuous Element of K6(K7(REAL,REAL))
dom exp_R is V46() V47() V48() Element of K6(REAL)
PI is non empty V11() real ext-real positive non negative set
tan is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
sin / cos is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
4 is non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() Element of NAT
].0,4.[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
PI is non empty V11() real ext-real positive non negative Element of REAL
PI / 4 is non empty V11() real ext-real positive non negative Element of REAL
4 " is non empty V11() real ext-real positive non negative set
PI * (4 ") is non empty V11() real ext-real positive non negative set
sin . (PI / 4) is V11() real ext-real Element of REAL
cos . (PI / 4) is V11() real ext-real Element of REAL
2 is non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() Element of NAT
PI / 2 is non empty V11() real ext-real positive non negative Element of REAL
2 " is non empty V11() real ext-real positive non negative set
PI * (2 ") is non empty V11() real ext-real positive non negative set
cos . (PI / 2) is V11() real ext-real Element of REAL
sin . (PI / 2) is V11() real ext-real Element of REAL
cos . PI is V11() real ext-real Element of REAL
- 1 is non empty V11() real ext-real non positive negative V63() Element of REAL
sin . PI is V11() real ext-real Element of REAL
PI + (PI / 2) is non empty V11() real ext-real positive non negative Element of REAL
cos . (PI + (PI / 2)) is V11() real ext-real Element of REAL
sin . (PI + (PI / 2)) is V11() real ext-real Element of REAL
2 * PI is non empty V11() real ext-real positive non negative Element of REAL
cos . (2 * PI) is V11() real ext-real Element of REAL
sin . (2 * PI) is V11() real ext-real Element of REAL
].0,PI.[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
- (PI / 2) is non empty V11() real ext-real non positive negative Element of REAL
].(- (PI / 2)),(PI / 2).[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
cos / sin is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
{0} is V46() V47() V48() V49() V50() V51() V84() set
ln is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
exp_R " is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ln is V46() V47() V48() Element of K6(REAL)
right_open_halfline 0 is V46() V47() V48() Element of K6(REAL)
+infty is non empty non real ext-real positive non negative set
K68(0,+infty) is V82() V83() V87() set
rng ln is V46() V47() V48() Element of K6(REAL)
#Z 2 is V19() V22( REAL ) V23( REAL ) Function-like V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
dom tan is V46() V47() V48() Element of K6(REAL)
cos " {0} is V46() V47() V48() Element of K6(REAL)
].(- (PI / 2)),(PI / 2).[ /\ (cos " {0}) is V46() V47() V48() set
Z is set
cos . Z is V11() real ext-real Element of REAL
].(- (PI / 2)),(PI / 2).[ \ (cos " {0}) is V46() V47() V48() set
(dom cos) \ (cos " {0}) is V46() V47() V48() set
(dom sin) /\ ((dom cos) \ (cos " {0})) is V46() V47() V48() set
cot is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom cot is V46() V47() V48() Element of K6(REAL)
sin " {0} is V46() V47() V48() Element of K6(REAL)
].0,PI.[ /\ (sin " {0}) is V46() V47() V48() set
Z is set
sin . Z is V11() real ext-real Element of REAL
].0,PI.[ \ (sin " {0}) is V46() V47() V48() set
(dom sin) \ (sin " {0}) is V46() V47() V48() set
(dom cos) /\ ((dom sin) \ (sin " {0})) is V46() V47() V48() set
Z is V11() real ext-real Element of REAL
cos . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
sin . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
diff (tan,Z) is V11() real ext-real Element of REAL
cos . Z is V11() real ext-real Element of REAL
(cos . Z) ^2 is V11() real ext-real Element of REAL
(cos . Z) * (cos . Z) is V11() real ext-real set
1 / ((cos . Z) ^2) is V11() real ext-real Element of REAL
((cos . Z) ^2) " is V11() real ext-real set
1 * (((cos . Z) ^2) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
diff (cot,Z) is V11() real ext-real Element of REAL
sin . Z is V11() real ext-real Element of REAL
(sin . Z) ^2 is V11() real ext-real Element of REAL
(sin . Z) * (sin . Z) is V11() real ext-real set
1 / ((sin . Z) ^2) is V11() real ext-real Element of REAL
((sin . Z) ^2) " is V11() real ext-real set
1 * (((sin . Z) ^2) ") is V11() real ext-real set
- (1 / ((sin . Z) ^2)) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
diff (tan,Z) is V11() real ext-real Element of REAL
cos . Z is V11() real ext-real Element of REAL
(cos . Z) ^2 is V11() real ext-real Element of REAL
(cos . Z) * (cos . Z) is V11() real ext-real set
1 / ((cos . Z) ^2) is V11() real ext-real Element of REAL
((cos . Z) ^2) " is V11() real ext-real set
1 * (((cos . Z) ^2) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
diff (cot,Z) is V11() real ext-real Element of REAL
sin . Z is V11() real ext-real Element of REAL
(sin . Z) ^2 is V11() real ext-real Element of REAL
(sin . Z) * (sin . Z) is V11() real ext-real set
1 / ((sin . Z) ^2) is V11() real ext-real Element of REAL
((sin . Z) ^2) " is V11() real ext-real set
1 * (((sin . Z) ^2) ") is V11() real ext-real set
- (1 / ((sin . Z) ^2)) is V11() real ext-real Element of REAL
tan | ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
cot | ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is V11() real ext-real Element of REAL
diff (tan,Z) is V11() real ext-real Element of REAL
cos . Z is V11() real ext-real Element of REAL
(cos . Z) ^2 is V11() real ext-real Element of REAL
(cos . Z) * (cos . Z) is V11() real ext-real set
1 / ((cos . Z) ^2) is V11() real ext-real Element of REAL
((cos . Z) ^2) " is V11() real ext-real set
1 * (((cos . Z) ^2) ") is V11() real ext-real set
0 / ((cos . Z) ^2) is V11() real ext-real Element of REAL
0 * (((cos . Z) ^2) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
diff (cot,Z) is V11() real ext-real Element of REAL
sin . Z is V11() real ext-real Element of REAL
(sin . Z) ^2 is V11() real ext-real Element of REAL
(sin . Z) * (sin . Z) is V11() real ext-real set
1 / ((sin . Z) ^2) is V11() real ext-real Element of REAL
((sin . Z) ^2) " is V11() real ext-real set
1 * (((sin . Z) ^2) ") is V11() real ext-real set
0 / ((sin . Z) ^2) is V11() real ext-real Element of REAL
0 * (((sin . Z) ^2) ") is V11() real ext-real set
- 0 is V11() real ext-real V63() Element of REAL
- (1 / ((sin . Z) ^2)) is V11() real ext-real Element of REAL
(tan | ].(- (PI / 2)),(PI / 2).[) " is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | ].0,PI.[) " is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
() is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is set
() . Z is V11() real ext-real Element of REAL
(Z) is set
() . Z is V11() real ext-real Element of REAL
() " is V19() Function-like set
() " is V19() Function-like set
rng () is V46() V47() V48() Element of K6(REAL)
dom (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
rng () is V46() V47() V48() Element of K6(REAL)
dom (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
- (PI / 4) is non empty V11() real ext-real non positive negative Element of REAL
- 4 is non empty V11() real ext-real non positive negative V63() Element of REAL
PI / (- 4) is non empty V11() real ext-real non positive negative Element of REAL
(- 4) " is non empty V11() real ext-real non positive negative set
PI * ((- 4) ") is non empty V11() real ext-real non positive negative set
- 2 is non empty V11() real ext-real non positive negative V63() Element of REAL
PI / (- 2) is non empty V11() real ext-real non positive negative Element of REAL
(- 2) " is non empty V11() real ext-real non positive negative set
PI * ((- 2) ") is non empty V11() real ext-real non positive negative set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= - (PI / 2) & not PI / 2 <= b₁ ) } is set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= - (PI / 2) & not PI / 2 <= b₁ ) } is set
PI / 1 is non empty V11() real ext-real positive non negative Element of REAL
1 " is non empty V11() real ext-real positive non negative set
PI * (1 ") is non empty V11() real ext-real positive non negative set
0 / 4 is V11() real ext-real Element of REAL
0 * (4 ") is V11() real ext-real set
3 is non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() Element of NAT
3 / 4 is non empty V11() real ext-real positive non negative Element of REAL
3 * (4 ") is non empty V11() real ext-real positive non negative set
(3 / 4) * PI is non empty V11() real ext-real positive non negative Element of REAL
(3 / 4) * 0 is V11() real ext-real Element of REAL
[.(- (PI / 4)),(PI / 4).] is closed V46() V47() V48() V87() Element of K6(REAL)
tan | [.(- (PI / 4)),(PI / 4).] is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (tan | [.(- (PI / 4)),(PI / 4).]) is V46() V47() V48() Element of K6(REAL)
[.(PI / 4),((3 / 4) * PI).] is closed V46() V47() V48() V87() Element of K6(REAL)
cot | [.(PI / 4),((3 / 4) * PI).] is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (cot | [.(PI / 4),((3 / 4) * PI).]) is V46() V47() V48() Element of K6(REAL)
Z is V11() real ext-real set
tan . Z is V11() real ext-real Element of REAL
tan Z is V11() real ext-real Element of REAL
sin Z is V11() real ext-real set
sin . Z is V11() real ext-real Element of REAL
cos Z is V11() real ext-real set
cos . Z is V11() real ext-real Element of REAL
(sin Z) / (cos Z) is V11() real ext-real set
(cos Z) " is V11() real ext-real set
(sin Z) * ((cos Z) ") is V11() real ext-real set
Z is V11() real ext-real set
cot . Z is V11() real ext-real Element of REAL
cot Z is V11() real ext-real Element of REAL
cos Z is V11() real ext-real set
cos . Z is V11() real ext-real Element of REAL
sin Z is V11() real ext-real set
sin . Z is V11() real ext-real Element of REAL
(cos Z) / (sin Z) is V11() real ext-real set
(sin Z) " is V11() real ext-real set
(cos Z) * ((sin Z) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
cos . Z is V11() real ext-real Element of REAL
tan . Z is V11() real ext-real Element of REAL
tan Z is V11() real ext-real Element of REAL
cos " {0} is V46() V47() V48() Element of K6(REAL)
(dom cos) \ (cos " {0}) is V46() V47() V48() set
(dom sin) /\ ((dom cos) \ (cos " {0})) is V46() V47() V48() set
dom (sin / cos) is V46() V47() V48() Element of K6(REAL)
sin Z is V11() real ext-real Element of REAL
sin . Z is V11() real ext-real Element of REAL
cos Z is V11() real ext-real Element of REAL
(sin Z) / (cos Z) is V11() real ext-real Element of REAL
(cos Z) " is V11() real ext-real set
(sin Z) * ((cos Z) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
sin . Z is V11() real ext-real Element of REAL
cot . Z is V11() real ext-real Element of REAL
cot Z is V11() real ext-real Element of REAL
sin " {0} is V46() V47() V48() Element of K6(REAL)
(dom sin) \ (sin " {0}) is V46() V47() V48() set
(dom cos) /\ ((dom sin) \ (sin " {0})) is V46() V47() V48() set
dom (cos / sin) is V46() V47() V48() Element of K6(REAL)
cos Z is V11() real ext-real Element of REAL
cos . Z is V11() real ext-real Element of REAL
sin Z is V11() real ext-real Element of REAL
(cos Z) / (sin Z) is V11() real ext-real Element of REAL
(sin Z) " is V11() real ext-real set
(cos Z) * ((sin Z) ") is V11() real ext-real set
tan . (- (PI / 4)) is V11() real ext-real Element of REAL
tan (- (PI / 4)) is V11() real ext-real Element of REAL
(sin . (PI / 4)) / (cos . (PI / 4)) is V11() real ext-real Element of REAL
(cos . (PI / 4)) " is V11() real ext-real set
(sin . (PI / 4)) * ((cos . (PI / 4)) ") is V11() real ext-real set
sin . (- (PI / 4)) is V11() real ext-real Element of REAL
cos . (- (PI / 4)) is V11() real ext-real Element of REAL
(sin . (- (PI / 4))) / (cos . (- (PI / 4))) is V11() real ext-real Element of REAL
(cos . (- (PI / 4))) " is V11() real ext-real set
(sin . (- (PI / 4))) * ((cos . (- (PI / 4))) ") is V11() real ext-real set
- (sin . (PI / 4)) is V11() real ext-real Element of REAL
(- (sin . (PI / 4))) / (cos . (- (PI / 4))) is V11() real ext-real Element of REAL
(- (sin . (PI / 4))) * ((cos . (- (PI / 4))) ") is V11() real ext-real set
(- (sin . (PI / 4))) / (cos . (PI / 4)) is V11() real ext-real Element of REAL
(- (sin . (PI / 4))) * ((cos . (PI / 4)) ") is V11() real ext-real set
cot . (PI / 4) is V11() real ext-real Element of REAL
cot (PI / 4) is V11() real ext-real Element of REAL
cot . ((3 / 4) * PI) is V11() real ext-real Element of REAL
cot ((3 / 4) * PI) is V11() real ext-real Element of REAL
cos . ((3 / 4) * PI) is V11() real ext-real Element of REAL
sin . ((3 / 4) * PI) is V11() real ext-real Element of REAL
(sin . ((3 / 4) * PI)) " is V11() real ext-real Element of REAL
(cos . ((3 / 4) * PI)) * ((sin . ((3 / 4) * PI)) ") is V11() real ext-real Element of REAL
- (sin . (PI / 4)) is V11() real ext-real Element of REAL
(PI / 2) + (PI / 4) is non empty V11() real ext-real positive non negative Element of REAL
sin . ((PI / 2) + (PI / 4)) is V11() real ext-real Element of REAL
(- (sin . (PI / 4))) / (sin . ((PI / 2) + (PI / 4))) is V11() real ext-real Element of REAL
(sin . ((PI / 2) + (PI / 4))) " is V11() real ext-real set
(- (sin . (PI / 4))) * ((sin . ((PI / 2) + (PI / 4))) ") is V11() real ext-real set
(- (sin . (PI / 4))) / (cos . (PI / 4)) is V11() real ext-real Element of REAL
(cos . (PI / 4)) " is V11() real ext-real set
(- (sin . (PI / 4))) * ((cos . (PI / 4)) ") is V11() real ext-real set
(sin . (PI / 4)) / (cos . (PI / 4)) is V11() real ext-real Element of REAL
(sin . (PI / 4)) * ((cos . (PI / 4)) ") is V11() real ext-real set
- ((sin . (PI / 4)) / (cos . (PI / 4))) is V11() real ext-real Element of REAL
(cos . (PI / 4)) / (sin . (PI / 4)) is V11() real ext-real Element of REAL
(sin . (PI / 4)) " is V11() real ext-real set
(cos . (PI / 4)) * ((sin . (PI / 4)) ") is V11() real ext-real set
[.(- 1),1.] is closed V46() V47() V48() V87() Element of K6(REAL)
Z is set
tan . Z is V11() real ext-real Element of REAL
].(- (PI / 4)),(PI / 4).[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
{(- (PI / 4)),(PI / 4)} is V46() V47() V48() set
].(- (PI / 4)),(PI / 4).[ \/ {(- (PI / 4)),(PI / 4)} is V46() V47() V48() set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= - (PI / 4) & not PI / 4 <= b₁ ) } is set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( - (PI / 4) <= b₁ & b₁ <= PI / 4 ) } is set
[.(- (PI / 4)),(PI / 4).] /\ (dom tan) is V46() V47() V48() set
tan . (PI / 4) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= - 1 & not 1 <= b₁ ) } is set
].(- 1),1.[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( - 1 <= b₁ & b₁ <= 1 ) } is set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( - 1 <= b₁ & b₁ <= 1 ) } is set
Z is set
cot . Z is V11() real ext-real Element of REAL
0 / 4 is V11() real ext-real Element of REAL
0 * (4 ") is V11() real ext-real set
(PI / 4) * 3 is non empty V11() real ext-real positive non negative Element of REAL
].(PI / 4),((3 / 4) * PI).[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
{(PI / 4),((3 / 4) * PI)} is V46() V47() V48() set
].(PI / 4),((3 / 4) * PI).[ \/ {(PI / 4),((3 / 4) * PI)} is V46() V47() V48() set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= PI / 4 & not (3 / 4) * PI <= b₁ ) } is set
[.(PI / 4),((3 / 4) * PI).] /\ (dom cot) is V46() V47() V48() set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( PI / 4 <= b₁ & b₁ <= (3 / 4) * PI ) } is set
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= - 1 & not 1 <= b₁ ) } is set
].(- 1),1.[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( - 1 <= b₁ & b₁ <= 1 ) } is set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( - 1 <= b₁ & b₁ <= 1 ) } is set
rng (tan | [.(- (PI / 4)),(PI / 4).]) is V46() V47() V48() Element of K6(REAL)
Z is set
f is V11() real ext-real Element of REAL
tan . (PI / 4) is V11() real ext-real Element of REAL
[.(tan . (- (PI / 4))),(tan . (PI / 4)).] is closed V46() V47() V48() V87() Element of K6(REAL)
[.(tan . (PI / 4)),(tan . (- (PI / 4))).] is closed V46() V47() V48() V87() Element of K6(REAL)
[.(tan . (- (PI / 4))),(tan . (PI / 4)).] \/ [.(tan . (PI / 4)),(tan . (- (PI / 4))).] is V46() V47() V48() set
x is V11() real ext-real Element of REAL
tan . x is V11() real ext-real Element of REAL
(tan | [.(- (PI / 4)),(PI / 4).]) . x is V11() real ext-real Element of REAL
f is set
(tan | [.(- (PI / 4)),(PI / 4).]) . f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
tan . x is V11() real ext-real Element of REAL
rng (cot | [.(PI / 4),((3 / 4) * PI).]) is V46() V47() V48() Element of K6(REAL)
Z is set
0 / 4 is V11() real ext-real Element of REAL
0 * (4 ") is V11() real ext-real set
(PI / 4) * 3 is non empty V11() real ext-real positive non negative Element of REAL
f is V11() real ext-real Element of REAL
[.(cot . ((3 / 4) * PI)),(cot . (PI / 4)).] is closed V46() V47() V48() V87() Element of K6(REAL)
[.(cot . (PI / 4)),(cot . ((3 / 4) * PI)).] is closed V46() V47() V48() V87() Element of K6(REAL)
[.(cot . ((3 / 4) * PI)),(cot . (PI / 4)).] \/ [.(cot . (PI / 4)),(cot . ((3 / 4) * PI)).] is V46() V47() V48() set
x is V11() real ext-real Element of REAL
cot . x is V11() real ext-real Element of REAL
(cot | [.(PI / 4),((3 / 4) * PI).]) . x is V11() real ext-real Element of REAL
f is set
(cot | [.(PI / 4),((3 / 4) * PI).]) . f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
cot . x is V11() real ext-real Element of REAL
dom () is V46() V47() V48() Element of K6(REAL)
rng (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
Z is set
tan .: [.(- (PI / 4)),(PI / 4).] is V46() V47() V48() Element of K6(REAL)
tan .: ].(- (PI / 2)),(PI / 2).[ is V46() V47() V48() Element of K6(REAL)
f is set
tan . f is V11() real ext-real Element of REAL
dom () is V46() V47() V48() Element of K6(REAL)
rng (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
Z is set
cot .: [.(PI / 4),((3 / 4) * PI).] is V46() V47() V48() Element of K6(REAL)
cot .: ].0,PI.[ is V46() V47() V48() Element of K6(REAL)
f is set
cot . f is V11() real ext-real Element of REAL
(tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).] is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).] is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
() | [.(- 1),1.] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan | [.(- (PI / 4)),(PI / 4).]) " is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
((tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).]) " is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan | ].(- (PI / 2)),(PI / 2).[) .: [.(- (PI / 4)),(PI / 4).] is V46() V47() V48() Element of K6(REAL)
((tan | ].(- (PI / 2)),(PI / 2).[) ") | ((tan | ].(- (PI / 2)),(PI / 2).[) .: [.(- (PI / 4)),(PI / 4).]) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
rng ((tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).]) is V46() V47() V48() Element of K6(REAL)
((tan | ].(- (PI / 2)),(PI / 2).[) ") | (rng ((tan | ].(- (PI / 2)),(PI / 2).[) | [.(- (PI / 4)),(PI / 4).])) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
((tan | ].(- (PI / 2)),(PI / 2).[) ") | [.(- 1),1.] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
() | [.(- 1),1.] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | [.(PI / 4),((3 / 4) * PI).]) " is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
((cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).]) " is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | ].0,PI.[) .: [.(PI / 4),((3 / 4) * PI).] is V46() V47() V48() Element of K6(REAL)
((cot | ].0,PI.[) ") | ((cot | ].0,PI.[) .: [.(PI / 4),((3 / 4) * PI).]) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
rng ((cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).]) is V46() V47() V48() Element of K6(REAL)
((cot | ].0,PI.[) ") | (rng ((cot | ].0,PI.[) | [.(PI / 4),((3 / 4) * PI).])) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
((cot | ].0,PI.[) ") | [.(- 1),1.] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan | [.(- (PI / 4)),(PI / 4).]) * (() | [.(- 1),1.]) is V19() Function-like one-to-one V36() V37() V38() set
id [.(- 1),1.] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | [.(PI / 4),((3 / 4) * PI).]) * (() | [.(- 1),1.]) is V19() Function-like one-to-one V36() V37() V38() set
(tan | [.(- (PI / 4)),(PI / 4).]) * (() | [.(- 1),1.]) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | [.(PI / 4),((3 / 4) * PI).]) * (() | [.(- 1),1.]) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
() * (tan | ].(- (PI / 2)),(PI / 2).[) is V19() Function-like one-to-one V36() V37() V38() set
id ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
() * (cot | ].0,PI.[) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
id ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
() * (cot | ].0,PI.[) is V19() Function-like one-to-one V36() V37() V38() set
Z is V11() real ext-real Element of REAL
tan . Z is V11() real ext-real Element of REAL
((tan . Z)) is V11() real ext-real Element of REAL
() . (tan . Z) is V11() real ext-real Element of REAL
tan Z is V11() real ext-real Element of REAL
((tan Z)) is V11() real ext-real Element of REAL
() . (tan Z) is V11() real ext-real Element of REAL
dom (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
(tan | ].(- (PI / 2)),(PI / 2).[) . Z is V11() real ext-real Element of REAL
() . ((tan | ].(- (PI / 2)),(PI / 2).[) . Z) is V11() real ext-real Element of REAL
(id ].(- (PI / 2)),(PI / 2).[) . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
cot . Z is V11() real ext-real Element of REAL
((cot . Z)) is V11() real ext-real Element of REAL
() . (cot . Z) is V11() real ext-real Element of REAL
cot Z is V11() real ext-real Element of REAL
((cot Z)) is V11() real ext-real Element of REAL
() . (cot Z) is V11() real ext-real Element of REAL
dom (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
(cot | ].0,PI.[) . Z is V11() real ext-real Element of REAL
() . ((cot | ].0,PI.[) . Z) is V11() real ext-real Element of REAL
(id ].0,PI.[) . Z is V11() real ext-real Element of REAL
((- 1)) is V11() real ext-real Element of REAL
() . (- 1) is V11() real ext-real Element of REAL
((- 1)) is V11() real ext-real Element of REAL
() . (- 1) is V11() real ext-real Element of REAL
(1) is V11() real ext-real Element of REAL
() . 1 is V11() real ext-real Element of REAL
tan . (PI / 4) is V11() real ext-real Element of REAL
((tan . (PI / 4))) is V11() real ext-real Element of REAL
() . (tan . (PI / 4)) is V11() real ext-real Element of REAL
(1) is V11() real ext-real Element of REAL
() . 1 is V11() real ext-real Element of REAL
tan . 0 is V11() real ext-real Element of REAL
tan 0 is V11() real ext-real Element of REAL
0 / 2 is V11() real ext-real Element of REAL
0 * (2 ") is V11() real ext-real set
0 - (PI / 2) is V11() real ext-real Element of REAL
- (PI / 2) is non empty V11() real ext-real non positive negative set
0 + (- (PI / 2)) is V11() real ext-real set
0 / (cos . 0) is V11() real ext-real Element of REAL
(cos . 0) " is V11() real ext-real set
0 * ((cos . 0) ") is V11() real ext-real set
cot . (PI / 2) is V11() real ext-real Element of REAL
cot (PI / 2) is V11() real ext-real Element of REAL
PI / 1 is non empty V11() real ext-real positive non negative Element of REAL
1 " is non empty V11() real ext-real positive non negative set
PI * (1 ") is non empty V11() real ext-real positive non negative set
0 / 2 is V11() real ext-real Element of REAL
0 * (2 ") is V11() real ext-real set
0 / (sin . (PI / 2)) is V11() real ext-real Element of REAL
(sin . (PI / 2)) " is V11() real ext-real set
0 * ((sin . (PI / 2)) ") is V11() real ext-real set
(0) is V11() real ext-real Element of REAL
() . 0 is V11() real ext-real Element of REAL
0 / 2 is V11() real ext-real Element of REAL
0 * (2 ") is V11() real ext-real set
0 - (PI / 2) is V11() real ext-real Element of REAL
- (PI / 2) is non empty V11() real ext-real non positive negative set
0 + (- (PI / 2)) is V11() real ext-real set
(0) is V11() real ext-real Element of REAL
() . 0 is V11() real ext-real Element of REAL
PI / 1 is non empty V11() real ext-real positive non negative Element of REAL
1 " is non empty V11() real ext-real positive non negative set
PI * (1 ") is non empty V11() real ext-real positive non negative set
0 / 2 is V11() real ext-real Element of REAL
0 * (2 ") is V11() real ext-real set
tan .: ].(- (PI / 2)),(PI / 2).[ is V46() V47() V48() Element of K6(REAL)
() | (tan .: ].(- (PI / 2)),(PI / 2).[) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan | ].(- (PI / 2)),(PI / 2).[) .: ].(- (PI / 2)),(PI / 2).[ is V46() V47() V48() Element of K6(REAL)
(tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
rng ((tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
rng (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
cot .: ].0,PI.[ is V46() V47() V48() Element of K6(REAL)
() | (cot .: ].0,PI.[) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | ].0,PI.[) .: ].0,PI.[ is V46() V47() V48() Element of K6(REAL)
(cot | ].0,PI.[) | ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
rng ((cot | ].0,PI.[) | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
rng (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
tan .: [.(- (PI / 4)),(PI / 4).] is V46() V47() V48() Element of K6(REAL)
cot .: [.(PI / 4),((3 / 4) * PI).] is V46() V47() V48() Element of K6(REAL)
Z is set
() . Z is V11() real ext-real Element of REAL
].(- 1),1.[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
{(- 1),1} is V46() V47() V48() V49() V50() set
].(- 1),1.[ \/ {(- 1),1} is V46() V47() V48() set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= - 1 & not 1 <= b₁ ) } is set
[.(- 1),1.] /\ (dom ()) is V46() V47() V48() set
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
Z is set
() . Z is V11() real ext-real Element of REAL
].(- 1),1.[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
{(- 1),1} is V46() V47() V48() V49() V50() set
].(- 1),1.[ \/ {(- 1),1} is V46() V47() V48() set
{ b₁ where b₁ is V11() real ext-real Element of REAL : ( not b₁ <= - 1 & not 1 <= b₁ ) } is set
[.(- 1),1.] /\ (dom ()) is V46() V47() V48() set
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
0 / 4 is V11() real ext-real Element of REAL
0 * (4 ") is V11() real ext-real set
(PI / 4) * 3 is non empty V11() real ext-real positive non negative Element of REAL
0 / 4 is V11() real ext-real Element of REAL
0 * (4 ") is V11() real ext-real set
(PI / 4) * 3 is non empty V11() real ext-real positive non negative Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
tan (Z) is V11() real ext-real Element of REAL
dom (() | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
tan . (() . Z) is V11() real ext-real Element of REAL
(tan | [.(- (PI / 4)),(PI / 4).]) . (() . Z) is V11() real ext-real set
(() | [.(- 1),1.]) . Z is V11() real ext-real Element of REAL
(tan | [.(- (PI / 4)),(PI / 4).]) . ((() | [.(- 1),1.]) . Z) is V11() real ext-real set
((tan | [.(- (PI / 4)),(PI / 4).]) * (() | [.(- 1),1.])) . Z is V11() real ext-real Element of REAL
(id [.(- 1),1.]) . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
cot (Z) is V11() real ext-real Element of REAL
dom (() | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
cot . (() . Z) is V11() real ext-real Element of REAL
(cot | [.(PI / 4),((3 / 4) * PI).]) . (() . Z) is V11() real ext-real set
(() | [.(- 1),1.]) . Z is V11() real ext-real Element of REAL
(cot | [.(PI / 4),((3 / 4) * PI).]) . ((() | [.(- 1),1.]) . Z) is V11() real ext-real set
((cot | [.(PI / 4),((3 / 4) * PI).]) * (() | [.(- 1),1.])) . Z is V11() real ext-real Element of REAL
(id [.(- 1),1.]) . Z is V11() real ext-real Element of REAL
((tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).]) " is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan | [.(- (PI / 4)),(PI / 4).]) .: [.(- (PI / 4)),(PI / 4).] is V46() V47() V48() Element of K6(REAL)
(((tan | [.(- (PI / 4)),(PI / 4).]) | [.(- (PI / 4)),(PI / 4).]) ") | ((tan | [.(- (PI / 4)),(PI / 4).]) .: [.(- (PI / 4)),(PI / 4).]) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(() | [.(- 1),1.]) | [.(- 1),1.] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
0 / 4 is V11() real ext-real Element of REAL
0 * (4 ") is V11() real ext-real set
(PI / 4) * 3 is non empty V11() real ext-real positive non negative Element of REAL
((cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).]) " is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot | [.(PI / 4),((3 / 4) * PI).]) .: [.(PI / 4),((3 / 4) * PI).] is V46() V47() V48() Element of K6(REAL)
(((cot | [.(PI / 4),((3 / 4) * PI).]) | [.(PI / 4),((3 / 4) * PI).]) ") | ((cot | [.(PI / 4),((3 / 4) * PI).]) .: [.(PI / 4),((3 / 4) * PI).]) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(() | [.(- 1),1.]) | [.(- 1),1.] is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
rng (() | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
Z is set
dom (() | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
[.(() . (- 1)),(() . 1).] is closed V46() V47() V48() V87() Element of K6(REAL)
[.(() . 1),(() . (- 1)).] is closed V46() V47() V48() V87() Element of K6(REAL)
[.(() . (- 1)),(() . 1).] \/ [.(() . 1),(() . (- 1)).] is V46() V47() V48() set
x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
(() | [.(- 1),1.]) . x is V11() real ext-real Element of REAL
f is set
(() | [.(- 1),1.]) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
rng (() | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
Z is set
dom (() | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
[.(() . 1),(() . (- 1)).] is closed V46() V47() V48() V87() Element of K6(REAL)
[.(() . (- 1)),(() . 1).] is closed V46() V47() V48() V87() Element of K6(REAL)
[.(() . 1),(() . (- 1)).] \/ [.(() . (- 1)),(() . 1).] is V46() V47() V48() set
x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
(() | [.(- 1),1.]) . x is V11() real ext-real Element of REAL
f is set
(() | [.(- 1),1.]) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
tan (PI / 4) is V11() real ext-real Element of REAL
tan . (PI / 4) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
dom (() | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
(() | [.(- 1),1.]) . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
dom (() | [.(- 1),1.]) is V46() V47() V48() Element of K6(REAL)
(() | [.(- 1),1.]) . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
tan (PI / 4) is V11() real ext-real Element of REAL
tan . (PI / 4) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
- Z is V11() real ext-real Element of REAL
((- Z)) is V11() real ext-real Element of REAL
() . (- Z) is V11() real ext-real Element of REAL
- ((- Z)) is V11() real ext-real Element of REAL
- (- 1) is non empty V11() real ext-real positive non negative V63() Element of REAL
- (- (PI / 4)) is non empty V11() real ext-real positive non negative Element of REAL
cos ((- Z)) is V11() real ext-real Element of REAL
cos . ((- Z)) is V11() real ext-real Element of REAL
tan ((- Z)) is V11() real ext-real Element of REAL
(tan 0) - (tan ((- Z))) is V11() real ext-real Element of REAL
- (tan ((- Z))) is V11() real ext-real set
(tan 0) + (- (tan ((- Z)))) is V11() real ext-real set
(tan 0) * (tan ((- Z))) is V11() real ext-real Element of REAL
1 + ((tan 0) * (tan ((- Z)))) is V11() real ext-real Element of REAL
((tan 0) - (tan ((- Z)))) / (1 + ((tan 0) * (tan ((- Z))))) is V11() real ext-real Element of REAL
(1 + ((tan 0) * (tan ((- Z))))) " is V11() real ext-real set
((tan 0) - (tan ((- Z)))) * ((1 + ((tan 0) * (tan ((- Z))))) ") is V11() real ext-real set
0 - ((- Z)) is V11() real ext-real Element of REAL
- ((- Z)) is V11() real ext-real set
0 + (- ((- Z))) is V11() real ext-real set
tan (0 - ((- Z))) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
- Z is V11() real ext-real Element of REAL
((- Z)) is V11() real ext-real Element of REAL
() . (- Z) is V11() real ext-real Element of REAL
PI - ((- Z)) is V11() real ext-real Element of REAL
- ((- Z)) is V11() real ext-real set
PI + (- ((- Z))) is V11() real ext-real set
- (- 1) is non empty V11() real ext-real positive non negative V63() Element of REAL
cot ((- Z)) is V11() real ext-real Element of REAL
- (cot ((- Z))) is V11() real ext-real Element of REAL
cos ((- Z)) is V11() real ext-real Element of REAL
cos . ((- Z)) is V11() real ext-real Element of REAL
sin ((- Z)) is V11() real ext-real Element of REAL
sin . ((- Z)) is V11() real ext-real Element of REAL
(cos ((- Z))) / (sin ((- Z))) is V11() real ext-real Element of REAL
(sin ((- Z))) " is V11() real ext-real set
(cos ((- Z))) * ((sin ((- Z))) ") is V11() real ext-real set
- ((cos ((- Z))) / (sin ((- Z)))) is V11() real ext-real Element of REAL
- (sin ((- Z))) is V11() real ext-real Element of REAL
(cos ((- Z))) / (- (sin ((- Z)))) is V11() real ext-real Element of REAL
(- (sin ((- Z)))) " is V11() real ext-real set
(cos ((- Z))) * ((- (sin ((- Z)))) ") is V11() real ext-real set
- ((- Z)) is V11() real ext-real Element of REAL
sin (- ((- Z))) is V11() real ext-real Element of REAL
sin . (- ((- Z))) is V11() real ext-real Element of REAL
(cos ((- Z))) / (sin (- ((- Z)))) is V11() real ext-real Element of REAL
(sin (- ((- Z)))) " is V11() real ext-real set
(cos ((- Z))) * ((sin (- ((- Z)))) ") is V11() real ext-real set
cos (- ((- Z))) is V11() real ext-real Element of REAL
cos . (- ((- Z))) is V11() real ext-real Element of REAL
(cos (- ((- Z)))) / (sin (- ((- Z)))) is V11() real ext-real Element of REAL
(cos (- ((- Z)))) * ((sin (- ((- Z)))) ") is V11() real ext-real set
cot (- ((- Z))) is V11() real ext-real Element of REAL
- ((3 / 4) * PI) is non empty V11() real ext-real non positive negative Element of REAL
PI + (- ((3 / 4) * PI)) is V11() real ext-real Element of REAL
PI + (- ((- Z))) is V11() real ext-real Element of REAL
PI + (- (PI / 4)) is V11() real ext-real Element of REAL
cot (PI + (- ((- Z)))) is V11() real ext-real Element of REAL
cos (PI + (- ((- Z)))) is V11() real ext-real Element of REAL
cos . (PI + (- ((- Z)))) is V11() real ext-real Element of REAL
sin (PI + (- ((- Z)))) is V11() real ext-real Element of REAL
sin . (PI + (- ((- Z)))) is V11() real ext-real Element of REAL
(cos (PI + (- ((- Z))))) / (sin (PI + (- ((- Z))))) is V11() real ext-real Element of REAL
(sin (PI + (- ((- Z))))) " is V11() real ext-real set
(cos (PI + (- ((- Z))))) * ((sin (PI + (- ((- Z))))) ") is V11() real ext-real set
- (cos (- ((- Z)))) is V11() real ext-real Element of REAL
(- (cos (- ((- Z))))) / (sin (PI + (- ((- Z))))) is V11() real ext-real Element of REAL
(- (cos (- ((- Z))))) * ((sin (PI + (- ((- Z))))) ") is V11() real ext-real set
- (sin (- ((- Z)))) is V11() real ext-real Element of REAL
(- (cos (- ((- Z))))) / (- (sin (- ((- Z))))) is V11() real ext-real Element of REAL
(- (sin (- ((- Z))))) " is V11() real ext-real set
(- (cos (- ((- Z))))) * ((- (sin (- ((- Z))))) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
cot (Z) is V11() real ext-real Element of REAL
1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
1 * (Z ") is V11() real ext-real set
sin (Z) is V11() real ext-real Element of REAL
sin . (Z) is V11() real ext-real Element of REAL
cos (Z) is V11() real ext-real Element of REAL
cos . (Z) is V11() real ext-real Element of REAL
(sin (Z)) / (cos (Z)) is V11() real ext-real Element of REAL
(cos (Z)) " is V11() real ext-real set
(sin (Z)) * ((cos (Z)) ") is V11() real ext-real set
tan (Z) is V11() real ext-real Element of REAL
(cos (Z)) / (sin (Z)) is V11() real ext-real Element of REAL
(sin (Z)) " is V11() real ext-real set
(cos (Z)) * ((sin (Z)) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
tan (Z) is V11() real ext-real Element of REAL
1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
1 * (Z ") is V11() real ext-real set
cos (Z) is V11() real ext-real Element of REAL
cos . (Z) is V11() real ext-real Element of REAL
sin (Z) is V11() real ext-real Element of REAL
sin . (Z) is V11() real ext-real Element of REAL
(cos (Z)) / (sin (Z)) is V11() real ext-real Element of REAL
(sin (Z)) " is V11() real ext-real set
(cos (Z)) * ((sin (Z)) ") is V11() real ext-real set
cot (Z) is V11() real ext-real Element of REAL
(sin (Z)) / (cos (Z)) is V11() real ext-real Element of REAL
(cos (Z)) " is V11() real ext-real set
(sin (Z)) * ((cos (Z)) ") is V11() real ext-real set
dom ((tan | ].(- (PI / 2)),(PI / 2).[) ") is V46() V47() V48() Element of K6(REAL)
rng (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
dom (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
(dom tan) /\ ].(- (PI / 2)),(PI / 2).[ is V46() V47() V48() set
f is V11() real ext-real Element of REAL
cos . f is V11() real ext-real Element of REAL
(cos . f) ^2 is V11() real ext-real Element of REAL
(cos . f) * (cos . f) is V11() real ext-real set
1 / ((cos . f) ^2) is V11() real ext-real Element of REAL
((cos . f) ^2) " is V11() real ext-real set
1 * (((cos . f) ^2) ") is V11() real ext-real set
0 / ((cos . f) ^2) is V11() real ext-real Element of REAL
0 * (((cos . f) ^2) ") is V11() real ext-real set
f is V11() real ext-real Element of REAL
diff ((tan | ].(- (PI / 2)),(PI / 2).[),f) is V11() real ext-real Element of REAL
(tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . f is V11() real ext-real Element of REAL
tan `| ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan `| ].(- (PI / 2)),(PI / 2).[) . f is V11() real ext-real Element of REAL
diff (tan,f) is V11() real ext-real Element of REAL
cos . f is V11() real ext-real Element of REAL
(cos . f) ^2 is V11() real ext-real Element of REAL
(cos . f) * (cos . f) is V11() real ext-real set
1 / ((cos . f) ^2) is V11() real ext-real Element of REAL
((cos . f) ^2) " is V11() real ext-real set
1 * (((cos . f) ^2) ") is V11() real ext-real set
(tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((cot | ].0,PI.[) ") is V46() V47() V48() Element of K6(REAL)
rng (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
dom (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
(dom cot) /\ ].0,PI.[ is V46() V47() V48() set
f is V11() real ext-real Element of REAL
sin . f is V11() real ext-real Element of REAL
(sin . f) ^2 is V11() real ext-real Element of REAL
(sin . f) * (sin . f) is V11() real ext-real set
1 / ((sin . f) ^2) is V11() real ext-real Element of REAL
((sin . f) ^2) " is V11() real ext-real set
1 * (((sin . f) ^2) ") is V11() real ext-real set
- (1 / ((sin . f) ^2)) is V11() real ext-real Element of REAL
0 / ((sin . f) ^2) is V11() real ext-real Element of REAL
0 * (((sin . f) ^2) ") is V11() real ext-real set
- 0 is V11() real ext-real V63() Element of REAL
f is V11() real ext-real Element of REAL
diff ((cot | ].0,PI.[),f) is V11() real ext-real Element of REAL
(cot | ].0,PI.[) `| ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((cot | ].0,PI.[) `| ].0,PI.[) . f is V11() real ext-real Element of REAL
cot `| ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot `| ].0,PI.[) . f is V11() real ext-real Element of REAL
diff (cot,f) is V11() real ext-real Element of REAL
sin . f is V11() real ext-real Element of REAL
(sin . f) ^2 is V11() real ext-real Element of REAL
(sin . f) * (sin . f) is V11() real ext-real set
1 / ((sin . f) ^2) is V11() real ext-real Element of REAL
((sin . f) ^2) " is V11() real ext-real set
1 * (((sin . f) ^2) ") is V11() real ext-real set
- (1 / ((sin . f) ^2)) is V11() real ext-real Element of REAL
(cot | ].0,PI.[) | ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
].(- 1),1.[ is open V46() V47() V48() V82() V83() V87() Element of K6(REAL)
Z is V11() real ext-real Element of REAL
rng (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
() | (dom ()) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is V11() real ext-real Element of REAL
rng (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
() | (dom ()) is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is V11() real ext-real Element of REAL
diff ((),Z) is V11() real ext-real Element of REAL
Z ^2 is V11() real ext-real Element of REAL
Z * Z is V11() real ext-real set
1 + (Z ^2) is V11() real ext-real Element of REAL
1 / (1 + (Z ^2)) is V11() real ext-real Element of REAL
(1 + (Z ^2)) " is V11() real ext-real set
1 * ((1 + (Z ^2)) ") is V11() real ext-real set
() . Z is V11() real ext-real Element of REAL
sin . (() . Z) is V11() real ext-real Element of REAL
(sin . (() . Z)) ^2 is V11() real ext-real Element of REAL
(sin . (() . Z)) * (sin . (() . Z)) is V11() real ext-real set
cos . (() . Z) is V11() real ext-real Element of REAL
(cos . (() . Z)) ^2 is V11() real ext-real Element of REAL
(cos . (() . Z)) * (cos . (() . Z)) is V11() real ext-real set
((sin . (() . Z)) ^2) + ((cos . (() . Z)) ^2) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
cos . x is V11() real ext-real Element of REAL
(cos . x) ^2 is V11() real ext-real Element of REAL
(cos . x) * (cos . x) is V11() real ext-real set
1 / ((cos . x) ^2) is V11() real ext-real Element of REAL
((cos . x) ^2) " is V11() real ext-real set
1 * (((cos . x) ^2) ") is V11() real ext-real set
0 / ((cos . x) ^2) is V11() real ext-real Element of REAL
0 * (((cos . x) ^2) ") is V11() real ext-real set
x is V11() real ext-real Element of REAL
diff ((tan | ].(- (PI / 2)),(PI / 2).[),x) is V11() real ext-real Element of REAL
(tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . x is V11() real ext-real Element of REAL
tan `| ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(tan `| ].(- (PI / 2)),(PI / 2).[) . x is V11() real ext-real Element of REAL
diff (tan,x) is V11() real ext-real Element of REAL
cos . x is V11() real ext-real Element of REAL
(cos . x) ^2 is V11() real ext-real Element of REAL
(cos . x) * (cos . x) is V11() real ext-real set
1 / ((cos . x) ^2) is V11() real ext-real Element of REAL
((cos . x) ^2) " is V11() real ext-real set
1 * (((cos . x) ^2) ") is V11() real ext-real set
(Z) is V11() real ext-real Element of REAL
tan (() . Z) is V11() real ext-real Element of REAL
sin (() . Z) is V11() real ext-real Element of REAL
cos (() . Z) is V11() real ext-real Element of REAL
(sin (() . Z)) / (cos (() . Z)) is V11() real ext-real Element of REAL
(cos (() . Z)) " is V11() real ext-real set
(sin (() . Z)) * ((cos (() . Z)) ") is V11() real ext-real set
dom (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
(dom tan) /\ ].(- (PI / 2)),(PI / 2).[ is V46() V47() V48() set
(tan | ].(- (PI / 2)),(PI / 2).[) | ].(- (PI / 2)),(PI / 2).[ is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
Z * (cos (() . Z)) is V11() real ext-real Element of REAL
(cos (() . Z)) ^2 is V11() real ext-real Element of REAL
(cos (() . Z)) * (cos (() . Z)) is V11() real ext-real set
(Z ^2) + 1 is V11() real ext-real Element of REAL
((cos (() . Z)) ^2) * ((Z ^2) + 1) is V11() real ext-real Element of REAL
diff ((tan | ].(- (PI / 2)),(PI / 2).[),(() . Z)) is V11() real ext-real Element of REAL
((tan | ].(- (PI / 2)),(PI / 2).[) `| ].(- (PI / 2)),(PI / 2).[) . (() . Z) is V11() real ext-real Element of REAL
(tan `| ].(- (PI / 2)),(PI / 2).[) . (() . Z) is V11() real ext-real Element of REAL
diff (tan,(() . Z)) is V11() real ext-real Element of REAL
1 / ((cos (() . Z)) ^2) is V11() real ext-real Element of REAL
((cos (() . Z)) ^2) " is V11() real ext-real set
1 * (((cos (() . Z)) ^2) ") is V11() real ext-real set
1 / (1 / ((cos (() . Z)) ^2)) is V11() real ext-real Element of REAL
(1 / ((cos (() . Z)) ^2)) " is V11() real ext-real set
1 * ((1 / ((cos (() . Z)) ^2)) ") is V11() real ext-real set
1 / ((Z ^2) + 1) is V11() real ext-real Element of REAL
((Z ^2) + 1) " is V11() real ext-real set
1 * (((Z ^2) + 1) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
diff ((),Z) is V11() real ext-real Element of REAL
Z ^2 is V11() real ext-real Element of REAL
Z * Z is V11() real ext-real set
1 + (Z ^2) is V11() real ext-real Element of REAL
1 / (1 + (Z ^2)) is V11() real ext-real Element of REAL
(1 + (Z ^2)) " is V11() real ext-real set
1 * ((1 + (Z ^2)) ") is V11() real ext-real set
- (1 / (1 + (Z ^2))) is V11() real ext-real Element of REAL
() . Z is V11() real ext-real Element of REAL
sin . (() . Z) is V11() real ext-real Element of REAL
(sin . (() . Z)) ^2 is V11() real ext-real Element of REAL
(sin . (() . Z)) * (sin . (() . Z)) is V11() real ext-real set
cos . (() . Z) is V11() real ext-real Element of REAL
(cos . (() . Z)) ^2 is V11() real ext-real Element of REAL
(cos . (() . Z)) * (cos . (() . Z)) is V11() real ext-real set
((sin . (() . Z)) ^2) + ((cos . (() . Z)) ^2) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
sin . x is V11() real ext-real Element of REAL
(sin . x) ^2 is V11() real ext-real Element of REAL
(sin . x) * (sin . x) is V11() real ext-real set
1 / ((sin . x) ^2) is V11() real ext-real Element of REAL
((sin . x) ^2) " is V11() real ext-real set
1 * (((sin . x) ^2) ") is V11() real ext-real set
- (1 / ((sin . x) ^2)) is V11() real ext-real Element of REAL
0 / ((sin . x) ^2) is V11() real ext-real Element of REAL
0 * (((sin . x) ^2) ") is V11() real ext-real set
- 0 is V11() real ext-real V63() Element of REAL
x is V11() real ext-real Element of REAL
diff ((cot | ].0,PI.[),x) is V11() real ext-real Element of REAL
(cot | ].0,PI.[) `| ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((cot | ].0,PI.[) `| ].0,PI.[) . x is V11() real ext-real Element of REAL
cot `| ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(cot `| ].0,PI.[) . x is V11() real ext-real Element of REAL
diff (cot,x) is V11() real ext-real Element of REAL
sin . x is V11() real ext-real Element of REAL
(sin . x) ^2 is V11() real ext-real Element of REAL
(sin . x) * (sin . x) is V11() real ext-real set
1 / ((sin . x) ^2) is V11() real ext-real Element of REAL
((sin . x) ^2) " is V11() real ext-real set
1 * (((sin . x) ^2) ") is V11() real ext-real set
- (1 / ((sin . x) ^2)) is V11() real ext-real Element of REAL
(Z) is V11() real ext-real Element of REAL
cot (() . Z) is V11() real ext-real Element of REAL
cos (() . Z) is V11() real ext-real Element of REAL
sin (() . Z) is V11() real ext-real Element of REAL
(cos (() . Z)) / (sin (() . Z)) is V11() real ext-real Element of REAL
(sin (() . Z)) " is V11() real ext-real set
(cos (() . Z)) * ((sin (() . Z)) ") is V11() real ext-real set
dom (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
(dom cot) /\ ].0,PI.[ is V46() V47() V48() set
(cot | ].0,PI.[) | ].0,PI.[ is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
Z * (sin (() . Z)) is V11() real ext-real Element of REAL
(sin (() . Z)) ^2 is V11() real ext-real Element of REAL
(sin (() . Z)) * (sin (() . Z)) is V11() real ext-real set
(Z ^2) + 1 is V11() real ext-real Element of REAL
((sin (() . Z)) ^2) * ((Z ^2) + 1) is V11() real ext-real Element of REAL
diff ((cot | ].0,PI.[),(() . Z)) is V11() real ext-real Element of REAL
((cot | ].0,PI.[) `| ].0,PI.[) . (() . Z) is V11() real ext-real Element of REAL
(cot `| ].0,PI.[) . (() . Z) is V11() real ext-real Element of REAL
diff (cot,(() . Z)) is V11() real ext-real Element of REAL
1 / ((sin (() . Z)) ^2) is V11() real ext-real Element of REAL
((sin (() . Z)) ^2) " is V11() real ext-real set
1 * (((sin (() . Z)) ^2) ") is V11() real ext-real set
- (1 / ((sin (() . Z)) ^2)) is V11() real ext-real Element of REAL
1 / (- (1 / ((sin (() . Z)) ^2))) is V11() real ext-real Element of REAL
(- (1 / ((sin (() . Z)) ^2))) " is V11() real ext-real set
1 * ((- (1 / ((sin (() . Z)) ^2))) ") is V11() real ext-real set
1 / (1 / ((sin (() . Z)) ^2)) is V11() real ext-real Element of REAL
(1 / ((sin (() . Z)) ^2)) " is V11() real ext-real set
1 * ((1 / ((sin (() . Z)) ^2)) ") is V11() real ext-real set
- (1 / (1 / ((sin (() . Z)) ^2))) is V11() real ext-real Element of REAL
1 / ((Z ^2) + 1) is V11() real ext-real Element of REAL
((Z ^2) + 1) " is V11() real ext-real set
1 * (((Z ^2) + 1) ") is V11() real ext-real set
- (1 / ((Z ^2) + 1)) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
diff (tan,Z) is V11() real ext-real Element of REAL
cos . Z is V11() real ext-real Element of REAL
(cos . Z) ^2 is V11() real ext-real Element of REAL
(cos . Z) * (cos . Z) is V11() real ext-real set
1 / ((cos . Z) ^2) is V11() real ext-real Element of REAL
((cos . Z) ^2) " is V11() real ext-real set
1 * (((cos . Z) ^2) ") is V11() real ext-real set
0 / ((cos . Z) ^2) is V11() real ext-real Element of REAL
0 * (((cos . Z) ^2) ") is V11() real ext-real set
rng (tan | ].(- (PI / 2)),(PI / 2).[) is V46() V47() V48() Element of K6(REAL)
Z is V11() real ext-real Element of REAL
diff (cot,Z) is V11() real ext-real Element of REAL
sin . Z is V11() real ext-real Element of REAL
(sin . Z) ^2 is V11() real ext-real Element of REAL
(sin . Z) * (sin . Z) is V11() real ext-real set
1 / ((sin . Z) ^2) is V11() real ext-real Element of REAL
((sin . Z) ^2) " is V11() real ext-real set
1 * (((sin . Z) ^2) ") is V11() real ext-real set
0 / ((sin . Z) ^2) is V11() real ext-real Element of REAL
0 * (((sin . Z) ^2) ") is V11() real ext-real set
- 0 is V11() real ext-real V63() Element of REAL
- (1 / ((sin . Z) ^2)) is V11() real ext-real Element of REAL
rng (cot | ].0,PI.[) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
(() `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
1 * ((1 + (f ^2)) ") is V11() real ext-real set
diff ((),f) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
(() `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
1 * ((1 + (f ^2)) ") is V11() real ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
(() `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
1 * ((1 + (f ^2)) ") is V11() real ext-real set
- (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
(() `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
1 * ((1 + (f ^2)) ") is V11() real ext-real set
- (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
Z (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is open V46() V47() V48() Element of K6(REAL)
(Z (#) ()) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (Z (#) ()) is V46() V47() V48() Element of K6(REAL)
x is V11() real ext-real Element of REAL
((Z (#) ()) `| f) . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
Z / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
Z * ((1 + (x ^2)) ") is V11() real ext-real set
diff ((),x) is V11() real ext-real Element of REAL
Z * (diff ((),x)) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (x ^2)) ") is V11() real ext-real set
Z * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((Z (#) ()) `| f) . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
Z / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
Z * ((1 + (x ^2)) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
Z (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is open V46() V47() V48() Element of K6(REAL)
(Z (#) ()) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (Z (#) ()) is V46() V47() V48() Element of K6(REAL)
x is V11() real ext-real Element of REAL
((Z (#) ()) `| f) . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
Z / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
Z * ((1 + (x ^2)) ") is V11() real ext-real set
- (Z / (1 + (x ^2))) is V11() real ext-real Element of REAL
diff ((),x) is V11() real ext-real Element of REAL
Z * (diff ((),x)) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (x ^2)) ") is V11() real ext-real set
- (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
Z * (- (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((Z (#) ()) `| f) . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
Z / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
Z * ((1 + (x ^2)) ") is V11() real ext-real set
- (Z / (1 + (x ^2))) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f . Z is V11() real ext-real Element of REAL
() * f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
diff ((() * f),Z) is V11() real ext-real Element of REAL
diff (f,Z) is V11() real ext-real Element of REAL
(f . Z) ^2 is V11() real ext-real Element of REAL
(f . Z) * (f . Z) is V11() real ext-real set
1 + ((f . Z) ^2) is V11() real ext-real Element of REAL
(diff (f,Z)) / (1 + ((f . Z) ^2)) is V11() real ext-real Element of REAL
(1 + ((f . Z) ^2)) " is V11() real ext-real set
(diff (f,Z)) * ((1 + ((f . Z) ^2)) ") is V11() real ext-real set
diff ((),(f . Z)) is V11() real ext-real Element of REAL
(diff ((),(f . Z))) * (diff (f,Z)) is V11() real ext-real Element of REAL
1 / (1 + ((f . Z) ^2)) is V11() real ext-real Element of REAL
1 * ((1 + ((f . Z) ^2)) ") is V11() real ext-real set
(diff (f,Z)) * (1 / (1 + ((f . Z) ^2))) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f . Z is V11() real ext-real Element of REAL
() * f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
diff ((() * f),Z) is V11() real ext-real Element of REAL
diff (f,Z) is V11() real ext-real Element of REAL
(f . Z) ^2 is V11() real ext-real Element of REAL
(f . Z) * (f . Z) is V11() real ext-real set
1 + ((f . Z) ^2) is V11() real ext-real Element of REAL
(diff (f,Z)) / (1 + ((f . Z) ^2)) is V11() real ext-real Element of REAL
(1 + ((f . Z) ^2)) " is V11() real ext-real set
(diff (f,Z)) * ((1 + ((f . Z) ^2)) ") is V11() real ext-real set
- ((diff (f,Z)) / (1 + ((f . Z) ^2))) is V11() real ext-real Element of REAL
diff ((),(f . Z)) is V11() real ext-real Element of REAL
(diff ((),(f . Z))) * (diff (f,Z)) is V11() real ext-real Element of REAL
1 / (1 + ((f . Z) ^2)) is V11() real ext-real Element of REAL
1 * ((1 + ((f . Z) ^2)) ") is V11() real ext-real set
- (1 / (1 + ((f . Z) ^2))) is V11() real ext-real Element of REAL
(diff (f,Z)) * (- (1 / (1 + ((f . Z) ^2)))) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
x is open V46() V47() V48() Element of K6(REAL)
g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * g) is V46() V47() V48() Element of K6(REAL)
(() * g) `| x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom g is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() real ext-real Element of REAL
g . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
g . x is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * g) `| x) . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
((Z * x) + f) ^2 is V11() real ext-real Element of REAL
((Z * x) + f) * ((Z * x) + f) is V11() real ext-real set
1 + (((Z * x) + f) ^2) is V11() real ext-real Element of REAL
Z / (1 + (((Z * x) + f) ^2)) is V11() real ext-real Element of REAL
(1 + (((Z * x) + f) ^2)) " is V11() real ext-real set
Z * ((1 + (((Z * x) + f) ^2)) ") is V11() real ext-real set
g . x is V11() real ext-real Element of REAL
diff ((() * g),x) is V11() real ext-real Element of REAL
diff (g,x) is V11() real ext-real Element of REAL
(g . x) ^2 is V11() real ext-real Element of REAL
(g . x) * (g . x) is V11() real ext-real set
1 + ((g . x) ^2) is V11() real ext-real Element of REAL
(diff (g,x)) / (1 + ((g . x) ^2)) is V11() real ext-real Element of REAL
(1 + ((g . x) ^2)) " is V11() real ext-real set
(diff (g,x)) * ((1 + ((g . x) ^2)) ") is V11() real ext-real set
g `| x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(g `| x) . x is V11() real ext-real Element of REAL
((g `| x) . x) / (1 + ((g . x) ^2)) is V11() real ext-real Element of REAL
((g `| x) . x) * ((1 + ((g . x) ^2)) ") is V11() real ext-real set
Z / (1 + ((g . x) ^2)) is V11() real ext-real Element of REAL
Z * ((1 + ((g . x) ^2)) ") is V11() real ext-real set
x is V11() real ext-real Element of REAL
((() * g) `| x) . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
((Z * x) + f) ^2 is V11() real ext-real Element of REAL
((Z * x) + f) * ((Z * x) + f) is V11() real ext-real set
1 + (((Z * x) + f) ^2) is V11() real ext-real Element of REAL
Z / (1 + (((Z * x) + f) ^2)) is V11() real ext-real Element of REAL
(1 + (((Z * x) + f) ^2)) " is V11() real ext-real set
Z * ((1 + (((Z * x) + f) ^2)) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
x is open V46() V47() V48() Element of K6(REAL)
g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * g) is V46() V47() V48() Element of K6(REAL)
(() * g) `| x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom g is V46() V47() V48() Element of K6(REAL)
x is set
x is V11() real ext-real Element of REAL
g . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
g . x is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * g) `| x) . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
((Z * x) + f) ^2 is V11() real ext-real Element of REAL
((Z * x) + f) * ((Z * x) + f) is V11() real ext-real set
1 + (((Z * x) + f) ^2) is V11() real ext-real Element of REAL
Z / (1 + (((Z * x) + f) ^2)) is V11() real ext-real Element of REAL
(1 + (((Z * x) + f) ^2)) " is V11() real ext-real set
Z * ((1 + (((Z * x) + f) ^2)) ") is V11() real ext-real set
- (Z / (1 + (((Z * x) + f) ^2))) is V11() real ext-real Element of REAL
g . x is V11() real ext-real Element of REAL
diff ((() * g),x) is V11() real ext-real Element of REAL
diff (g,x) is V11() real ext-real Element of REAL
(g . x) ^2 is V11() real ext-real Element of REAL
(g . x) * (g . x) is V11() real ext-real set
1 + ((g . x) ^2) is V11() real ext-real Element of REAL
(diff (g,x)) / (1 + ((g . x) ^2)) is V11() real ext-real Element of REAL
(1 + ((g . x) ^2)) " is V11() real ext-real set
(diff (g,x)) * ((1 + ((g . x) ^2)) ") is V11() real ext-real set
- ((diff (g,x)) / (1 + ((g . x) ^2))) is V11() real ext-real Element of REAL
g `| x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(g `| x) . x is V11() real ext-real Element of REAL
((g `| x) . x) / (1 + ((g . x) ^2)) is V11() real ext-real Element of REAL
((g `| x) . x) * ((1 + ((g . x) ^2)) ") is V11() real ext-real set
- (((g `| x) . x) / (1 + ((g . x) ^2))) is V11() real ext-real Element of REAL
Z / (1 + ((g . x) ^2)) is V11() real ext-real Element of REAL
Z * ((1 + ((g . x) ^2)) ") is V11() real ext-real set
- (Z / (1 + ((g . x) ^2))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * g) `| x) . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
((Z * x) + f) ^2 is V11() real ext-real Element of REAL
((Z * x) + f) * ((Z * x) + f) is V11() real ext-real set
1 + (((Z * x) + f) ^2) is V11() real ext-real Element of REAL
Z / (1 + (((Z * x) + f) ^2)) is V11() real ext-real Element of REAL
(1 + (((Z * x) + f) ^2)) " is V11() real ext-real set
Z * ((1 + (((Z * x) + f) ^2)) ") is V11() real ext-real set
- (Z / (1 + (((Z * x) + f) ^2))) is V11() real ext-real Element of REAL
ln * () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (ln * ()) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(ln * ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((ln * ()) `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(1 + (f ^2)) * (() . f) is V11() real ext-real Element of REAL
1 / ((1 + (f ^2)) * (() . f)) is V11() real ext-real Element of REAL
((1 + (f ^2)) * (() . f)) " is V11() real ext-real set
1 * (((1 + (f ^2)) * (() . f)) ") is V11() real ext-real set
diff ((ln * ()),f) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
(diff ((),f)) / (() . f) is V11() real ext-real Element of REAL
(() . f) " is V11() real ext-real set
(diff ((),f)) * ((() . f) ") is V11() real ext-real set
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
1 * ((1 + (f ^2)) ") is V11() real ext-real set
(1 / (1 + (f ^2))) / (() . f) is V11() real ext-real Element of REAL
(1 / (1 + (f ^2))) * ((() . f) ") is V11() real ext-real set
f is V11() real ext-real Element of REAL
((ln * ()) `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(1 + (f ^2)) * (() . f) is V11() real ext-real Element of REAL
1 / ((1 + (f ^2)) * (() . f)) is V11() real ext-real Element of REAL
((1 + (f ^2)) * (() . f)) " is V11() real ext-real set
1 * (((1 + (f ^2)) * (() . f)) ") is V11() real ext-real set
ln * () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (ln * ()) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(ln * ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((ln * ()) `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(1 + (f ^2)) * (() . f) is V11() real ext-real Element of REAL
1 / ((1 + (f ^2)) * (() . f)) is V11() real ext-real Element of REAL
((1 + (f ^2)) * (() . f)) " is V11() real ext-real set
1 * (((1 + (f ^2)) * (() . f)) ") is V11() real ext-real set
- (1 / ((1 + (f ^2)) * (() . f))) is V11() real ext-real Element of REAL
diff ((ln * ()),f) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
(diff ((),f)) / (() . f) is V11() real ext-real Element of REAL
(() . f) " is V11() real ext-real set
(diff ((),f)) * ((() . f) ") is V11() real ext-real set
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
1 * ((1 + (f ^2)) ") is V11() real ext-real set
- (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
(- (1 / (1 + (f ^2)))) / (() . f) is V11() real ext-real Element of REAL
(- (1 / (1 + (f ^2)))) * ((() . f) ") is V11() real ext-real set
(1 / (1 + (f ^2))) / (() . f) is V11() real ext-real Element of REAL
(1 / (1 + (f ^2))) * ((() . f) ") is V11() real ext-real set
- ((1 / (1 + (f ^2))) / (() . f)) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((ln * ()) `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(1 + (f ^2)) * (() . f) is V11() real ext-real Element of REAL
1 / ((1 + (f ^2)) * (() . f)) is V11() real ext-real Element of REAL
((1 + (f ^2)) * (() . f)) " is V11() real ext-real set
1 * (((1 + (f ^2)) * (() . f)) ") is V11() real ext-real set
- (1 / ((1 + (f ^2)) * (() . f))) is V11() real ext-real Element of REAL
Z is natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() Element of NAT
#Z Z is V19() V22( REAL ) V23( REAL ) Function-like V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
(#Z Z) * () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((#Z Z) * ()) is V46() V47() V48() Element of K6(REAL)
Z - 1 is V11() real ext-real V63() Element of REAL
- 1 is non empty V11() real ext-real non positive negative V63() set
Z + (- 1) is V11() real ext-real V63() set
f is open V46() V47() V48() Element of K6(REAL)
((#Z Z) * ()) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
(((#Z Z) * ()) `| f) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
(() . x) #Z (Z - 1) is V11() real ext-real Element of REAL
Z * ((() . x) #Z (Z - 1)) is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
(Z * ((() . x) #Z (Z - 1))) / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
(Z * ((() . x) #Z (Z - 1))) * ((1 + (x ^2)) ") is V11() real ext-real set
diff (((#Z Z) * ()),x) is V11() real ext-real Element of REAL
diff ((),x) is V11() real ext-real Element of REAL
(Z * ((() . x) #Z (Z - 1))) * (diff ((),x)) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (x ^2)) ") is V11() real ext-real set
(Z * ((() . x) #Z (Z - 1))) * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
(((#Z Z) * ()) `| f) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
(() . x) #Z (Z - 1) is V11() real ext-real Element of REAL
Z * ((() . x) #Z (Z - 1)) is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
(Z * ((() . x) #Z (Z - 1))) / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
(Z * ((() . x) #Z (Z - 1))) * ((1 + (x ^2)) ") is V11() real ext-real set
Z is natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() Element of NAT
#Z Z is V19() V22( REAL ) V23( REAL ) Function-like V33( REAL , REAL ) V36() V37() V38() Element of K6(K7(REAL,REAL))
(#Z Z) * () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((#Z Z) * ()) is V46() V47() V48() Element of K6(REAL)
Z - 1 is V11() real ext-real V63() Element of REAL
- 1 is non empty V11() real ext-real non positive negative V63() set
Z + (- 1) is V11() real ext-real V63() set
f is open V46() V47() V48() Element of K6(REAL)
((#Z Z) * ()) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
(((#Z Z) * ()) `| f) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
(() . x) #Z (Z - 1) is V11() real ext-real Element of REAL
Z * ((() . x) #Z (Z - 1)) is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
(Z * ((() . x) #Z (Z - 1))) / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
(Z * ((() . x) #Z (Z - 1))) * ((1 + (x ^2)) ") is V11() real ext-real set
- ((Z * ((() . x) #Z (Z - 1))) / (1 + (x ^2))) is V11() real ext-real Element of REAL
diff (((#Z Z) * ()),x) is V11() real ext-real Element of REAL
diff ((),x) is V11() real ext-real Element of REAL
(Z * ((() . x) #Z (Z - 1))) * (diff ((),x)) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (x ^2)) ") is V11() real ext-real set
- (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(Z * ((() . x) #Z (Z - 1))) * (- (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
(((#Z Z) * ()) `| f) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
(() . x) #Z (Z - 1) is V11() real ext-real Element of REAL
Z * ((() . x) #Z (Z - 1)) is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
(Z * ((() . x) #Z (Z - 1))) / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
(Z * ((() . x) #Z (Z - 1))) * ((1 + (x ^2)) ") is V11() real ext-real set
- ((Z * ((() . x) #Z (Z - 1))) / (1 + (x ^2))) is V11() real ext-real Element of REAL
(#Z 2) * () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
1 / 2 is non empty V11() real ext-real positive non negative Element of REAL
1 * (2 ") is non empty V11() real ext-real positive non negative set
(1 / 2) (#) ((#Z 2) * ()) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((1 / 2) (#) ((#Z 2) * ())) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
((1 / 2) (#) ((#Z 2) * ())) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((#Z 2) * ()) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
(((1 / 2) (#) ((#Z 2) * ())) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(() . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(() . f) * ((1 + (f ^2)) ") is V11() real ext-real set
diff (((#Z 2) * ()),f) is V11() real ext-real Element of REAL
(1 / 2) * (diff (((#Z 2) * ()),f)) is V11() real ext-real Element of REAL
((#Z 2) * ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((#Z 2) * ()) `| Z) . f is V11() real ext-real Element of REAL
(1 / 2) * ((((#Z 2) * ()) `| Z) . f) is V11() real ext-real Element of REAL
2 - 1 is V11() real ext-real V63() Element of REAL
- 1 is non empty V11() real ext-real non positive negative V63() set
2 + (- 1) is V11() real ext-real V63() set
(() . f) #Z (2 - 1) is V11() real ext-real Element of REAL
2 * ((() . f) #Z (2 - 1)) is V11() real ext-real Element of REAL
(2 * ((() . f) #Z (2 - 1))) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(2 * ((() . f) #Z (2 - 1))) * ((1 + (f ^2)) ") is V11() real ext-real set
(1 / 2) * ((2 * ((() . f) #Z (2 - 1))) / (1 + (f ^2))) is V11() real ext-real Element of REAL
2 * (() . f) is V11() real ext-real Element of REAL
(2 * (() . f)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(2 * (() . f)) * ((1 + (f ^2)) ") is V11() real ext-real set
(1 / 2) * ((2 * (() . f)) / (1 + (f ^2))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
(((1 / 2) (#) ((#Z 2) * ())) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(() . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(() . f) * ((1 + (f ^2)) ") is V11() real ext-real set
(#Z 2) * () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(1 / 2) (#) ((#Z 2) * ()) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((1 / 2) (#) ((#Z 2) * ())) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
((1 / 2) (#) ((#Z 2) * ())) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((#Z 2) * ()) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
(((1 / 2) (#) ((#Z 2) * ())) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(() . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(() . f) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((() . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
diff (((#Z 2) * ()),f) is V11() real ext-real Element of REAL
(1 / 2) * (diff (((#Z 2) * ()),f)) is V11() real ext-real Element of REAL
((#Z 2) * ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((#Z 2) * ()) `| Z) . f is V11() real ext-real Element of REAL
(1 / 2) * ((((#Z 2) * ()) `| Z) . f) is V11() real ext-real Element of REAL
2 - 1 is V11() real ext-real V63() Element of REAL
- 1 is non empty V11() real ext-real non positive negative V63() set
2 + (- 1) is V11() real ext-real V63() set
(() . f) #Z (2 - 1) is V11() real ext-real Element of REAL
2 * ((() . f) #Z (2 - 1)) is V11() real ext-real Element of REAL
(2 * ((() . f) #Z (2 - 1))) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(2 * ((() . f) #Z (2 - 1))) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((2 * ((() . f) #Z (2 - 1))) / (1 + (f ^2))) is V11() real ext-real Element of REAL
(1 / 2) * (- ((2 * ((() . f) #Z (2 - 1))) / (1 + (f ^2)))) is V11() real ext-real Element of REAL
(() . f) #Z 1 is V11() real ext-real Element of REAL
2 * ((() . f) #Z 1) is V11() real ext-real Element of REAL
(2 * ((() . f) #Z 1)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(2 * ((() . f) #Z 1)) * ((1 + (f ^2)) ") is V11() real ext-real set
(1 / 2) * ((2 * ((() . f) #Z 1)) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((1 / 2) * ((2 * ((() . f) #Z 1)) / (1 + (f ^2)))) is V11() real ext-real Element of REAL
2 * (() . f) is V11() real ext-real Element of REAL
(2 * (() . f)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(2 * (() . f)) * ((1 + (f ^2)) ") is V11() real ext-real set
(1 / 2) * ((2 * (() . f)) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((1 / 2) * ((2 * (() . f)) / (1 + (f ^2)))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
(((1 / 2) (#) ((#Z 2) * ())) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(() . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(() . f) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((() . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (id Z) is V46() V47() V48() Element of K6(REAL)
(dom (id Z)) /\ (dom ()) is V46() V47() V48() set
dom ((id Z) (#) ()) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
(id Z) . f is V11() real ext-real Element of REAL
1 * f is V11() real ext-real Element of REAL
(1 * f) + 0 is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
(((id Z) (#) ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
f / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
f * ((1 + (f ^2)) ") is V11() real ext-real set
(() . f) + (f / (1 + (f ^2))) is V11() real ext-real Element of REAL
diff ((id Z),f) is V11() real ext-real Element of REAL
(() . f) * (diff ((id Z),f)) is V11() real ext-real Element of REAL
(id Z) . f is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
((id Z) . f) * (diff ((),f)) is V11() real ext-real Element of REAL
((() . f) * (diff ((id Z),f))) + (((id Z) . f) * (diff ((),f))) is V11() real ext-real Element of REAL
(id Z) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) `| Z) . f is V11() real ext-real Element of REAL
(() . f) * (((id Z) `| Z) . f) is V11() real ext-real Element of REAL
((() . f) * (((id Z) `| Z) . f)) + (((id Z) . f) * (diff ((),f))) is V11() real ext-real Element of REAL
(() . f) * 1 is V11() real ext-real Element of REAL
((() . f) * 1) + (((id Z) . f) * (diff ((),f))) is V11() real ext-real Element of REAL
f * (diff ((),f)) is V11() real ext-real Element of REAL
(() . f) + (f * (diff ((),f))) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
f * (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
(() . f) + (f * (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
(((id Z) (#) ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
f / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
f * ((1 + (f ^2)) ") is V11() real ext-real set
(() . f) + (f / (1 + (f ^2))) is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (id Z) is V46() V47() V48() Element of K6(REAL)
(dom (id Z)) /\ (dom ()) is V46() V47() V48() set
dom ((id Z) (#) ()) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
(id Z) . f is V11() real ext-real Element of REAL
1 * f is V11() real ext-real Element of REAL
(1 * f) + 0 is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
(((id Z) (#) ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
f / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
f * ((1 + (f ^2)) ") is V11() real ext-real set
(() . f) - (f / (1 + (f ^2))) is V11() real ext-real Element of REAL
- (f / (1 + (f ^2))) is V11() real ext-real set
(() . f) + (- (f / (1 + (f ^2)))) is V11() real ext-real set
diff ((id Z),f) is V11() real ext-real Element of REAL
(() . f) * (diff ((id Z),f)) is V11() real ext-real Element of REAL
(id Z) . f is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
((id Z) . f) * (diff ((),f)) is V11() real ext-real Element of REAL
((() . f) * (diff ((id Z),f))) + (((id Z) . f) * (diff ((),f))) is V11() real ext-real Element of REAL
(id Z) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) `| Z) . f is V11() real ext-real Element of REAL
(() . f) * (((id Z) `| Z) . f) is V11() real ext-real Element of REAL
((() . f) * (((id Z) `| Z) . f)) + (((id Z) . f) * (diff ((),f))) is V11() real ext-real Element of REAL
(() . f) * 1 is V11() real ext-real Element of REAL
((() . f) * 1) + (((id Z) . f) * (diff ((),f))) is V11() real ext-real Element of REAL
f * (diff ((),f)) is V11() real ext-real Element of REAL
(() . f) + (f * (diff ((),f))) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
- (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
f * (- (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
(() . f) + (f * (- (1 / (1 + (f ^2))))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
(((id Z) (#) ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
f / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
f * ((1 + (f ^2)) ") is V11() real ext-real set
(() . f) - (f / (1 + (f ^2))) is V11() real ext-real Element of REAL
- (f / (1 + (f ^2))) is V11() real ext-real set
(() . f) + (- (f / (1 + (f ^2)))) is V11() real ext-real set
Z is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
x is open V46() V47() V48() Element of K6(REAL)
g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (g (#) ()) is V46() V47() V48() Element of K6(REAL)
(g (#) ()) `| x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom g is V46() V47() V48() Element of K6(REAL)
(dom g) /\ (dom ()) is V46() V47() V48() set
x is V11() real ext-real Element of REAL
((g (#) ()) `| x) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
Z * (() . x) is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
((Z * x) + f) / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
((Z * x) + f) * ((1 + (x ^2)) ") is V11() real ext-real set
(Z * (() . x)) + (((Z * x) + f) / (1 + (x ^2))) is V11() real ext-real Element of REAL
diff (g,x) is V11() real ext-real Element of REAL
(() . x) * (diff (g,x)) is V11() real ext-real Element of REAL
g . x is V11() real ext-real Element of REAL
diff ((),x) is V11() real ext-real Element of REAL
(g . x) * (diff ((),x)) is V11() real ext-real Element of REAL
((() . x) * (diff (g,x))) + ((g . x) * (diff ((),x))) is V11() real ext-real Element of REAL
g `| x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(g `| x) . x is V11() real ext-real Element of REAL
(() . x) * ((g `| x) . x) is V11() real ext-real Element of REAL
((() . x) * ((g `| x) . x)) + ((g . x) * (diff ((),x))) is V11() real ext-real Element of REAL
(() . x) * Z is V11() real ext-real Element of REAL
((() . x) * Z) + ((g . x) * (diff ((),x))) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (x ^2)) ") is V11() real ext-real set
(g . x) * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
((() . x) * Z) + ((g . x) * (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((g (#) ()) `| x) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
Z * (() . x) is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
((Z * x) + f) / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
((Z * x) + f) * ((1 + (x ^2)) ") is V11() real ext-real set
(Z * (() . x)) + (((Z * x) + f) / (1 + (x ^2))) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
x is open V46() V47() V48() Element of K6(REAL)
g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (g (#) ()) is V46() V47() V48() Element of K6(REAL)
(g (#) ()) `| x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom g is V46() V47() V48() Element of K6(REAL)
(dom g) /\ (dom ()) is V46() V47() V48() set
x is V11() real ext-real Element of REAL
((g (#) ()) `| x) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
Z * (() . x) is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
((Z * x) + f) / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
((Z * x) + f) * ((1 + (x ^2)) ") is V11() real ext-real set
(Z * (() . x)) - (((Z * x) + f) / (1 + (x ^2))) is V11() real ext-real Element of REAL
- (((Z * x) + f) / (1 + (x ^2))) is V11() real ext-real set
(Z * (() . x)) + (- (((Z * x) + f) / (1 + (x ^2)))) is V11() real ext-real set
diff (g,x) is V11() real ext-real Element of REAL
(() . x) * (diff (g,x)) is V11() real ext-real Element of REAL
g . x is V11() real ext-real Element of REAL
diff ((),x) is V11() real ext-real Element of REAL
(g . x) * (diff ((),x)) is V11() real ext-real Element of REAL
((() . x) * (diff (g,x))) + ((g . x) * (diff ((),x))) is V11() real ext-real Element of REAL
g `| x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(g `| x) . x is V11() real ext-real Element of REAL
(() . x) * ((g `| x) . x) is V11() real ext-real Element of REAL
((() . x) * ((g `| x) . x)) + ((g . x) * (diff ((),x))) is V11() real ext-real Element of REAL
(() . x) * Z is V11() real ext-real Element of REAL
((() . x) * Z) + ((g . x) * (diff ((),x))) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (x ^2)) ") is V11() real ext-real set
- (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(g . x) * (- (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
((() . x) * Z) + ((g . x) * (- (1 / (1 + (x ^2))))) is V11() real ext-real Element of REAL
(g . x) * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
((() . x) * Z) - ((g . x) * (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
- ((g . x) * (1 / (1 + (x ^2)))) is V11() real ext-real set
((() . x) * Z) + (- ((g . x) * (1 / (1 + (x ^2))))) is V11() real ext-real set
x is V11() real ext-real Element of REAL
((g (#) ()) `| x) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
Z * (() . x) is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) + f is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
((Z * x) + f) / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
((Z * x) + f) * ((1 + (x ^2)) ") is V11() real ext-real set
(Z * (() . x)) - (((Z * x) + f) / (1 + (x ^2))) is V11() real ext-real Element of REAL
- (((Z * x) + f) / (1 + (x ^2))) is V11() real ext-real set
(Z * (() . x)) + (- (((Z * x) + f) / (1 + (x ^2)))) is V11() real ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(1 / 2) (#) (() * f) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((1 / 2) (#) (() * f)) is V46() V47() V48() Element of K6(REAL)
((1 / 2) (#) (() * f)) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() real ext-real Element of REAL
f . x is V11() real ext-real Element of REAL
2 * x is V11() real ext-real Element of REAL
(2 * x) + 0 is V11() real ext-real Element of REAL
dom (() * f) is V46() V47() V48() Element of K6(REAL)
x is V11() real ext-real Element of REAL
(((1 / 2) (#) (() * f)) `| Z) . x is V11() real ext-real Element of REAL
2 * x is V11() real ext-real Element of REAL
(2 * x) ^2 is V11() real ext-real Element of REAL
(2 * x) * (2 * x) is V11() real ext-real set
1 + ((2 * x) ^2) is V11() real ext-real Element of REAL
1 / (1 + ((2 * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((2 * x) ^2)) " is V11() real ext-real set
1 * ((1 + ((2 * x) ^2)) ") is V11() real ext-real set
diff ((() * f),x) is V11() real ext-real Element of REAL
(1 / 2) * (diff ((() * f),x)) is V11() real ext-real Element of REAL
(() * f) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((() * f) `| Z) . x is V11() real ext-real Element of REAL
(1 / 2) * (((() * f) `| Z) . x) is V11() real ext-real Element of REAL
(2 * x) + 0 is V11() real ext-real Element of REAL
((2 * x) + 0) ^2 is V11() real ext-real Element of REAL
((2 * x) + 0) * ((2 * x) + 0) is V11() real ext-real set
1 + (((2 * x) + 0) ^2) is V11() real ext-real Element of REAL
2 / (1 + (((2 * x) + 0) ^2)) is V11() real ext-real Element of REAL
(1 + (((2 * x) + 0) ^2)) " is V11() real ext-real set
2 * ((1 + (((2 * x) + 0) ^2)) ") is V11() real ext-real set
(1 / 2) * (2 / (1 + (((2 * x) + 0) ^2))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
(((1 / 2) (#) (() * f)) `| Z) . x is V11() real ext-real Element of REAL
2 * x is V11() real ext-real Element of REAL
(2 * x) ^2 is V11() real ext-real Element of REAL
(2 * x) * (2 * x) is V11() real ext-real set
1 + ((2 * x) ^2) is V11() real ext-real Element of REAL
1 / (1 + ((2 * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((2 * x) ^2)) " is V11() real ext-real set
1 * ((1 + ((2 * x) ^2)) ") is V11() real ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(1 / 2) (#) (() * f) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((1 / 2) (#) (() * f)) is V46() V47() V48() Element of K6(REAL)
((1 / 2) (#) (() * f)) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V11() real ext-real Element of REAL
f . x is V11() real ext-real Element of REAL
2 * x is V11() real ext-real Element of REAL
(2 * x) + 0 is V11() real ext-real Element of REAL
dom (() * f) is V46() V47() V48() Element of K6(REAL)
x is V11() real ext-real Element of REAL
(((1 / 2) (#) (() * f)) `| Z) . x is V11() real ext-real Element of REAL
2 * x is V11() real ext-real Element of REAL
(2 * x) ^2 is V11() real ext-real Element of REAL
(2 * x) * (2 * x) is V11() real ext-real set
1 + ((2 * x) ^2) is V11() real ext-real Element of REAL
1 / (1 + ((2 * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((2 * x) ^2)) " is V11() real ext-real set
1 * ((1 + ((2 * x) ^2)) ") is V11() real ext-real set
- (1 / (1 + ((2 * x) ^2))) is V11() real ext-real Element of REAL
diff ((() * f),x) is V11() real ext-real Element of REAL
(1 / 2) * (diff ((() * f),x)) is V11() real ext-real Element of REAL
(() * f) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((() * f) `| Z) . x is V11() real ext-real Element of REAL
(1 / 2) * (((() * f) `| Z) . x) is V11() real ext-real Element of REAL
(2 * x) + 0 is V11() real ext-real Element of REAL
((2 * x) + 0) ^2 is V11() real ext-real Element of REAL
((2 * x) + 0) * ((2 * x) + 0) is V11() real ext-real set
1 + (((2 * x) + 0) ^2) is V11() real ext-real Element of REAL
2 / (1 + (((2 * x) + 0) ^2)) is V11() real ext-real Element of REAL
(1 + (((2 * x) + 0) ^2)) " is V11() real ext-real set
2 * ((1 + (((2 * x) + 0) ^2)) ") is V11() real ext-real set
- (2 / (1 + (((2 * x) + 0) ^2))) is V11() real ext-real Element of REAL
(1 / 2) * (- (2 / (1 + (((2 * x) + 0) ^2)))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
(((1 / 2) (#) (() * f)) `| Z) . x is V11() real ext-real Element of REAL
2 * x is V11() real ext-real Element of REAL
(2 * x) ^2 is V11() real ext-real Element of REAL
(2 * x) * (2 * x) is V11() real ext-real set
1 + ((2 * x) ^2) is V11() real ext-real Element of REAL
1 / (1 + ((2 * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((2 * x) ^2)) " is V11() real ext-real set
1 * ((1 + ((2 * x) ^2)) ") is V11() real ext-real set
- (1 / (1 + ((2 * x) ^2))) is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f + x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (f + x) is V46() V47() V48() Element of K6(REAL)
(f + x) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V11() real ext-real Element of REAL
dom f is V46() V47() V48() Element of K6(REAL)
dom x is V46() V47() V48() Element of K6(REAL)
(dom f) /\ (dom x) is V46() V47() V48() set
g is V11() real ext-real Element of REAL
f . g is V11() real ext-real Element of REAL
0 * g is V11() real ext-real Element of REAL
(0 * g) + 1 is V11() real ext-real Element of REAL
x `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V11() real ext-real Element of REAL
(x `| Z) . g is V11() real ext-real Element of REAL
2 * g is V11() real ext-real Element of REAL
2 - 1 is V11() real ext-real V63() Element of REAL
- 1 is non empty V11() real ext-real non positive negative V63() set
2 + (- 1) is V11() real ext-real V63() set
g #Z (2 - 1) is V11() real ext-real Element of REAL
2 * (g #Z (2 - 1)) is V11() real ext-real Element of REAL
diff (x,g) is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
((f + x) `| Z) . g is V11() real ext-real Element of REAL
2 * g is V11() real ext-real Element of REAL
diff (f,g) is V11() real ext-real Element of REAL
diff (x,g) is V11() real ext-real Element of REAL
(diff (f,g)) + (diff (x,g)) is V11() real ext-real Element of REAL
f `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(f `| Z) . g is V11() real ext-real Element of REAL
((f `| Z) . g) + (diff (x,g)) is V11() real ext-real Element of REAL
(x `| Z) . g is V11() real ext-real Element of REAL
((f `| Z) . g) + ((x `| Z) . g) is V11() real ext-real Element of REAL
0 + ((x `| Z) . g) is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
((f + x) `| Z) . g is V11() real ext-real Element of REAL
2 * g is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f + x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
ln * (f + x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(1 / 2) (#) (ln * (f + x)) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((1 / 2) (#) (ln * (f + x))) is V46() V47() V48() Element of K6(REAL)
((1 / 2) (#) (ln * (f + x))) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (ln * (f + x)) is V46() V47() V48() Element of K6(REAL)
dom (f + x) is V46() V47() V48() Element of K6(REAL)
g is set
g is V11() real ext-real Element of REAL
(f + x) . g is V11() real ext-real Element of REAL
f . g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
(f . g) + (x . g) is V11() real ext-real Element of REAL
1 + (x . g) is V11() real ext-real Element of REAL
1 + 1 is non empty V11() real ext-real positive non negative V63() Element of REAL
g #Z (1 + 1) is V11() real ext-real Element of REAL
1 + (g #Z (1 + 1)) is V11() real ext-real Element of REAL
g #Z 1 is V11() real ext-real Element of REAL
(g #Z 1) * (g #Z 1) is V11() real ext-real Element of REAL
1 + ((g #Z 1) * (g #Z 1)) is V11() real ext-real Element of REAL
g * (g #Z 1) is V11() real ext-real Element of REAL
1 + (g * (g #Z 1)) is V11() real ext-real Element of REAL
g * g is V11() real ext-real Element of REAL
1 + (g * g) is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
(((1 / 2) (#) (ln * (f + x))) `| Z) . g is V11() real ext-real Element of REAL
g ^2 is V11() real ext-real Element of REAL
g * g is V11() real ext-real set
1 + (g ^2) is V11() real ext-real Element of REAL
g / (1 + (g ^2)) is V11() real ext-real Element of REAL
(1 + (g ^2)) " is V11() real ext-real set
g * ((1 + (g ^2)) ") is V11() real ext-real set
(f + x) . g is V11() real ext-real Element of REAL
f . g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
(f . g) + (x . g) is V11() real ext-real Element of REAL
1 + (x . g) is V11() real ext-real Element of REAL
1 + 1 is non empty V11() real ext-real positive non negative V63() Element of REAL
g #Z (1 + 1) is V11() real ext-real Element of REAL
1 + (g #Z (1 + 1)) is V11() real ext-real Element of REAL
g #Z 1 is V11() real ext-real Element of REAL
(g #Z 1) * (g #Z 1) is V11() real ext-real Element of REAL
1 + ((g #Z 1) * (g #Z 1)) is V11() real ext-real Element of REAL
g * (g #Z 1) is V11() real ext-real Element of REAL
1 + (g * (g #Z 1)) is V11() real ext-real Element of REAL
g * g is V11() real ext-real Element of REAL
1 + (g * g) is V11() real ext-real Element of REAL
diff ((ln * (f + x)),g) is V11() real ext-real Element of REAL
diff ((f + x),g) is V11() real ext-real Element of REAL
(diff ((f + x),g)) / ((f + x) . g) is V11() real ext-real Element of REAL
((f + x) . g) " is V11() real ext-real set
(diff ((f + x),g)) * (((f + x) . g) ") is V11() real ext-real set
(f + x) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((f + x) `| Z) . g is V11() real ext-real Element of REAL
(((f + x) `| Z) . g) / ((f + x) . g) is V11() real ext-real Element of REAL
(((f + x) `| Z) . g) * (((f + x) . g) ") is V11() real ext-real set
2 * g is V11() real ext-real Element of REAL
(2 * g) / (1 + (g ^2)) is V11() real ext-real Element of REAL
(2 * g) * ((1 + (g ^2)) ") is V11() real ext-real set
(1 / 2) * ((2 * g) / (1 + (g ^2))) is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
(((1 / 2) (#) (ln * (f + x))) `| Z) . g is V11() real ext-real Element of REAL
g ^2 is V11() real ext-real Element of REAL
g * g is V11() real ext-real set
1 + (g ^2) is V11() real ext-real Element of REAL
g / (1 + (g ^2)) is V11() real ext-real Element of REAL
(1 + (g ^2)) " is V11() real ext-real set
g * ((1 + (g ^2)) ") is V11() real ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f + x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
ln * (f + x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(1 / 2) (#) (ln * (f + x)) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) (#) ()) - ((1 / 2) (#) (ln * (f + x))) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
- ((1 / 2) (#) (ln * (f + x))) is V19() Function-like V36() set
- 1 is non empty V11() real ext-real non positive negative V63() set
(- 1) (#) ((1 / 2) (#) (ln * (f + x))) is V19() Function-like set
((id Z) (#) ()) + (- ((1 / 2) (#) (ln * (f + x)))) is V19() Function-like set
dom (((id Z) (#) ()) - ((1 / 2) (#) (ln * (f + x)))) is V46() V47() V48() Element of K6(REAL)
(((id Z) (#) ()) - ((1 / 2) (#) (ln * (f + x)))) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id Z) (#) ()) is V46() V47() V48() Element of K6(REAL)
dom ((1 / 2) (#) (ln * (f + x))) is V46() V47() V48() Element of K6(REAL)
(dom ((id Z) (#) ())) /\ (dom ((1 / 2) (#) (ln * (f + x)))) is V46() V47() V48() set
g is V11() real ext-real Element of REAL
((((id Z) (#) ()) - ((1 / 2) (#) (ln * (f + x)))) `| Z) . g is V11() real ext-real Element of REAL
() . g is V11() real ext-real Element of REAL
diff (((id Z) (#) ()),g) is V11() real ext-real Element of REAL
diff (((1 / 2) (#) (ln * (f + x))),g) is V11() real ext-real Element of REAL
(diff (((id Z) (#) ()),g)) - (diff (((1 / 2) (#) (ln * (f + x))),g)) is V11() real ext-real Element of REAL
- (diff (((1 / 2) (#) (ln * (f + x))),g)) is V11() real ext-real set
(diff (((id Z) (#) ()),g)) + (- (diff (((1 / 2) (#) (ln * (f + x))),g))) is V11() real ext-real set
((id Z) (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((id Z) (#) ()) `| Z) . g is V11() real ext-real Element of REAL
((((id Z) (#) ()) `| Z) . g) - (diff (((1 / 2) (#) (ln * (f + x))),g)) is V11() real ext-real Element of REAL
((((id Z) (#) ()) `| Z) . g) + (- (diff (((1 / 2) (#) (ln * (f + x))),g))) is V11() real ext-real set
((1 / 2) (#) (ln * (f + x))) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((1 / 2) (#) (ln * (f + x))) `| Z) . g is V11() real ext-real Element of REAL
((((id Z) (#) ()) `| Z) . g) - ((((1 / 2) (#) (ln * (f + x))) `| Z) . g) is V11() real ext-real Element of REAL
- ((((1 / 2) (#) (ln * (f + x))) `| Z) . g) is V11() real ext-real set
((((id Z) (#) ()) `| Z) . g) + (- ((((1 / 2) (#) (ln * (f + x))) `| Z) . g)) is V11() real ext-real set
g ^2 is V11() real ext-real Element of REAL
g * g is V11() real ext-real set
1 + (g ^2) is V11() real ext-real Element of REAL
g / (1 + (g ^2)) is V11() real ext-real Element of REAL
(1 + (g ^2)) " is V11() real ext-real set
g * ((1 + (g ^2)) ") is V11() real ext-real set
(() . g) + (g / (1 + (g ^2))) is V11() real ext-real Element of REAL
((() . g) + (g / (1 + (g ^2)))) - ((((1 / 2) (#) (ln * (f + x))) `| Z) . g) is V11() real ext-real Element of REAL
((() . g) + (g / (1 + (g ^2)))) + (- ((((1 / 2) (#) (ln * (f + x))) `| Z) . g)) is V11() real ext-real set
((() . g) + (g / (1 + (g ^2)))) - (g / (1 + (g ^2))) is V11() real ext-real Element of REAL
- (g / (1 + (g ^2))) is V11() real ext-real set
((() . g) + (g / (1 + (g ^2)))) + (- (g / (1 + (g ^2)))) is V11() real ext-real set
g is V11() real ext-real Element of REAL
((((id Z) (#) ()) - ((1 / 2) (#) (ln * (f + x)))) `| Z) . g is V11() real ext-real Element of REAL
() . g is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f + x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
ln * (f + x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(1 / 2) (#) (ln * (f + x)) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) (#) ()) + ((1 / 2) (#) (ln * (f + x))) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (((id Z) (#) ()) + ((1 / 2) (#) (ln * (f + x)))) is V46() V47() V48() Element of K6(REAL)
(((id Z) (#) ()) + ((1 / 2) (#) (ln * (f + x)))) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id Z) (#) ()) is V46() V47() V48() Element of K6(REAL)
dom ((1 / 2) (#) (ln * (f + x))) is V46() V47() V48() Element of K6(REAL)
(dom ((id Z) (#) ())) /\ (dom ((1 / 2) (#) (ln * (f + x)))) is V46() V47() V48() set
g is V11() real ext-real Element of REAL
((((id Z) (#) ()) + ((1 / 2) (#) (ln * (f + x)))) `| Z) . g is V11() real ext-real Element of REAL
() . g is V11() real ext-real Element of REAL
diff (((id Z) (#) ()),g) is V11() real ext-real Element of REAL
diff (((1 / 2) (#) (ln * (f + x))),g) is V11() real ext-real Element of REAL
(diff (((id Z) (#) ()),g)) + (diff (((1 / 2) (#) (ln * (f + x))),g)) is V11() real ext-real Element of REAL
((id Z) (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((id Z) (#) ()) `| Z) . g is V11() real ext-real Element of REAL
((((id Z) (#) ()) `| Z) . g) + (diff (((1 / 2) (#) (ln * (f + x))),g)) is V11() real ext-real Element of REAL
((1 / 2) (#) (ln * (f + x))) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((1 / 2) (#) (ln * (f + x))) `| Z) . g is V11() real ext-real Element of REAL
((((id Z) (#) ()) `| Z) . g) + ((((1 / 2) (#) (ln * (f + x))) `| Z) . g) is V11() real ext-real Element of REAL
g ^2 is V11() real ext-real Element of REAL
g * g is V11() real ext-real set
1 + (g ^2) is V11() real ext-real Element of REAL
g / (1 + (g ^2)) is V11() real ext-real Element of REAL
(1 + (g ^2)) " is V11() real ext-real set
g * ((1 + (g ^2)) ") is V11() real ext-real set
(() . g) - (g / (1 + (g ^2))) is V11() real ext-real Element of REAL
- (g / (1 + (g ^2))) is V11() real ext-real set
(() . g) + (- (g / (1 + (g ^2)))) is V11() real ext-real set
((() . g) - (g / (1 + (g ^2)))) + ((((1 / 2) (#) (ln * (f + x))) `| Z) . g) is V11() real ext-real Element of REAL
((() . g) - (g / (1 + (g ^2)))) + (g / (1 + (g ^2))) is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
((((id Z) (#) ()) + ((1 / 2) (#) (ln * (f + x)))) `| Z) . g is V11() real ext-real Element of REAL
() . g is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
f is open V46() V47() V48() Element of K6(REAL)
id f is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(id f) (#) (() * x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id f) (#) (() * x)) is V46() V47() V48() Element of K6(REAL)
((id f) (#) (() * x)) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (id f) is V46() V47() V48() Element of K6(REAL)
dom (() * x) is V46() V47() V48() Element of K6(REAL)
(dom (id f)) /\ (dom (() * x)) is V46() V47() V48() set
1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
1 * (Z ") is V11() real ext-real set
g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
(1 / Z) * g is V11() real ext-real Element of REAL
((1 / Z) * g) + 0 is V11() real ext-real Element of REAL
g / Z is V11() real ext-real Element of REAL
g * (Z ") is V11() real ext-real set
g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
(1 / Z) * g is V11() real ext-real Element of REAL
((1 / Z) * g) + 0 is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
(id f) . g is V11() real ext-real Element of REAL
1 * g is V11() real ext-real Element of REAL
(1 * g) + 0 is V11() real ext-real Element of REAL
(() * x) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V11() real ext-real Element of REAL
((() * x) `| f) . g is V11() real ext-real Element of REAL
g / Z is V11() real ext-real Element of REAL
g * (Z ") is V11() real ext-real set
(g / Z) ^2 is V11() real ext-real Element of REAL
(g / Z) * (g / Z) is V11() real ext-real set
1 + ((g / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((g / Z) ^2)) is V11() real ext-real Element of REAL
1 / (Z * (1 + ((g / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((g / Z) ^2))) " is V11() real ext-real set
1 * ((Z * (1 + ((g / Z) ^2))) ") is V11() real ext-real set
(1 / Z) * g is V11() real ext-real Element of REAL
((1 / Z) * g) + 0 is V11() real ext-real Element of REAL
(((1 / Z) * g) + 0) ^2 is V11() real ext-real Element of REAL
(((1 / Z) * g) + 0) * (((1 / Z) * g) + 0) is V11() real ext-real set
1 + ((((1 / Z) * g) + 0) ^2) is V11() real ext-real Element of REAL
(1 / Z) / (1 + ((((1 / Z) * g) + 0) ^2)) is V11() real ext-real Element of REAL
(1 + ((((1 / Z) * g) + 0) ^2)) " is V11() real ext-real set
(1 / Z) * ((1 + ((((1 / Z) * g) + 0) ^2)) ") is V11() real ext-real set
g is V11() real ext-real Element of REAL
(((id f) (#) (() * x)) `| f) . g is V11() real ext-real Element of REAL
g / Z is V11() real ext-real Element of REAL
g * (Z ") is V11() real ext-real set
() . (g / Z) is V11() real ext-real Element of REAL
(g / Z) ^2 is V11() real ext-real Element of REAL
(g / Z) * (g / Z) is V11() real ext-real set
1 + ((g / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((g / Z) ^2)) is V11() real ext-real Element of REAL
g / (Z * (1 + ((g / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((g / Z) ^2))) " is V11() real ext-real set
g * ((Z * (1 + ((g / Z) ^2))) ") is V11() real ext-real set
(() . (g / Z)) + (g / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real Element of REAL
(() * x) . g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
() . (x . g) is V11() real ext-real Element of REAL
diff ((id f),g) is V11() real ext-real Element of REAL
((() * x) . g) * (diff ((id f),g)) is V11() real ext-real Element of REAL
(id f) . g is V11() real ext-real Element of REAL
diff ((() * x),g) is V11() real ext-real Element of REAL
((id f) . g) * (diff ((() * x),g)) is V11() real ext-real Element of REAL
(((() * x) . g) * (diff ((id f),g))) + (((id f) . g) * (diff ((() * x),g))) is V11() real ext-real Element of REAL
(id f) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id f) `| f) . g is V11() real ext-real Element of REAL
((() * x) . g) * (((id f) `| f) . g) is V11() real ext-real Element of REAL
(((() * x) . g) * (((id f) `| f) . g)) + (((id f) . g) * (diff ((() * x),g))) is V11() real ext-real Element of REAL
((() * x) . g) * 1 is V11() real ext-real Element of REAL
(((() * x) . g) * 1) + (((id f) . g) * (diff ((() * x),g))) is V11() real ext-real Element of REAL
((() * x) `| f) . g is V11() real ext-real Element of REAL
((id f) . g) * (((() * x) `| f) . g) is V11() real ext-real Element of REAL
(((() * x) . g) * 1) + (((id f) . g) * (((() * x) `| f) . g)) is V11() real ext-real Element of REAL
g * (((() * x) `| f) . g) is V11() real ext-real Element of REAL
((() * x) . g) + (g * (((() * x) `| f) . g)) is V11() real ext-real Element of REAL
1 / (Z * (1 + ((g / Z) ^2))) is V11() real ext-real Element of REAL
1 * ((Z * (1 + ((g / Z) ^2))) ") is V11() real ext-real set
g * (1 / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real Element of REAL
(() . (g / Z)) + (g * (1 / (Z * (1 + ((g / Z) ^2))))) is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
(((id f) (#) (() * x)) `| f) . g is V11() real ext-real Element of REAL
g / Z is V11() real ext-real Element of REAL
g * (Z ") is V11() real ext-real set
() . (g / Z) is V11() real ext-real Element of REAL
(g / Z) ^2 is V11() real ext-real Element of REAL
(g / Z) * (g / Z) is V11() real ext-real set
1 + ((g / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((g / Z) ^2)) is V11() real ext-real Element of REAL
g / (Z * (1 + ((g / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((g / Z) ^2))) " is V11() real ext-real set
g * ((Z * (1 + ((g / Z) ^2))) ") is V11() real ext-real set
(() . (g / Z)) + (g / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
f is open V46() V47() V48() Element of K6(REAL)
id f is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(id f) (#) (() * x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id f) (#) (() * x)) is V46() V47() V48() Element of K6(REAL)
((id f) (#) (() * x)) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (id f) is V46() V47() V48() Element of K6(REAL)
dom (() * x) is V46() V47() V48() Element of K6(REAL)
(dom (id f)) /\ (dom (() * x)) is V46() V47() V48() set
1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
1 * (Z ") is V11() real ext-real set
g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
(1 / Z) * g is V11() real ext-real Element of REAL
((1 / Z) * g) + 0 is V11() real ext-real Element of REAL
g / Z is V11() real ext-real Element of REAL
g * (Z ") is V11() real ext-real set
g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
(1 / Z) * g is V11() real ext-real Element of REAL
((1 / Z) * g) + 0 is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
(id f) . g is V11() real ext-real Element of REAL
1 * g is V11() real ext-real Element of REAL
(1 * g) + 0 is V11() real ext-real Element of REAL
(() * x) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V11() real ext-real Element of REAL
((() * x) `| f) . g is V11() real ext-real Element of REAL
g / Z is V11() real ext-real Element of REAL
g * (Z ") is V11() real ext-real set
(g / Z) ^2 is V11() real ext-real Element of REAL
(g / Z) * (g / Z) is V11() real ext-real set
1 + ((g / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((g / Z) ^2)) is V11() real ext-real Element of REAL
1 / (Z * (1 + ((g / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((g / Z) ^2))) " is V11() real ext-real set
1 * ((Z * (1 + ((g / Z) ^2))) ") is V11() real ext-real set
- (1 / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real Element of REAL
(1 / Z) * g is V11() real ext-real Element of REAL
((1 / Z) * g) + 0 is V11() real ext-real Element of REAL
(((1 / Z) * g) + 0) ^2 is V11() real ext-real Element of REAL
(((1 / Z) * g) + 0) * (((1 / Z) * g) + 0) is V11() real ext-real set
1 + ((((1 / Z) * g) + 0) ^2) is V11() real ext-real Element of REAL
(1 / Z) / (1 + ((((1 / Z) * g) + 0) ^2)) is V11() real ext-real Element of REAL
(1 + ((((1 / Z) * g) + 0) ^2)) " is V11() real ext-real set
(1 / Z) * ((1 + ((((1 / Z) * g) + 0) ^2)) ") is V11() real ext-real set
- ((1 / Z) / (1 + ((((1 / Z) * g) + 0) ^2))) is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
(((id f) (#) (() * x)) `| f) . g is V11() real ext-real Element of REAL
g / Z is V11() real ext-real Element of REAL
g * (Z ") is V11() real ext-real set
() . (g / Z) is V11() real ext-real Element of REAL
(g / Z) ^2 is V11() real ext-real Element of REAL
(g / Z) * (g / Z) is V11() real ext-real set
1 + ((g / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((g / Z) ^2)) is V11() real ext-real Element of REAL
g / (Z * (1 + ((g / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((g / Z) ^2))) " is V11() real ext-real set
g * ((Z * (1 + ((g / Z) ^2))) ") is V11() real ext-real set
(() . (g / Z)) - (g / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real Element of REAL
- (g / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real set
(() . (g / Z)) + (- (g / (Z * (1 + ((g / Z) ^2))))) is V11() real ext-real set
(() * x) . g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
() . (x . g) is V11() real ext-real Element of REAL
diff ((id f),g) is V11() real ext-real Element of REAL
((() * x) . g) * (diff ((id f),g)) is V11() real ext-real Element of REAL
(id f) . g is V11() real ext-real Element of REAL
diff ((() * x),g) is V11() real ext-real Element of REAL
((id f) . g) * (diff ((() * x),g)) is V11() real ext-real Element of REAL
(((() * x) . g) * (diff ((id f),g))) + (((id f) . g) * (diff ((() * x),g))) is V11() real ext-real Element of REAL
(id f) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id f) `| f) . g is V11() real ext-real Element of REAL
((() * x) . g) * (((id f) `| f) . g) is V11() real ext-real Element of REAL
(((() * x) . g) * (((id f) `| f) . g)) + (((id f) . g) * (diff ((() * x),g))) is V11() real ext-real Element of REAL
((() * x) . g) * 1 is V11() real ext-real Element of REAL
(((() * x) . g) * 1) + (((id f) . g) * (diff ((() * x),g))) is V11() real ext-real Element of REAL
((() * x) `| f) . g is V11() real ext-real Element of REAL
((id f) . g) * (((() * x) `| f) . g) is V11() real ext-real Element of REAL
(((() * x) . g) * 1) + (((id f) . g) * (((() * x) `| f) . g)) is V11() real ext-real Element of REAL
g * (((() * x) `| f) . g) is V11() real ext-real Element of REAL
((() * x) . g) + (g * (((() * x) `| f) . g)) is V11() real ext-real Element of REAL
1 / (Z * (1 + ((g / Z) ^2))) is V11() real ext-real Element of REAL
1 * ((Z * (1 + ((g / Z) ^2))) ") is V11() real ext-real set
- (1 / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real Element of REAL
g * (- (1 / (Z * (1 + ((g / Z) ^2))))) is V11() real ext-real Element of REAL
(() . (g / Z)) + (g * (- (1 / (Z * (1 + ((g / Z) ^2)))))) is V11() real ext-real Element of REAL
g is V11() real ext-real Element of REAL
(((id f) (#) (() * x)) `| f) . g is V11() real ext-real Element of REAL
g / Z is V11() real ext-real Element of REAL
g * (Z ") is V11() real ext-real set
() . (g / Z) is V11() real ext-real Element of REAL
(g / Z) ^2 is V11() real ext-real Element of REAL
(g / Z) * (g / Z) is V11() real ext-real set
1 + ((g / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((g / Z) ^2)) is V11() real ext-real Element of REAL
g / (Z * (1 + ((g / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((g / Z) ^2))) " is V11() real ext-real set
g * ((Z * (1 + ((g / Z) ^2))) ") is V11() real ext-real set
(() . (g / Z)) - (g / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real Element of REAL
- (g / (Z * (1 + ((g / Z) ^2)))) is V11() real ext-real set
(() . (g / Z)) + (- (g / (Z * (1 + ((g / Z) ^2))))) is V11() real ext-real set
Z is V11() real ext-real Element of REAL
Z ^2 is V11() real ext-real Element of REAL
Z * Z is V11() real ext-real set
f is open V46() V47() V48() Element of K6(REAL)
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x + g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (x + g) is V46() V47() V48() Element of K6(REAL)
(x + g) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(#Z 2) * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f1 is V11() real ext-real Element of REAL
x . f1 is V11() real ext-real Element of REAL
0 * f1 is V11() real ext-real Element of REAL
(0 * f1) + 1 is V11() real ext-real Element of REAL
dom x is V46() V47() V48() Element of K6(REAL)
dom g is V46() V47() V48() Element of K6(REAL)
(dom x) /\ (dom g) is V46() V47() V48() set
1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
1 * (Z ") is V11() real ext-real set
f1 is V11() real ext-real Element of REAL
x . f1 is V11() real ext-real Element of REAL
(1 / Z) * f1 is V11() real ext-real Element of REAL
((1 / Z) * f1) + 0 is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
f1 * (Z ") is V11() real ext-real set
f1 is V11() real ext-real Element of REAL
dom ((#Z 2) * x) is V46() V47() V48() Element of K6(REAL)
dom x is V46() V47() V48() Element of K6(REAL)
f2 is set
g `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f1 is V11() real ext-real Element of REAL
(g `| f) . f1 is V11() real ext-real Element of REAL
2 * f1 is V11() real ext-real Element of REAL
(2 * f1) / (Z ^2) is V11() real ext-real Element of REAL
(Z ^2) " is V11() real ext-real set
(2 * f1) * ((Z ^2) ") is V11() real ext-real set
dom ((#Z 2) * x) is V46() V47() V48() Element of K6(REAL)
dom x is V46() V47() V48() Element of K6(REAL)
f2 is set
diff (((#Z 2) * x),f1) is V11() real ext-real Element of REAL
x . f1 is V11() real ext-real Element of REAL
2 - 1 is V11() real ext-real V63() Element of REAL
- 1 is non empty V11() real ext-real non positive negative V63() set
2 + (- 1) is V11() real ext-real V63() set
(x . f1) #Z (2 - 1) is V11() real ext-real Element of REAL
2 * ((x . f1) #Z (2 - 1)) is V11() real ext-real Element of REAL
diff (x,f1) is V11() real ext-real Element of REAL
(2 * ((x . f1) #Z (2 - 1))) * (diff (x,f1)) is V11() real ext-real Element of REAL
2 * (x . f1) is V11() real ext-real Element of REAL
(2 * (x . f1)) * (diff (x,f1)) is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
f1 * (Z ") is V11() real ext-real set
2 * (f1 / Z) is V11() real ext-real Element of REAL
(2 * (f1 / Z)) * (diff (x,f1)) is V11() real ext-real Element of REAL
x `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(x `| f) . f1 is V11() real ext-real Element of REAL
(2 * (f1 / Z)) * ((x `| f) . f1) is V11() real ext-real Element of REAL
(2 * (f1 / Z)) * (1 / Z) is V11() real ext-real Element of REAL
(f1 / Z) * (1 / Z) is V11() real ext-real Element of REAL
2 * ((f1 / Z) * (1 / Z)) is V11() real ext-real Element of REAL
f1 * 1 is V11() real ext-real Element of REAL
Z * Z is V11() real ext-real Element of REAL
(f1 * 1) / (Z * Z) is V11() real ext-real Element of REAL
(Z * Z) " is V11() real ext-real set
(f1 * 1) * ((Z * Z) ") is V11() real ext-real set
2 * ((f1 * 1) / (Z * Z)) is V11() real ext-real Element of REAL
f1 is V11() real ext-real Element of REAL
((x + g) `| f) . f1 is V11() real ext-real Element of REAL
2 * f1 is V11() real ext-real Element of REAL
(2 * f1) / (Z ^2) is V11() real ext-real Element of REAL
(Z ^2) " is V11() real ext-real set
(2 * f1) * ((Z ^2) ") is V11() real ext-real set
diff (x,f1) is V11() real ext-real Element of REAL
diff (g,f1) is V11() real ext-real Element of REAL
(diff (x,f1)) + (diff (g,f1)) is V11() real ext-real Element of REAL
x `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(x `| f) . f1 is V11() real ext-real Element of REAL
((x `| f) . f1) + (diff (g,f1)) is V11() real ext-real Element of REAL
(g `| f) . f1 is V11() real ext-real Element of REAL
((x `| f) . f1) + ((g `| f) . f1) is V11() real ext-real Element of REAL
0 + ((g `| f) . f1) is V11() real ext-real Element of REAL
f1 is V11() real ext-real Element of REAL
((x + g) `| f) . f1 is V11() real ext-real Element of REAL
2 * f1 is V11() real ext-real Element of REAL
(2 * f1) / (Z ^2) is V11() real ext-real Element of REAL
(Z ^2) " is V11() real ext-real set
(2 * f1) * ((Z ^2) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
Z / 2 is V11() real ext-real Element of REAL
Z * (2 ") is V11() real ext-real set
f is open V46() V47() V48() Element of K6(REAL)
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x + g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
ln * (x + g) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(Z / 2) (#) (ln * (x + g)) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((Z / 2) (#) (ln * (x + g))) is V46() V47() V48() Element of K6(REAL)
((Z / 2) (#) (ln * (x + g))) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(#Z 2) * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (ln * (x + g)) is V46() V47() V48() Element of K6(REAL)
dom (x + g) is V46() V47() V48() Element of K6(REAL)
f1 is set
dom x is V46() V47() V48() Element of K6(REAL)
dom g is V46() V47() V48() Element of K6(REAL)
(dom x) /\ (dom g) is V46() V47() V48() set
f1 is V11() real ext-real Element of REAL
(x + g) . f1 is V11() real ext-real Element of REAL
x . f1 is V11() real ext-real Element of REAL
g . f1 is V11() real ext-real Element of REAL
(x . f1) + (g . f1) is V11() real ext-real Element of REAL
((#Z 2) * x) . f1 is V11() real ext-real Element of REAL
1 + (((#Z 2) * x) . f1) is V11() real ext-real Element of REAL
x . f1 is V11() real ext-real Element of REAL
(#Z 2) . (x . f1) is V11() real ext-real Element of REAL
1 + ((#Z 2) . (x . f1)) is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
f1 * (Z ") is V11() real ext-real set
(#Z 2) . (f1 / Z) is V11() real ext-real Element of REAL
1 + ((#Z 2) . (f1 / Z)) is V11() real ext-real Element of REAL
1 + 1 is non empty V11() real ext-real positive non negative V63() Element of REAL
(f1 / Z) #Z (1 + 1) is V11() real ext-real Element of REAL
1 + ((f1 / Z) #Z (1 + 1)) is V11() real ext-real Element of REAL
(f1 / Z) #Z 1 is V11() real ext-real Element of REAL
((f1 / Z) #Z 1) * ((f1 / Z) #Z 1) is V11() real ext-real Element of REAL
1 + (((f1 / Z) #Z 1) * ((f1 / Z) #Z 1)) is V11() real ext-real Element of REAL
(f1 / Z) * ((f1 / Z) #Z 1) is V11() real ext-real Element of REAL
1 + ((f1 / Z) * ((f1 / Z) #Z 1)) is V11() real ext-real Element of REAL
(f1 / Z) * (f1 / Z) is V11() real ext-real Element of REAL
1 + ((f1 / Z) * (f1 / Z)) is V11() real ext-real Element of REAL
f1 is V11() real ext-real Element of REAL
(((Z / 2) (#) (ln * (x + g))) `| f) . f1 is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
f1 * (Z ") is V11() real ext-real set
(f1 / Z) ^2 is V11() real ext-real Element of REAL
(f1 / Z) * (f1 / Z) is V11() real ext-real set
1 + ((f1 / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((f1 / Z) ^2)) is V11() real ext-real Element of REAL
f1 / (Z * (1 + ((f1 / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((f1 / Z) ^2))) " is V11() real ext-real set
f1 * ((Z * (1 + ((f1 / Z) ^2))) ") is V11() real ext-real set
(x + g) . f1 is V11() real ext-real Element of REAL
x . f1 is V11() real ext-real Element of REAL
g . f1 is V11() real ext-real Element of REAL
(x . f1) + (g . f1) is V11() real ext-real Element of REAL
((#Z 2) * x) . f1 is V11() real ext-real Element of REAL
1 + (((#Z 2) * x) . f1) is V11() real ext-real Element of REAL
x . f1 is V11() real ext-real Element of REAL
(#Z 2) . (x . f1) is V11() real ext-real Element of REAL
1 + ((#Z 2) . (x . f1)) is V11() real ext-real Element of REAL
(#Z 2) . (f1 / Z) is V11() real ext-real Element of REAL
1 + ((#Z 2) . (f1 / Z)) is V11() real ext-real Element of REAL
1 + 1 is non empty V11() real ext-real positive non negative V63() Element of REAL
(f1 / Z) #Z (1 + 1) is V11() real ext-real Element of REAL
1 + ((f1 / Z) #Z (1 + 1)) is V11() real ext-real Element of REAL
(f1 / Z) #Z 1 is V11() real ext-real Element of REAL
((f1 / Z) #Z 1) * ((f1 / Z) #Z 1) is V11() real ext-real Element of REAL
1 + (((f1 / Z) #Z 1) * ((f1 / Z) #Z 1)) is V11() real ext-real Element of REAL
(f1 / Z) * ((f1 / Z) #Z 1) is V11() real ext-real Element of REAL
1 + ((f1 / Z) * ((f1 / Z) #Z 1)) is V11() real ext-real Element of REAL
(f1 / Z) * (f1 / Z) is V11() real ext-real Element of REAL
1 + ((f1 / Z) * (f1 / Z)) is V11() real ext-real Element of REAL
diff ((ln * (x + g)),f1) is V11() real ext-real Element of REAL
diff ((x + g),f1) is V11() real ext-real Element of REAL
(diff ((x + g),f1)) / ((x + g) . f1) is V11() real ext-real Element of REAL
((x + g) . f1) " is V11() real ext-real set
(diff ((x + g),f1)) * (((x + g) . f1) ") is V11() real ext-real set
(x + g) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((x + g) `| f) . f1 is V11() real ext-real Element of REAL
(((x + g) `| f) . f1) / ((x + g) . f1) is V11() real ext-real Element of REAL
(((x + g) `| f) . f1) * (((x + g) . f1) ") is V11() real ext-real set
2 * f1 is V11() real ext-real Element of REAL
Z ^2 is V11() real ext-real Element of REAL
Z * Z is V11() real ext-real set
(2 * f1) / (Z ^2) is V11() real ext-real Element of REAL
(Z ^2) " is V11() real ext-real set
(2 * f1) * ((Z ^2) ") is V11() real ext-real set
((2 * f1) / (Z ^2)) / (1 + ((f1 / Z) ^2)) is V11() real ext-real Element of REAL
(1 + ((f1 / Z) ^2)) " is V11() real ext-real set
((2 * f1) / (Z ^2)) * ((1 + ((f1 / Z) ^2)) ") is V11() real ext-real set
(Z / 2) * (diff ((ln * (x + g)),f1)) is V11() real ext-real Element of REAL
Z * f1 is V11() real ext-real Element of REAL
(Z * f1) / (Z ^2) is V11() real ext-real Element of REAL
(Z * f1) * ((Z ^2) ") is V11() real ext-real set
((Z * f1) / (Z ^2)) / (1 + ((f1 / Z) ^2)) is V11() real ext-real Element of REAL
((Z * f1) / (Z ^2)) * ((1 + ((f1 / Z) ^2)) ") is V11() real ext-real set
Z / Z is V11() real ext-real Element of REAL
Z * (Z ") is V11() real ext-real set
(Z / Z) * (f1 / Z) is V11() real ext-real Element of REAL
((Z / Z) * (f1 / Z)) / (1 + ((f1 / Z) ^2)) is V11() real ext-real Element of REAL
((Z / Z) * (f1 / Z)) * ((1 + ((f1 / Z) ^2)) ") is V11() real ext-real set
1 * (f1 / Z) is V11() real ext-real Element of REAL
(1 * (f1 / Z)) / (1 + ((f1 / Z) ^2)) is V11() real ext-real Element of REAL
(1 * (f1 / Z)) * ((1 + ((f1 / Z) ^2)) ") is V11() real ext-real set
f1 is V11() real ext-real Element of REAL
(((Z / 2) (#) (ln * (x + g))) `| f) . f1 is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
f1 * (Z ") is V11() real ext-real set
(f1 / Z) ^2 is V11() real ext-real Element of REAL
(f1 / Z) * (f1 / Z) is V11() real ext-real set
1 + ((f1 / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((f1 / Z) ^2)) is V11() real ext-real Element of REAL
f1 / (Z * (1 + ((f1 / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((f1 / Z) ^2))) " is V11() real ext-real set
f1 * ((Z * (1 + ((f1 / Z) ^2))) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
Z / 2 is V11() real ext-real Element of REAL
Z * (2 ") is V11() real ext-real set
f is open V46() V47() V48() Element of K6(REAL)
id f is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(id f) (#) (() * x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(#Z 2) * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g + x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
ln * (g + x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(Z / 2) (#) (ln * (g + x)) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id f) (#) (() * x)) - ((Z / 2) (#) (ln * (g + x))) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
- ((Z / 2) (#) (ln * (g + x))) is V19() Function-like V36() set
- 1 is non empty V11() real ext-real non positive negative V63() set
(- 1) (#) ((Z / 2) (#) (ln * (g + x))) is V19() Function-like set
((id f) (#) (() * x)) + (- ((Z / 2) (#) (ln * (g + x)))) is V19() Function-like set
dom (((id f) (#) (() * x)) - ((Z / 2) (#) (ln * (g + x)))) is V46() V47() V48() Element of K6(REAL)
(((id f) (#) (() * x)) - ((Z / 2) (#) (ln * (g + x)))) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id f) (#) (() * x)) is V46() V47() V48() Element of K6(REAL)
dom ((Z / 2) (#) (ln * (g + x))) is V46() V47() V48() Element of K6(REAL)
(dom ((id f) (#) (() * x))) /\ (dom ((Z / 2) (#) (ln * (g + x)))) is V46() V47() V48() set
f1 is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) - ((Z / 2) (#) (ln * (g + x)))) `| f) . f1 is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
f1 * (Z ") is V11() real ext-real set
() . (f1 / Z) is V11() real ext-real Element of REAL
diff (((id f) (#) (() * x)),f1) is V11() real ext-real Element of REAL
diff (((Z / 2) (#) (ln * (g + x))),f1) is V11() real ext-real Element of REAL
(diff (((id f) (#) (() * x)),f1)) - (diff (((Z / 2) (#) (ln * (g + x))),f1)) is V11() real ext-real Element of REAL
- (diff (((Z / 2) (#) (ln * (g + x))),f1)) is V11() real ext-real set
(diff (((id f) (#) (() * x)),f1)) + (- (diff (((Z / 2) (#) (ln * (g + x))),f1))) is V11() real ext-real set
((id f) (#) (() * x)) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((id f) (#) (() * x)) `| f) . f1 is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) `| f) . f1) - (diff (((Z / 2) (#) (ln * (g + x))),f1)) is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) `| f) . f1) + (- (diff (((Z / 2) (#) (ln * (g + x))),f1))) is V11() real ext-real set
((Z / 2) (#) (ln * (g + x))) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((Z / 2) (#) (ln * (g + x))) `| f) . f1 is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) `| f) . f1) - ((((Z / 2) (#) (ln * (g + x))) `| f) . f1) is V11() real ext-real Element of REAL
- ((((Z / 2) (#) (ln * (g + x))) `| f) . f1) is V11() real ext-real set
((((id f) (#) (() * x)) `| f) . f1) + (- ((((Z / 2) (#) (ln * (g + x))) `| f) . f1)) is V11() real ext-real set
(f1 / Z) ^2 is V11() real ext-real Element of REAL
(f1 / Z) * (f1 / Z) is V11() real ext-real set
1 + ((f1 / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((f1 / Z) ^2)) is V11() real ext-real Element of REAL
f1 / (Z * (1 + ((f1 / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((f1 / Z) ^2))) " is V11() real ext-real set
f1 * ((Z * (1 + ((f1 / Z) ^2))) ") is V11() real ext-real set
(() . (f1 / Z)) + (f1 / (Z * (1 + ((f1 / Z) ^2)))) is V11() real ext-real Element of REAL
((() . (f1 / Z)) + (f1 / (Z * (1 + ((f1 / Z) ^2))))) - ((((Z / 2) (#) (ln * (g + x))) `| f) . f1) is V11() real ext-real Element of REAL
((() . (f1 / Z)) + (f1 / (Z * (1 + ((f1 / Z) ^2))))) + (- ((((Z / 2) (#) (ln * (g + x))) `| f) . f1)) is V11() real ext-real set
((() . (f1 / Z)) + (f1 / (Z * (1 + ((f1 / Z) ^2))))) - (f1 / (Z * (1 + ((f1 / Z) ^2)))) is V11() real ext-real Element of REAL
- (f1 / (Z * (1 + ((f1 / Z) ^2)))) is V11() real ext-real set
((() . (f1 / Z)) + (f1 / (Z * (1 + ((f1 / Z) ^2))))) + (- (f1 / (Z * (1 + ((f1 / Z) ^2))))) is V11() real ext-real set
f1 is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) - ((Z / 2) (#) (ln * (g + x)))) `| f) . f1 is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
f1 * (Z ") is V11() real ext-real set
() . (f1 / Z) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
Z / 2 is V11() real ext-real Element of REAL
Z * (2 ") is V11() real ext-real set
f is open V46() V47() V48() Element of K6(REAL)
id f is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(id f) (#) (() * x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(#Z 2) * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g + x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
ln * (g + x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(Z / 2) (#) (ln * (g + x)) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id f) (#) (() * x)) + ((Z / 2) (#) (ln * (g + x))) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (((id f) (#) (() * x)) + ((Z / 2) (#) (ln * (g + x)))) is V46() V47() V48() Element of K6(REAL)
(((id f) (#) (() * x)) + ((Z / 2) (#) (ln * (g + x)))) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id f) (#) (() * x)) is V46() V47() V48() Element of K6(REAL)
dom ((Z / 2) (#) (ln * (g + x))) is V46() V47() V48() Element of K6(REAL)
(dom ((id f) (#) (() * x))) /\ (dom ((Z / 2) (#) (ln * (g + x)))) is V46() V47() V48() set
f1 is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) + ((Z / 2) (#) (ln * (g + x)))) `| f) . f1 is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
f1 * (Z ") is V11() real ext-real set
() . (f1 / Z) is V11() real ext-real Element of REAL
diff (((id f) (#) (() * x)),f1) is V11() real ext-real Element of REAL
diff (((Z / 2) (#) (ln * (g + x))),f1) is V11() real ext-real Element of REAL
(diff (((id f) (#) (() * x)),f1)) + (diff (((Z / 2) (#) (ln * (g + x))),f1)) is V11() real ext-real Element of REAL
((id f) (#) (() * x)) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((id f) (#) (() * x)) `| f) . f1 is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) `| f) . f1) + (diff (((Z / 2) (#) (ln * (g + x))),f1)) is V11() real ext-real Element of REAL
((Z / 2) (#) (ln * (g + x))) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((Z / 2) (#) (ln * (g + x))) `| f) . f1 is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) `| f) . f1) + ((((Z / 2) (#) (ln * (g + x))) `| f) . f1) is V11() real ext-real Element of REAL
(f1 / Z) ^2 is V11() real ext-real Element of REAL
(f1 / Z) * (f1 / Z) is V11() real ext-real set
1 + ((f1 / Z) ^2) is V11() real ext-real Element of REAL
Z * (1 + ((f1 / Z) ^2)) is V11() real ext-real Element of REAL
f1 / (Z * (1 + ((f1 / Z) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((f1 / Z) ^2))) " is V11() real ext-real set
f1 * ((Z * (1 + ((f1 / Z) ^2))) ") is V11() real ext-real set
(() . (f1 / Z)) - (f1 / (Z * (1 + ((f1 / Z) ^2)))) is V11() real ext-real Element of REAL
- (f1 / (Z * (1 + ((f1 / Z) ^2)))) is V11() real ext-real set
(() . (f1 / Z)) + (- (f1 / (Z * (1 + ((f1 / Z) ^2))))) is V11() real ext-real set
((() . (f1 / Z)) - (f1 / (Z * (1 + ((f1 / Z) ^2))))) + ((((Z / 2) (#) (ln * (g + x))) `| f) . f1) is V11() real ext-real Element of REAL
((() . (f1 / Z)) - (f1 / (Z * (1 + ((f1 / Z) ^2))))) + (f1 / (Z * (1 + ((f1 / Z) ^2)))) is V11() real ext-real Element of REAL
f1 is V11() real ext-real Element of REAL
((((id f) (#) (() * x)) + ((Z / 2) (#) (ln * (g + x)))) `| f) . f1 is V11() real ext-real Element of REAL
f1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
f1 * (Z ") is V11() real ext-real set
() . (f1 / Z) is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * ((id Z) ^) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * ((id Z) ^)) is V46() V47() V48() Element of K6(REAL)
(() * ((id Z) ^)) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
x is V11() real ext-real Element of REAL
((id Z) ^) . x is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * ((id Z) ^)) `| Z) . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
1 * ((1 + (x ^2)) ") is V11() real ext-real set
- (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
((id Z) ^) . x is V11() real ext-real Element of REAL
(id Z) . x is V11() real ext-real Element of REAL
diff ((() * ((id Z) ^)),x) is V11() real ext-real Element of REAL
diff (((id Z) ^),x) is V11() real ext-real Element of REAL
(((id Z) ^) . x) ^2 is V11() real ext-real Element of REAL
(((id Z) ^) . x) * (((id Z) ^) . x) is V11() real ext-real set
1 + ((((id Z) ^) . x) ^2) is V11() real ext-real Element of REAL
(diff (((id Z) ^),x)) / (1 + ((((id Z) ^) . x) ^2)) is V11() real ext-real Element of REAL
(1 + ((((id Z) ^) . x) ^2)) " is V11() real ext-real set
(diff (((id Z) ^),x)) * ((1 + ((((id Z) ^) . x) ^2)) ") is V11() real ext-real set
((id Z) ^) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((id Z) ^) `| Z) . x is V11() real ext-real Element of REAL
((((id Z) ^) `| Z) . x) / (1 + ((((id Z) ^) . x) ^2)) is V11() real ext-real Element of REAL
((((id Z) ^) `| Z) . x) * ((1 + ((((id Z) ^) . x) ^2)) ") is V11() real ext-real set
1 / (x ^2) is V11() real ext-real Element of REAL
(x ^2) " is V11() real ext-real set
1 * ((x ^2) ") is V11() real ext-real set
- (1 / (x ^2)) is V11() real ext-real Element of REAL
(- (1 / (x ^2))) / (1 + ((((id Z) ^) . x) ^2)) is V11() real ext-real Element of REAL
(- (1 / (x ^2))) * ((1 + ((((id Z) ^) . x) ^2)) ") is V11() real ext-real set
((id Z) . x) " is V11() real ext-real Element of REAL
(((id Z) . x) ") ^2 is V11() real ext-real Element of REAL
(((id Z) . x) ") * (((id Z) . x) ") is V11() real ext-real set
1 + ((((id Z) . x) ") ^2) is V11() real ext-real Element of REAL
(- (1 / (x ^2))) / (1 + ((((id Z) . x) ") ^2)) is V11() real ext-real Element of REAL
(1 + ((((id Z) . x) ") ^2)) " is V11() real ext-real set
(- (1 / (x ^2))) * ((1 + ((((id Z) . x) ") ^2)) ") is V11() real ext-real set
1 / x is V11() real ext-real Element of REAL
x " is V11() real ext-real set
1 * (x ") is V11() real ext-real set
(1 / x) ^2 is V11() real ext-real Element of REAL
(1 / x) * (1 / x) is V11() real ext-real set
1 + ((1 / x) ^2) is V11() real ext-real Element of REAL
(- (1 / (x ^2))) / (1 + ((1 / x) ^2)) is V11() real ext-real Element of REAL
(1 + ((1 / x) ^2)) " is V11() real ext-real set
(- (1 / (x ^2))) * ((1 + ((1 / x) ^2)) ") is V11() real ext-real set
(1 / (x ^2)) / (1 + ((1 / x) ^2)) is V11() real ext-real Element of REAL
(1 / (x ^2)) * ((1 + ((1 / x) ^2)) ") is V11() real ext-real set
- ((1 / (x ^2)) / (1 + ((1 / x) ^2))) is V11() real ext-real Element of REAL
(x ^2) * (1 + ((1 / x) ^2)) is V11() real ext-real Element of REAL
1 / ((x ^2) * (1 + ((1 / x) ^2))) is V11() real ext-real Element of REAL
((x ^2) * (1 + ((1 / x) ^2))) " is V11() real ext-real set
1 * (((x ^2) * (1 + ((1 / x) ^2))) ") is V11() real ext-real set
- (1 / ((x ^2) * (1 + ((1 / x) ^2)))) is V11() real ext-real Element of REAL
(x ^2) * 1 is V11() real ext-real Element of REAL
(x ^2) * ((1 / x) ^2) is V11() real ext-real Element of REAL
((x ^2) * 1) + ((x ^2) * ((1 / x) ^2)) is V11() real ext-real Element of REAL
1 / (((x ^2) * 1) + ((x ^2) * ((1 / x) ^2))) is V11() real ext-real Element of REAL
(((x ^2) * 1) + ((x ^2) * ((1 / x) ^2))) " is V11() real ext-real set
1 * ((((x ^2) * 1) + ((x ^2) * ((1 / x) ^2))) ") is V11() real ext-real set
- (1 / (((x ^2) * 1) + ((x ^2) * ((1 / x) ^2)))) is V11() real ext-real Element of REAL
x * x is V11() real ext-real Element of REAL
1 / (x * x) is V11() real ext-real Element of REAL
(x * x) " is V11() real ext-real set
1 * ((x * x) ") is V11() real ext-real set
(x ^2) * (1 / (x * x)) is V11() real ext-real Element of REAL
(x ^2) + ((x ^2) * (1 / (x * x))) is V11() real ext-real Element of REAL
1 / ((x ^2) + ((x ^2) * (1 / (x * x)))) is V11() real ext-real Element of REAL
((x ^2) + ((x ^2) * (1 / (x * x)))) " is V11() real ext-real set
1 * (((x ^2) + ((x ^2) * (1 / (x * x)))) ") is V11() real ext-real set
- (1 / ((x ^2) + ((x ^2) * (1 / (x * x))))) is V11() real ext-real Element of REAL
((x ^2) * 1) / (x ^2) is V11() real ext-real Element of REAL
((x ^2) * 1) * ((x ^2) ") is V11() real ext-real set
(x ^2) + (((x ^2) * 1) / (x ^2)) is V11() real ext-real Element of REAL
1 / ((x ^2) + (((x ^2) * 1) / (x ^2))) is V11() real ext-real Element of REAL
((x ^2) + (((x ^2) * 1) / (x ^2))) " is V11() real ext-real set
1 * (((x ^2) + (((x ^2) * 1) / (x ^2))) ") is V11() real ext-real set
- (1 / ((x ^2) + (((x ^2) * 1) / (x ^2)))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * ((id Z) ^)) `| Z) . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
1 * ((1 + (x ^2)) ") is V11() real ext-real set
- (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * ((id Z) ^) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * ((id Z) ^)) is V46() V47() V48() Element of K6(REAL)
(() * ((id Z) ^)) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
x is V11() real ext-real Element of REAL
((id Z) ^) . x is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * ((id Z) ^)) `| Z) . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
1 * ((1 + (x ^2)) ") is V11() real ext-real set
((id Z) ^) . x is V11() real ext-real Element of REAL
(id Z) . x is V11() real ext-real Element of REAL
diff ((() * ((id Z) ^)),x) is V11() real ext-real Element of REAL
diff (((id Z) ^),x) is V11() real ext-real Element of REAL
(((id Z) ^) . x) ^2 is V11() real ext-real Element of REAL
(((id Z) ^) . x) * (((id Z) ^) . x) is V11() real ext-real set
1 + ((((id Z) ^) . x) ^2) is V11() real ext-real Element of REAL
(diff (((id Z) ^),x)) / (1 + ((((id Z) ^) . x) ^2)) is V11() real ext-real Element of REAL
(1 + ((((id Z) ^) . x) ^2)) " is V11() real ext-real set
(diff (((id Z) ^),x)) * ((1 + ((((id Z) ^) . x) ^2)) ") is V11() real ext-real set
- ((diff (((id Z) ^),x)) / (1 + ((((id Z) ^) . x) ^2))) is V11() real ext-real Element of REAL
((id Z) ^) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((id Z) ^) `| Z) . x is V11() real ext-real Element of REAL
((((id Z) ^) `| Z) . x) / (1 + ((((id Z) ^) . x) ^2)) is V11() real ext-real Element of REAL
((((id Z) ^) `| Z) . x) * ((1 + ((((id Z) ^) . x) ^2)) ") is V11() real ext-real set
- (((((id Z) ^) `| Z) . x) / (1 + ((((id Z) ^) . x) ^2))) is V11() real ext-real Element of REAL
1 / (x ^2) is V11() real ext-real Element of REAL
(x ^2) " is V11() real ext-real set
1 * ((x ^2) ") is V11() real ext-real set
- (1 / (x ^2)) is V11() real ext-real Element of REAL
(- (1 / (x ^2))) / (1 + ((((id Z) ^) . x) ^2)) is V11() real ext-real Element of REAL
(- (1 / (x ^2))) * ((1 + ((((id Z) ^) . x) ^2)) ") is V11() real ext-real set
- ((- (1 / (x ^2))) / (1 + ((((id Z) ^) . x) ^2))) is V11() real ext-real Element of REAL
((id Z) . x) " is V11() real ext-real Element of REAL
(((id Z) . x) ") ^2 is V11() real ext-real Element of REAL
(((id Z) . x) ") * (((id Z) . x) ") is V11() real ext-real set
1 + ((((id Z) . x) ") ^2) is V11() real ext-real Element of REAL
(- (1 / (x ^2))) / (1 + ((((id Z) . x) ") ^2)) is V11() real ext-real Element of REAL
(1 + ((((id Z) . x) ") ^2)) " is V11() real ext-real set
(- (1 / (x ^2))) * ((1 + ((((id Z) . x) ") ^2)) ") is V11() real ext-real set
- ((- (1 / (x ^2))) / (1 + ((((id Z) . x) ") ^2))) is V11() real ext-real Element of REAL
1 / x is V11() real ext-real Element of REAL
x " is V11() real ext-real set
1 * (x ") is V11() real ext-real set
(1 / x) ^2 is V11() real ext-real Element of REAL
(1 / x) * (1 / x) is V11() real ext-real set
1 + ((1 / x) ^2) is V11() real ext-real Element of REAL
(- (1 / (x ^2))) / (1 + ((1 / x) ^2)) is V11() real ext-real Element of REAL
(1 + ((1 / x) ^2)) " is V11() real ext-real set
(- (1 / (x ^2))) * ((1 + ((1 / x) ^2)) ") is V11() real ext-real set
- ((- (1 / (x ^2))) / (1 + ((1 / x) ^2))) is V11() real ext-real Element of REAL
(1 / (x ^2)) / (1 + ((1 / x) ^2)) is V11() real ext-real Element of REAL
(1 / (x ^2)) * ((1 + ((1 / x) ^2)) ") is V11() real ext-real set
(x ^2) * (1 + ((1 / x) ^2)) is V11() real ext-real Element of REAL
1 / ((x ^2) * (1 + ((1 / x) ^2))) is V11() real ext-real Element of REAL
((x ^2) * (1 + ((1 / x) ^2))) " is V11() real ext-real set
1 * (((x ^2) * (1 + ((1 / x) ^2))) ") is V11() real ext-real set
(x ^2) * 1 is V11() real ext-real Element of REAL
(x ^2) * ((1 / x) ^2) is V11() real ext-real Element of REAL
((x ^2) * 1) + ((x ^2) * ((1 / x) ^2)) is V11() real ext-real Element of REAL
1 / (((x ^2) * 1) + ((x ^2) * ((1 / x) ^2))) is V11() real ext-real Element of REAL
(((x ^2) * 1) + ((x ^2) * ((1 / x) ^2))) " is V11() real ext-real set
1 * ((((x ^2) * 1) + ((x ^2) * ((1 / x) ^2))) ") is V11() real ext-real set
x * x is V11() real ext-real Element of REAL
1 / (x * x) is V11() real ext-real Element of REAL
(x * x) " is V11() real ext-real set
1 * ((x * x) ") is V11() real ext-real set
(x ^2) * (1 / (x * x)) is V11() real ext-real Element of REAL
(x ^2) + ((x ^2) * (1 / (x * x))) is V11() real ext-real Element of REAL
1 / ((x ^2) + ((x ^2) * (1 / (x * x)))) is V11() real ext-real Element of REAL
((x ^2) + ((x ^2) * (1 / (x * x)))) " is V11() real ext-real set
1 * (((x ^2) + ((x ^2) * (1 / (x * x)))) ") is V11() real ext-real set
((x ^2) * 1) / (x ^2) is V11() real ext-real Element of REAL
((x ^2) * 1) * ((x ^2) ") is V11() real ext-real set
(x ^2) + (((x ^2) * 1) / (x ^2)) is V11() real ext-real Element of REAL
1 / ((x ^2) + (((x ^2) * 1) / (x ^2))) is V11() real ext-real Element of REAL
((x ^2) + (((x ^2) * 1) / (x ^2))) " is V11() real ext-real set
1 * (((x ^2) + (((x ^2) * 1) / (x ^2))) ") is V11() real ext-real set
x is V11() real ext-real Element of REAL
((() * ((id Z) ^)) `| Z) . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
1 + (x ^2) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
1 * ((1 + (x ^2)) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
2 * Z is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
g is open V46() V47() V48() Element of K6(REAL)
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * x) is V46() V47() V48() Element of K6(REAL)
f1 is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f2 is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
Z (#) f2 is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f1 + (Z (#) f2) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * (f1 + (Z (#) f2)) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() * (f1 + (Z (#) f2))) `| g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom x is V46() V47() V48() Element of K6(REAL)
dom (f1 + (Z (#) f2)) is V46() V47() V48() Element of K6(REAL)
dom f1 is V46() V47() V48() Element of K6(REAL)
dom (Z (#) f2) is V46() V47() V48() Element of K6(REAL)
(dom f1) /\ (dom (Z (#) f2)) is V46() V47() V48() set
x is V11() real ext-real Element of REAL
x . x is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * (f1 + (Z (#) f2))) `| g) . x is V11() real ext-real Element of REAL
(2 * Z) * x is V11() real ext-real Element of REAL
x + ((2 * Z) * x) is V11() real ext-real Element of REAL
x * x is V11() real ext-real Element of REAL
f + (x * x) is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
Z * (x ^2) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (x ^2)) is V11() real ext-real Element of REAL
((f + (x * x)) + (Z * (x ^2))) ^2 is V11() real ext-real Element of REAL
((f + (x * x)) + (Z * (x ^2))) * ((f + (x * x)) + (Z * (x ^2))) is V11() real ext-real set
1 + (((f + (x * x)) + (Z * (x ^2))) ^2) is V11() real ext-real Element of REAL
(x + ((2 * Z) * x)) / (1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) is V11() real ext-real Element of REAL
(1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) " is V11() real ext-real set
(x + ((2 * Z) * x)) * ((1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) ") is V11() real ext-real set
(f1 + (Z (#) f2)) . x is V11() real ext-real Element of REAL
f1 . x is V11() real ext-real Element of REAL
(Z (#) f2) . x is V11() real ext-real Element of REAL
(f1 . x) + ((Z (#) f2) . x) is V11() real ext-real Element of REAL
f2 . x is V11() real ext-real Element of REAL
Z * (f2 . x) is V11() real ext-real Element of REAL
(f1 . x) + (Z * (f2 . x)) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (f2 . x)) is V11() real ext-real Element of REAL
1 + 1 is non empty V11() real ext-real positive non negative V63() Element of REAL
x #Z (1 + 1) is V11() real ext-real Element of REAL
Z * (x #Z (1 + 1)) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (x #Z (1 + 1))) is V11() real ext-real Element of REAL
x #Z 1 is V11() real ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V11() real ext-real Element of REAL
Z * ((x #Z 1) * (x #Z 1)) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * ((x #Z 1) * (x #Z 1))) is V11() real ext-real Element of REAL
x * (x #Z 1) is V11() real ext-real Element of REAL
Z * (x * (x #Z 1)) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (x * (x #Z 1))) is V11() real ext-real Element of REAL
x . x is V11() real ext-real Element of REAL
diff ((() * x),x) is V11() real ext-real Element of REAL
diff (x,x) is V11() real ext-real Element of REAL
(x . x) ^2 is V11() real ext-real Element of REAL
(x . x) * (x . x) is V11() real ext-real set
1 + ((x . x) ^2) is V11() real ext-real Element of REAL
(diff (x,x)) / (1 + ((x . x) ^2)) is V11() real ext-real Element of REAL
(1 + ((x . x) ^2)) " is V11() real ext-real set
(diff (x,x)) * ((1 + ((x . x) ^2)) ") is V11() real ext-real set
(f1 + (Z (#) f2)) `| g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((f1 + (Z (#) f2)) `| g) . x is V11() real ext-real Element of REAL
(((f1 + (Z (#) f2)) `| g) . x) / (1 + ((x . x) ^2)) is V11() real ext-real Element of REAL
(((f1 + (Z (#) f2)) `| g) . x) * ((1 + ((x . x) ^2)) ") is V11() real ext-real set
x is V11() real ext-real Element of REAL
((() * (f1 + (Z (#) f2))) `| g) . x is V11() real ext-real Element of REAL
(2 * Z) * x is V11() real ext-real Element of REAL
x + ((2 * Z) * x) is V11() real ext-real Element of REAL
x * x is V11() real ext-real Element of REAL
f + (x * x) is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
Z * (x ^2) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (x ^2)) is V11() real ext-real Element of REAL
((f + (x * x)) + (Z * (x ^2))) ^2 is V11() real ext-real Element of REAL
((f + (x * x)) + (Z * (x ^2))) * ((f + (x * x)) + (Z * (x ^2))) is V11() real ext-real set
1 + (((f + (x * x)) + (Z * (x ^2))) ^2) is V11() real ext-real Element of REAL
(x + ((2 * Z) * x)) / (1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) is V11() real ext-real Element of REAL
(1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) " is V11() real ext-real set
(x + ((2 * Z) * x)) * ((1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) ") is V11() real ext-real set
Z is V11() real ext-real Element of REAL
2 * Z is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
g is open V46() V47() V48() Element of K6(REAL)
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * x) is V46() V47() V48() Element of K6(REAL)
f1 is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f2 is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
Z (#) f2 is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f1 + (Z (#) f2) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * (f1 + (Z (#) f2)) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() * (f1 + (Z (#) f2))) `| g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom x is V46() V47() V48() Element of K6(REAL)
dom (f1 + (Z (#) f2)) is V46() V47() V48() Element of K6(REAL)
dom f1 is V46() V47() V48() Element of K6(REAL)
dom (Z (#) f2) is V46() V47() V48() Element of K6(REAL)
(dom f1) /\ (dom (Z (#) f2)) is V46() V47() V48() set
x is V11() real ext-real Element of REAL
x . x is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * (f1 + (Z (#) f2))) `| g) . x is V11() real ext-real Element of REAL
(2 * Z) * x is V11() real ext-real Element of REAL
x + ((2 * Z) * x) is V11() real ext-real Element of REAL
x * x is V11() real ext-real Element of REAL
f + (x * x) is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
Z * (x ^2) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (x ^2)) is V11() real ext-real Element of REAL
((f + (x * x)) + (Z * (x ^2))) ^2 is V11() real ext-real Element of REAL
((f + (x * x)) + (Z * (x ^2))) * ((f + (x * x)) + (Z * (x ^2))) is V11() real ext-real set
1 + (((f + (x * x)) + (Z * (x ^2))) ^2) is V11() real ext-real Element of REAL
(x + ((2 * Z) * x)) / (1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) is V11() real ext-real Element of REAL
(1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) " is V11() real ext-real set
(x + ((2 * Z) * x)) * ((1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) ") is V11() real ext-real set
- ((x + ((2 * Z) * x)) / (1 + (((f + (x * x)) + (Z * (x ^2))) ^2))) is V11() real ext-real Element of REAL
(f1 + (Z (#) f2)) . x is V11() real ext-real Element of REAL
f1 . x is V11() real ext-real Element of REAL
(Z (#) f2) . x is V11() real ext-real Element of REAL
(f1 . x) + ((Z (#) f2) . x) is V11() real ext-real Element of REAL
f2 . x is V11() real ext-real Element of REAL
Z * (f2 . x) is V11() real ext-real Element of REAL
(f1 . x) + (Z * (f2 . x)) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (f2 . x)) is V11() real ext-real Element of REAL
1 + 1 is non empty V11() real ext-real positive non negative V63() Element of REAL
x #Z (1 + 1) is V11() real ext-real Element of REAL
Z * (x #Z (1 + 1)) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (x #Z (1 + 1))) is V11() real ext-real Element of REAL
x #Z 1 is V11() real ext-real Element of REAL
(x #Z 1) * (x #Z 1) is V11() real ext-real Element of REAL
Z * ((x #Z 1) * (x #Z 1)) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * ((x #Z 1) * (x #Z 1))) is V11() real ext-real Element of REAL
x * (x #Z 1) is V11() real ext-real Element of REAL
Z * (x * (x #Z 1)) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (x * (x #Z 1))) is V11() real ext-real Element of REAL
x . x is V11() real ext-real Element of REAL
diff ((() * x),x) is V11() real ext-real Element of REAL
diff (x,x) is V11() real ext-real Element of REAL
(x . x) ^2 is V11() real ext-real Element of REAL
(x . x) * (x . x) is V11() real ext-real set
1 + ((x . x) ^2) is V11() real ext-real Element of REAL
(diff (x,x)) / (1 + ((x . x) ^2)) is V11() real ext-real Element of REAL
(1 + ((x . x) ^2)) " is V11() real ext-real set
(diff (x,x)) * ((1 + ((x . x) ^2)) ") is V11() real ext-real set
- ((diff (x,x)) / (1 + ((x . x) ^2))) is V11() real ext-real Element of REAL
(f1 + (Z (#) f2)) `| g is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((f1 + (Z (#) f2)) `| g) . x is V11() real ext-real Element of REAL
(((f1 + (Z (#) f2)) `| g) . x) / (1 + ((x . x) ^2)) is V11() real ext-real Element of REAL
(((f1 + (Z (#) f2)) `| g) . x) * ((1 + ((x . x) ^2)) ") is V11() real ext-real set
- ((((f1 + (Z (#) f2)) `| g) . x) / (1 + ((x . x) ^2))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((() * (f1 + (Z (#) f2))) `| g) . x is V11() real ext-real Element of REAL
(2 * Z) * x is V11() real ext-real Element of REAL
x + ((2 * Z) * x) is V11() real ext-real Element of REAL
x * x is V11() real ext-real Element of REAL
f + (x * x) is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
Z * (x ^2) is V11() real ext-real Element of REAL
(f + (x * x)) + (Z * (x ^2)) is V11() real ext-real Element of REAL
((f + (x * x)) + (Z * (x ^2))) ^2 is V11() real ext-real Element of REAL
((f + (x * x)) + (Z * (x ^2))) * ((f + (x * x)) + (Z * (x ^2))) is V11() real ext-real set
1 + (((f + (x * x)) + (Z * (x ^2))) ^2) is V11() real ext-real Element of REAL
(x + ((2 * Z) * x)) / (1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) is V11() real ext-real Element of REAL
(1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) " is V11() real ext-real set
(x + ((2 * Z) * x)) * ((1 + (((f + (x * x)) + (Z * (x ^2))) ^2)) ") is V11() real ext-real set
- ((x + ((2 * Z) * x)) / (1 + (((f + (x * x)) + (Z * (x ^2))) ^2))) is V11() real ext-real Element of REAL
() * exp_R is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * exp_R) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(() * exp_R) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
(exp_R . f) + 0 is V11() real ext-real Element of REAL
0 + (- 1) is V11() real ext-real V63() Element of REAL
f is V11() real ext-real Element of REAL
((() * exp_R) `| Z) . f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
(exp_R . f) ^2 is V11() real ext-real Element of REAL
(exp_R . f) * (exp_R . f) is V11() real ext-real set
1 + ((exp_R . f) ^2) is V11() real ext-real Element of REAL
(exp_R . f) / (1 + ((exp_R . f) ^2)) is V11() real ext-real Element of REAL
(1 + ((exp_R . f) ^2)) " is V11() real ext-real set
(exp_R . f) * ((1 + ((exp_R . f) ^2)) ") is V11() real ext-real set
(exp_R . f) + 0 is V11() real ext-real Element of REAL
0 + (- 1) is V11() real ext-real V63() Element of REAL
diff ((() * exp_R),f) is V11() real ext-real Element of REAL
diff (exp_R,f) is V11() real ext-real Element of REAL
(diff (exp_R,f)) / (1 + ((exp_R . f) ^2)) is V11() real ext-real Element of REAL
(diff (exp_R,f)) * ((1 + ((exp_R . f) ^2)) ") is V11() real ext-real set
f is V11() real ext-real Element of REAL
((() * exp_R) `| Z) . f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
(exp_R . f) ^2 is V11() real ext-real Element of REAL
(exp_R . f) * (exp_R . f) is V11() real ext-real set
1 + ((exp_R . f) ^2) is V11() real ext-real Element of REAL
(exp_R . f) / (1 + ((exp_R . f) ^2)) is V11() real ext-real Element of REAL
(1 + ((exp_R . f) ^2)) " is V11() real ext-real set
(exp_R . f) * ((1 + ((exp_R . f) ^2)) ") is V11() real ext-real set
() * exp_R is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * exp_R) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(() * exp_R) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
(exp_R . f) + 0 is V11() real ext-real Element of REAL
0 + (- 1) is V11() real ext-real V63() Element of REAL
f is V11() real ext-real Element of REAL
((() * exp_R) `| Z) . f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
(exp_R . f) ^2 is V11() real ext-real Element of REAL
(exp_R . f) * (exp_R . f) is V11() real ext-real set
1 + ((exp_R . f) ^2) is V11() real ext-real Element of REAL
(exp_R . f) / (1 + ((exp_R . f) ^2)) is V11() real ext-real Element of REAL
(1 + ((exp_R . f) ^2)) " is V11() real ext-real set
(exp_R . f) * ((1 + ((exp_R . f) ^2)) ") is V11() real ext-real set
- ((exp_R . f) / (1 + ((exp_R . f) ^2))) is V11() real ext-real Element of REAL
(exp_R . f) + 0 is V11() real ext-real Element of REAL
0 + (- 1) is V11() real ext-real V63() Element of REAL
diff ((() * exp_R),f) is V11() real ext-real Element of REAL
diff (exp_R,f) is V11() real ext-real Element of REAL
(diff (exp_R,f)) / (1 + ((exp_R . f) ^2)) is V11() real ext-real Element of REAL
(diff (exp_R,f)) * ((1 + ((exp_R . f) ^2)) ") is V11() real ext-real set
- ((diff (exp_R,f)) / (1 + ((exp_R . f) ^2))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((() * exp_R) `| Z) . f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
(exp_R . f) ^2 is V11() real ext-real Element of REAL
(exp_R . f) * (exp_R . f) is V11() real ext-real set
1 + ((exp_R . f) ^2) is V11() real ext-real Element of REAL
(exp_R . f) / (1 + ((exp_R . f) ^2)) is V11() real ext-real Element of REAL
(1 + ((exp_R . f) ^2)) " is V11() real ext-real set
(exp_R . f) * ((1 + ((exp_R . f) ^2)) ") is V11() real ext-real set
- ((exp_R . f) / (1 + ((exp_R . f) ^2))) is V11() real ext-real Element of REAL
() * ln is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * ln) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(() * ln) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
{ b<sub>1</sub> where b<sub>1</sub> is V11() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
ln . f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((() * ln) `| Z) . f is V11() real ext-real Element of REAL
ln . f is V11() real ext-real Element of REAL
(ln . f) ^2 is V11() real ext-real Element of REAL
(ln . f) * (ln . f) is V11() real ext-real set
1 + ((ln . f) ^2) is V11() real ext-real Element of REAL
f * (1 + ((ln . f) ^2)) is V11() real ext-real Element of REAL
1 / (f * (1 + ((ln . f) ^2))) is V11() real ext-real Element of REAL
(f * (1 + ((ln . f) ^2))) " is V11() real ext-real set
1 * ((f * (1 + ((ln . f) ^2))) ") is V11() real ext-real set
diff ((() * ln),f) is V11() real ext-real Element of REAL
diff (ln,f) is V11() real ext-real Element of REAL
(diff (ln,f)) / (1 + ((ln . f) ^2)) is V11() real ext-real Element of REAL
(1 + ((ln . f) ^2)) " is V11() real ext-real set
(diff (ln,f)) * ((1 + ((ln . f) ^2)) ") is V11() real ext-real set
1 / f is V11() real ext-real Element of REAL
f " is V11() real ext-real set
1 * (f ") is V11() real ext-real set
(1 / f) / (1 + ((ln . f) ^2)) is V11() real ext-real Element of REAL
(1 / f) * ((1 + ((ln . f) ^2)) ") is V11() real ext-real set
f is V11() real ext-real Element of REAL
((() * ln) `| Z) . f is V11() real ext-real Element of REAL
ln . f is V11() real ext-real Element of REAL
(ln . f) ^2 is V11() real ext-real Element of REAL
(ln . f) * (ln . f) is V11() real ext-real set
1 + ((ln . f) ^2) is V11() real ext-real Element of REAL
f * (1 + ((ln . f) ^2)) is V11() real ext-real Element of REAL
1 / (f * (1 + ((ln . f) ^2))) is V11() real ext-real Element of REAL
(f * (1 + ((ln . f) ^2))) " is V11() real ext-real set
1 * ((f * (1 + ((ln . f) ^2))) ") is V11() real ext-real set
() * ln is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (() * ln) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(() * ln) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
{ b₁ where b₁ is V11() real ext-real Element of REAL : not b₁ <= 0 } is set
f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
ln . f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((() * ln) `| Z) . f is V11() real ext-real Element of REAL
ln . f is V11() real ext-real Element of REAL
(ln . f) ^2 is V11() real ext-real Element of REAL
(ln . f) * (ln . f) is V11() real ext-real set
1 + ((ln . f) ^2) is V11() real ext-real Element of REAL
f * (1 + ((ln . f) ^2)) is V11() real ext-real Element of REAL
1 / (f * (1 + ((ln . f) ^2))) is V11() real ext-real Element of REAL
(f * (1 + ((ln . f) ^2))) " is V11() real ext-real set
1 * ((f * (1 + ((ln . f) ^2))) ") is V11() real ext-real set
- (1 / (f * (1 + ((ln . f) ^2)))) is V11() real ext-real Element of REAL
diff ((() * ln),f) is V11() real ext-real Element of REAL
diff (ln,f) is V11() real ext-real Element of REAL
(diff (ln,f)) / (1 + ((ln . f) ^2)) is V11() real ext-real Element of REAL
(1 + ((ln . f) ^2)) " is V11() real ext-real set
(diff (ln,f)) * ((1 + ((ln . f) ^2)) ") is V11() real ext-real set
- ((diff (ln,f)) / (1 + ((ln . f) ^2))) is V11() real ext-real Element of REAL
1 / f is V11() real ext-real Element of REAL
f " is V11() real ext-real set
1 * (f ") is V11() real ext-real set
(1 / f) / (1 + ((ln . f) ^2)) is V11() real ext-real Element of REAL
(1 / f) * ((1 + ((ln . f) ^2)) ") is V11() real ext-real set
- ((1 / f) / (1 + ((ln . f) ^2))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((() * ln) `| Z) . f is V11() real ext-real Element of REAL
ln . f is V11() real ext-real Element of REAL
(ln . f) ^2 is V11() real ext-real Element of REAL
(ln . f) * (ln . f) is V11() real ext-real set
1 + ((ln . f) ^2) is V11() real ext-real Element of REAL
f * (1 + ((ln . f) ^2)) is V11() real ext-real Element of REAL
1 / (f * (1 + ((ln . f) ^2))) is V11() real ext-real Element of REAL
(f * (1 + ((ln . f) ^2))) " is V11() real ext-real set
1 * ((f * (1 + ((ln . f) ^2))) ") is V11() real ext-real set
- (1 / (f * (1 + ((ln . f) ^2)))) is V11() real ext-real Element of REAL
exp_R * () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (exp_R * ()) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(exp_R * ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((exp_R * ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
exp_R . (() . f) is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(exp_R . (() . f)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(exp_R . (() . f)) * ((1 + (f ^2)) ") is V11() real ext-real set
diff ((exp_R * ()),f) is V11() real ext-real Element of REAL
diff (exp_R,(() . f)) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
(diff (exp_R,(() . f))) * (diff ((),f)) is V11() real ext-real Element of REAL
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . f is V11() real ext-real Element of REAL
(diff (exp_R,(() . f))) * ((() `| Z) . f) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
(diff (exp_R,(() . f))) * (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((exp_R * ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
exp_R . (() . f) is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(exp_R . (() . f)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(exp_R . (() . f)) * ((1 + (f ^2)) ") is V11() real ext-real set
exp_R * () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (exp_R * ()) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(exp_R * ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((exp_R * ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
exp_R . (() . f) is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(exp_R . (() . f)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(exp_R . (() . f)) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((exp_R . (() . f)) / (1 + (f ^2))) is V11() real ext-real Element of REAL
diff ((exp_R * ()),f) is V11() real ext-real Element of REAL
diff (exp_R,(() . f)) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
(diff (exp_R,(() . f))) * (diff ((),f)) is V11() real ext-real Element of REAL
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . f is V11() real ext-real Element of REAL
(diff (exp_R,(() . f))) * ((() `| Z) . f) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
- (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
(diff (exp_R,(() . f))) * (- (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
(diff (exp_R,(() . f))) * (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((diff (exp_R,(() . f))) * (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((exp_R * ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
exp_R . (() . f) is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(exp_R . (() . f)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(exp_R . (() . f)) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((exp_R . (() . f)) / (1 + (f ^2))) is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
() - (id Z) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
- (id Z) is V19() Function-like V36() set
- 1 is non empty V11() real ext-real non positive negative V63() set
(- 1) (#) (id Z) is V19() Function-like set
() + (- (id Z)) is V19() Function-like set
dom (() - (id Z)) is V46() V47() V48() Element of K6(REAL)
(() - (id Z)) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
f is V11() real ext-real Element of REAL
(id Z) . f is V11() real ext-real Element of REAL
1 * f is V11() real ext-real Element of REAL
(1 * f) + 0 is V11() real ext-real Element of REAL
dom (id Z) is V46() V47() V48() Element of K6(REAL)
(dom ()) /\ (dom (id Z)) is V46() V47() V48() set
f is V11() real ext-real Element of REAL
((() - (id Z)) `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(f ^2) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(f ^2) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((f ^2) / (1 + (f ^2))) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
diff ((id Z),f) is V11() real ext-real Element of REAL
(diff ((),f)) - (diff ((id Z),f)) is V11() real ext-real Element of REAL
- (diff ((id Z),f)) is V11() real ext-real set
(diff ((),f)) + (- (diff ((id Z),f))) is V11() real ext-real set
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . f is V11() real ext-real Element of REAL
((() `| Z) . f) - (diff ((id Z),f)) is V11() real ext-real Element of REAL
((() `| Z) . f) + (- (diff ((id Z),f))) is V11() real ext-real set
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
(1 / (1 + (f ^2))) - (diff ((id Z),f)) is V11() real ext-real Element of REAL
(1 / (1 + (f ^2))) + (- (diff ((id Z),f))) is V11() real ext-real set
(id Z) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) `| Z) . f is V11() real ext-real Element of REAL
(1 / (1 + (f ^2))) - (((id Z) `| Z) . f) is V11() real ext-real Element of REAL
- (((id Z) `| Z) . f) is V11() real ext-real set
(1 / (1 + (f ^2))) + (- (((id Z) `| Z) . f)) is V11() real ext-real set
(1 / (1 + (f ^2))) - 1 is V11() real ext-real Element of REAL
(1 / (1 + (f ^2))) + (- 1) is V11() real ext-real set
(1 + (f ^2)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) * ((1 + (f ^2)) ") is V11() real ext-real set
(1 / (1 + (f ^2))) - ((1 + (f ^2)) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((1 + (f ^2)) / (1 + (f ^2))) is V11() real ext-real set
(1 / (1 + (f ^2))) + (- ((1 + (f ^2)) / (1 + (f ^2)))) is V11() real ext-real set
f is V11() real ext-real Element of REAL
((() - (id Z)) `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(f ^2) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(f ^2) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((f ^2) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
- 1 is non empty V11() real ext-real non positive negative V63() set
(- 1) (#) () is V19() Function-like set
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(- ()) - (id Z) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
- (id Z) is V19() Function-like V36() set
(- 1) (#) (id Z) is V19() Function-like set
(- ()) + (- (id Z)) is V19() Function-like set
dom ((- ()) - (id Z)) is V46() V47() V48() Element of K6(REAL)
((- ()) - (id Z)) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (- ()) is V46() V47() V48() Element of K6(REAL)
dom (id Z) is V46() V47() V48() Element of K6(REAL)
(dom (- ())) /\ (dom (id Z)) is V46() V47() V48() set
f is V11() real ext-real Element of REAL
(id Z) . f is V11() real ext-real Element of REAL
1 * f is V11() real ext-real Element of REAL
(1 * f) + 0 is V11() real ext-real Element of REAL
(- 1) (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((- 1) (#) ()) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
(((- ()) - (id Z)) `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(f ^2) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(f ^2) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((f ^2) / (1 + (f ^2))) is V11() real ext-real Element of REAL
diff ((- ()),f) is V11() real ext-real Element of REAL
diff ((id Z),f) is V11() real ext-real Element of REAL
(diff ((- ()),f)) - (diff ((id Z),f)) is V11() real ext-real Element of REAL
- (diff ((id Z),f)) is V11() real ext-real set
(diff ((- ()),f)) + (- (diff ((id Z),f))) is V11() real ext-real set
(- ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((- ()) `| Z) . f is V11() real ext-real Element of REAL
(((- ()) `| Z) . f) - (diff ((id Z),f)) is V11() real ext-real Element of REAL
(((- ()) `| Z) . f) + (- (diff ((id Z),f))) is V11() real ext-real set
diff ((),f) is V11() real ext-real Element of REAL
(- 1) * (diff ((),f)) is V11() real ext-real Element of REAL
((- 1) * (diff ((),f))) - (diff ((id Z),f)) is V11() real ext-real Element of REAL
((- 1) * (diff ((),f))) + (- (diff ((id Z),f))) is V11() real ext-real set
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . f is V11() real ext-real Element of REAL
(- 1) * ((() `| Z) . f) is V11() real ext-real Element of REAL
((- 1) * ((() `| Z) . f)) - (diff ((id Z),f)) is V11() real ext-real Element of REAL
((- 1) * ((() `| Z) . f)) + (- (diff ((id Z),f))) is V11() real ext-real set
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
- (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
(- 1) * (- (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
((- 1) * (- (1 / (1 + (f ^2))))) - (diff ((id Z),f)) is V11() real ext-real Element of REAL
((- 1) * (- (1 / (1 + (f ^2))))) + (- (diff ((id Z),f))) is V11() real ext-real set
(id Z) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) `| Z) . f is V11() real ext-real Element of REAL
(1 / (1 + (f ^2))) - (((id Z) `| Z) . f) is V11() real ext-real Element of REAL
- (((id Z) `| Z) . f) is V11() real ext-real set
(1 / (1 + (f ^2))) + (- (((id Z) `| Z) . f)) is V11() real ext-real set
(1 / (1 + (f ^2))) - 1 is V11() real ext-real Element of REAL
(1 / (1 + (f ^2))) + (- 1) is V11() real ext-real set
(1 + (f ^2)) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) * ((1 + (f ^2)) ") is V11() real ext-real set
(1 / (1 + (f ^2))) - ((1 + (f ^2)) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((1 + (f ^2)) / (1 + (f ^2))) is V11() real ext-real set
(1 / (1 + (f ^2))) + (- ((1 + (f ^2)) / (1 + (f ^2)))) is V11() real ext-real set
f is V11() real ext-real Element of REAL
(((- ()) - (id Z)) `| Z) . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(f ^2) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(f ^2) * ((1 + (f ^2)) ") is V11() real ext-real set
- ((f ^2) / (1 + (f ^2))) is V11() real ext-real Element of REAL
exp_R (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(exp_R (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom exp_R) /\ (dom ()) is V46() V47() V48() set
dom (exp_R (#) ()) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((exp_R (#) ()) `| Z) . f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(exp_R . f) * (() . f) is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(exp_R . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(exp_R . f) * ((1 + (f ^2)) ") is V11() real ext-real set
((exp_R . f) * (() . f)) + ((exp_R . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
diff (exp_R,f) is V11() real ext-real Element of REAL
(() . f) * (diff (exp_R,f)) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
(exp_R . f) * (diff ((),f)) is V11() real ext-real Element of REAL
((() . f) * (diff (exp_R,f))) + ((exp_R . f) * (diff ((),f))) is V11() real ext-real Element of REAL
(() . f) * (exp_R . f) is V11() real ext-real Element of REAL
((() . f) * (exp_R . f)) + ((exp_R . f) * (diff ((),f))) is V11() real ext-real Element of REAL
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . f is V11() real ext-real Element of REAL
(exp_R . f) * ((() `| Z) . f) is V11() real ext-real Element of REAL
((exp_R . f) * (() . f)) + ((exp_R . f) * ((() `| Z) . f)) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
(exp_R . f) * (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
((exp_R . f) * (() . f)) + ((exp_R . f) * (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((exp_R (#) ()) `| Z) . f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(exp_R . f) * (() . f) is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(exp_R . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(exp_R . f) * ((1 + (f ^2)) ") is V11() real ext-real set
((exp_R . f) * (() . f)) + ((exp_R . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
exp_R (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
Z is open V46() V47() V48() Element of K6(REAL)
(exp_R (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(dom exp_R) /\ (dom ()) is V46() V47() V48() set
dom (exp_R (#) ()) is V46() V47() V48() Element of K6(REAL)
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((exp_R (#) ()) `| Z) . f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(exp_R . f) * (() . f) is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(exp_R . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(exp_R . f) * ((1 + (f ^2)) ") is V11() real ext-real set
((exp_R . f) * (() . f)) - ((exp_R . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((exp_R . f) / (1 + (f ^2))) is V11() real ext-real set
((exp_R . f) * (() . f)) + (- ((exp_R . f) / (1 + (f ^2)))) is V11() real ext-real set
diff (exp_R,f) is V11() real ext-real Element of REAL
(() . f) * (diff (exp_R,f)) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
(exp_R . f) * (diff ((),f)) is V11() real ext-real Element of REAL
((() . f) * (diff (exp_R,f))) + ((exp_R . f) * (diff ((),f))) is V11() real ext-real Element of REAL
(() . f) * (exp_R . f) is V11() real ext-real Element of REAL
((() . f) * (exp_R . f)) + ((exp_R . f) * (diff ((),f))) is V11() real ext-real Element of REAL
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . f is V11() real ext-real Element of REAL
(exp_R . f) * ((() `| Z) . f) is V11() real ext-real Element of REAL
((exp_R . f) * (() . f)) + ((exp_R . f) * ((() `| Z) . f)) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
- (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
(exp_R . f) * (- (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
((exp_R . f) * (() . f)) + ((exp_R . f) * (- (1 / (1 + (f ^2))))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((exp_R (#) ()) `| Z) . f is V11() real ext-real Element of REAL
exp_R . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(exp_R . f) * (() . f) is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(exp_R . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(exp_R . f) * ((1 + (f ^2)) ") is V11() real ext-real set
((exp_R . f) * (() . f)) - ((exp_R . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((exp_R . f) / (1 + (f ^2))) is V11() real ext-real set
((exp_R . f) * (() . f)) + (- ((exp_R . f) / (1 + (f ^2)))) is V11() real ext-real set
Z is V11() real ext-real Element of REAL
1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
1 * (Z ") is V11() real ext-real set
f is open V46() V47() V48() Element of K6(REAL)
id f is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(1 / Z) (#) (() * x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((1 / Z) (#) (() * x)) - (id f) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
- (id f) is V19() Function-like V36() set
(- 1) (#) (id f) is V19() Function-like set
((1 / Z) (#) (() * x)) + (- (id f)) is V19() Function-like set
dom (((1 / Z) (#) (() * x)) - (id f)) is V46() V47() V48() Element of K6(REAL)
(((1 / Z) (#) (() * x)) - (id f)) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
Z * g is V11() real ext-real Element of REAL
(Z * g) + 0 is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
(id f) . x is V11() real ext-real Element of REAL
1 * x is V11() real ext-real Element of REAL
(1 * x) + 0 is V11() real ext-real Element of REAL
dom ((1 / Z) (#) (() * x)) is V46() V47() V48() Element of K6(REAL)
dom (id f) is V46() V47() V48() Element of K6(REAL)
(dom ((1 / Z) (#) (() * x))) /\ (dom (id f)) is V46() V47() V48() set
dom (() * x) is V46() V47() V48() Element of K6(REAL)
x is V11() real ext-real Element of REAL
((((1 / Z) (#) (() * x)) - (id f)) `| f) . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) ^2 is V11() real ext-real Element of REAL
(Z * x) * (Z * x) is V11() real ext-real set
1 + ((Z * x) ^2) is V11() real ext-real Element of REAL
((Z * x) ^2) / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((Z * x) ^2)) " is V11() real ext-real set
((Z * x) ^2) * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
- (((Z * x) ^2) / (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
diff (((1 / Z) (#) (() * x)),x) is V11() real ext-real Element of REAL
diff ((id f),x) is V11() real ext-real Element of REAL
(diff (((1 / Z) (#) (() * x)),x)) - (diff ((id f),x)) is V11() real ext-real Element of REAL
- (diff ((id f),x)) is V11() real ext-real set
(diff (((1 / Z) (#) (() * x)),x)) + (- (diff ((id f),x))) is V11() real ext-real set
((1 / Z) (#) (() * x)) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((1 / Z) (#) (() * x)) `| f) . x is V11() real ext-real Element of REAL
((((1 / Z) (#) (() * x)) `| f) . x) - (diff ((id f),x)) is V11() real ext-real Element of REAL
((((1 / Z) (#) (() * x)) `| f) . x) + (- (diff ((id f),x))) is V11() real ext-real set
diff ((() * x),x) is V11() real ext-real Element of REAL
(1 / Z) * (diff ((() * x),x)) is V11() real ext-real Element of REAL
((1 / Z) * (diff ((() * x),x))) - (diff ((id f),x)) is V11() real ext-real Element of REAL
((1 / Z) * (diff ((() * x),x))) + (- (diff ((id f),x))) is V11() real ext-real set
(() * x) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((() * x) `| f) . x is V11() real ext-real Element of REAL
(1 / Z) * (((() * x) `| f) . x) is V11() real ext-real Element of REAL
((1 / Z) * (((() * x) `| f) . x)) - (diff ((id f),x)) is V11() real ext-real Element of REAL
((1 / Z) * (((() * x) `| f) . x)) + (- (diff ((id f),x))) is V11() real ext-real set
(id f) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id f) `| f) . x is V11() real ext-real Element of REAL
((1 / Z) * (((() * x) `| f) . x)) - (((id f) `| f) . x) is V11() real ext-real Element of REAL
- (((id f) `| f) . x) is V11() real ext-real set
((1 / Z) * (((() * x) `| f) . x)) + (- (((id f) `| f) . x)) is V11() real ext-real set
(Z * x) + 0 is V11() real ext-real Element of REAL
((Z * x) + 0) ^2 is V11() real ext-real Element of REAL
((Z * x) + 0) * ((Z * x) + 0) is V11() real ext-real set
1 + (((Z * x) + 0) ^2) is V11() real ext-real Element of REAL
Z / (1 + (((Z * x) + 0) ^2)) is V11() real ext-real Element of REAL
(1 + (((Z * x) + 0) ^2)) " is V11() real ext-real set
Z * ((1 + (((Z * x) + 0) ^2)) ") is V11() real ext-real set
(1 / Z) * (Z / (1 + (((Z * x) + 0) ^2))) is V11() real ext-real Element of REAL
((1 / Z) * (Z / (1 + (((Z * x) + 0) ^2)))) - (((id f) `| f) . x) is V11() real ext-real Element of REAL
((1 / Z) * (Z / (1 + (((Z * x) + 0) ^2)))) + (- (((id f) `| f) . x)) is V11() real ext-real set
Z / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
Z * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
(1 / Z) * (Z / (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
((1 / Z) * (Z / (1 + ((Z * x) ^2)))) - 1 is V11() real ext-real Element of REAL
((1 / Z) * (Z / (1 + ((Z * x) ^2)))) + (- 1) is V11() real ext-real set
1 * Z is V11() real ext-real Element of REAL
Z * (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
(1 * Z) / (Z * (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((Z * x) ^2))) " is V11() real ext-real set
(1 * Z) * ((Z * (1 + ((Z * x) ^2))) ") is V11() real ext-real set
((1 * Z) / (Z * (1 + ((Z * x) ^2)))) - 1 is V11() real ext-real Element of REAL
((1 * Z) / (Z * (1 + ((Z * x) ^2)))) + (- 1) is V11() real ext-real set
1 / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
1 * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
(1 / (1 + ((Z * x) ^2))) - 1 is V11() real ext-real Element of REAL
(1 / (1 + ((Z * x) ^2))) + (- 1) is V11() real ext-real set
(1 + ((Z * x) ^2)) / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((Z * x) ^2)) * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
(1 / (1 + ((Z * x) ^2))) - ((1 + ((Z * x) ^2)) / (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
- ((1 + ((Z * x) ^2)) / (1 + ((Z * x) ^2))) is V11() real ext-real set
(1 / (1 + ((Z * x) ^2))) + (- ((1 + ((Z * x) ^2)) / (1 + ((Z * x) ^2)))) is V11() real ext-real set
x is V11() real ext-real Element of REAL
((((1 / Z) (#) (() * x)) - (id f)) `| f) . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) ^2 is V11() real ext-real Element of REAL
(Z * x) * (Z * x) is V11() real ext-real set
1 + ((Z * x) ^2) is V11() real ext-real Element of REAL
((Z * x) ^2) / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((Z * x) ^2)) " is V11() real ext-real set
((Z * x) ^2) * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
- (((Z * x) ^2) / (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
Z is V11() real ext-real Element of REAL
1 / Z is V11() real ext-real Element of REAL
Z " is V11() real ext-real set
1 * (Z ") is V11() real ext-real set
- (1 / Z) is V11() real ext-real Element of REAL
f is open V46() V47() V48() Element of K6(REAL)
id f is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
() * x is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(- (1 / Z)) (#) (() * x) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((- (1 / Z)) (#) (() * x)) - (id f) is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
- (id f) is V19() Function-like V36() set
(- 1) (#) (id f) is V19() Function-like set
((- (1 / Z)) (#) (() * x)) + (- (id f)) is V19() Function-like set
dom (((- (1 / Z)) (#) (() * x)) - (id f)) is V46() V47() V48() Element of K6(REAL)
(((- (1 / Z)) (#) (() * x)) - (id f)) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
g is V11() real ext-real Element of REAL
x . g is V11() real ext-real Element of REAL
Z * g is V11() real ext-real Element of REAL
(Z * g) + 0 is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
(id f) . x is V11() real ext-real Element of REAL
1 * x is V11() real ext-real Element of REAL
(1 * x) + 0 is V11() real ext-real Element of REAL
dom ((- (1 / Z)) (#) (() * x)) is V46() V47() V48() Element of K6(REAL)
dom (id f) is V46() V47() V48() Element of K6(REAL)
(dom ((- (1 / Z)) (#) (() * x))) /\ (dom (id f)) is V46() V47() V48() set
dom (() * x) is V46() V47() V48() Element of K6(REAL)
x is V11() real ext-real Element of REAL
((((- (1 / Z)) (#) (() * x)) - (id f)) `| f) . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) ^2 is V11() real ext-real Element of REAL
(Z * x) * (Z * x) is V11() real ext-real set
1 + ((Z * x) ^2) is V11() real ext-real Element of REAL
((Z * x) ^2) / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((Z * x) ^2)) " is V11() real ext-real set
((Z * x) ^2) * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
- (((Z * x) ^2) / (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
diff (((- (1 / Z)) (#) (() * x)),x) is V11() real ext-real Element of REAL
diff ((id f),x) is V11() real ext-real Element of REAL
(diff (((- (1 / Z)) (#) (() * x)),x)) - (diff ((id f),x)) is V11() real ext-real Element of REAL
- (diff ((id f),x)) is V11() real ext-real set
(diff (((- (1 / Z)) (#) (() * x)),x)) + (- (diff ((id f),x))) is V11() real ext-real set
((- (1 / Z)) (#) (() * x)) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((- (1 / Z)) (#) (() * x)) `| f) . x is V11() real ext-real Element of REAL
((((- (1 / Z)) (#) (() * x)) `| f) . x) - (diff ((id f),x)) is V11() real ext-real Element of REAL
((((- (1 / Z)) (#) (() * x)) `| f) . x) + (- (diff ((id f),x))) is V11() real ext-real set
diff ((() * x),x) is V11() real ext-real Element of REAL
(- (1 / Z)) * (diff ((() * x),x)) is V11() real ext-real Element of REAL
((- (1 / Z)) * (diff ((() * x),x))) - (diff ((id f),x)) is V11() real ext-real Element of REAL
((- (1 / Z)) * (diff ((() * x),x))) + (- (diff ((id f),x))) is V11() real ext-real set
(() * x) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((() * x) `| f) . x is V11() real ext-real Element of REAL
(- (1 / Z)) * (((() * x) `| f) . x) is V11() real ext-real Element of REAL
((- (1 / Z)) * (((() * x) `| f) . x)) - (diff ((id f),x)) is V11() real ext-real Element of REAL
((- (1 / Z)) * (((() * x) `| f) . x)) + (- (diff ((id f),x))) is V11() real ext-real set
(id f) `| f is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id f) `| f) . x is V11() real ext-real Element of REAL
((- (1 / Z)) * (((() * x) `| f) . x)) - (((id f) `| f) . x) is V11() real ext-real Element of REAL
- (((id f) `| f) . x) is V11() real ext-real set
((- (1 / Z)) * (((() * x) `| f) . x)) + (- (((id f) `| f) . x)) is V11() real ext-real set
(Z * x) + 0 is V11() real ext-real Element of REAL
((Z * x) + 0) ^2 is V11() real ext-real Element of REAL
((Z * x) + 0) * ((Z * x) + 0) is V11() real ext-real set
1 + (((Z * x) + 0) ^2) is V11() real ext-real Element of REAL
Z / (1 + (((Z * x) + 0) ^2)) is V11() real ext-real Element of REAL
(1 + (((Z * x) + 0) ^2)) " is V11() real ext-real set
Z * ((1 + (((Z * x) + 0) ^2)) ") is V11() real ext-real set
- (Z / (1 + (((Z * x) + 0) ^2))) is V11() real ext-real Element of REAL
(- (1 / Z)) * (- (Z / (1 + (((Z * x) + 0) ^2)))) is V11() real ext-real Element of REAL
((- (1 / Z)) * (- (Z / (1 + (((Z * x) + 0) ^2))))) - (((id f) `| f) . x) is V11() real ext-real Element of REAL
((- (1 / Z)) * (- (Z / (1 + (((Z * x) + 0) ^2))))) + (- (((id f) `| f) . x)) is V11() real ext-real set
(- 1) / Z is V11() real ext-real Element of REAL
(- 1) * (Z ") is V11() real ext-real set
- Z is V11() real ext-real Element of REAL
(- Z) / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
(- Z) * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
((- 1) / Z) * ((- Z) / (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
(((- 1) / Z) * ((- Z) / (1 + ((Z * x) ^2)))) - 1 is V11() real ext-real Element of REAL
(((- 1) / Z) * ((- Z) / (1 + ((Z * x) ^2)))) + (- 1) is V11() real ext-real set
(- 1) * (- Z) is V11() real ext-real Element of REAL
Z * (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
((- 1) * (- Z)) / (Z * (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
(Z * (1 + ((Z * x) ^2))) " is V11() real ext-real set
((- 1) * (- Z)) * ((Z * (1 + ((Z * x) ^2))) ") is V11() real ext-real set
(((- 1) * (- Z)) / (Z * (1 + ((Z * x) ^2)))) - 1 is V11() real ext-real Element of REAL
(((- 1) * (- Z)) / (Z * (1 + ((Z * x) ^2)))) + (- 1) is V11() real ext-real set
1 * Z is V11() real ext-real Element of REAL
(1 * Z) / (Z * (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
(1 * Z) * ((Z * (1 + ((Z * x) ^2))) ") is V11() real ext-real set
((1 * Z) / (Z * (1 + ((Z * x) ^2)))) - 1 is V11() real ext-real Element of REAL
((1 * Z) / (Z * (1 + ((Z * x) ^2)))) + (- 1) is V11() real ext-real set
1 / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
1 * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
(1 / (1 + ((Z * x) ^2))) - 1 is V11() real ext-real Element of REAL
(1 / (1 + ((Z * x) ^2))) + (- 1) is V11() real ext-real set
(1 + ((Z * x) ^2)) / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((Z * x) ^2)) * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
(1 / (1 + ((Z * x) ^2))) - ((1 + ((Z * x) ^2)) / (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
- ((1 + ((Z * x) ^2)) / (1 + ((Z * x) ^2))) is V11() real ext-real set
(1 / (1 + ((Z * x) ^2))) + (- ((1 + ((Z * x) ^2)) / (1 + ((Z * x) ^2)))) is V11() real ext-real set
x is V11() real ext-real Element of REAL
((((- (1 / Z)) (#) (() * x)) - (id f)) `| f) . x is V11() real ext-real Element of REAL
Z * x is V11() real ext-real Element of REAL
(Z * x) ^2 is V11() real ext-real Element of REAL
(Z * x) * (Z * x) is V11() real ext-real set
1 + ((Z * x) ^2) is V11() real ext-real Element of REAL
((Z * x) ^2) / (1 + ((Z * x) ^2)) is V11() real ext-real Element of REAL
(1 + ((Z * x) ^2)) " is V11() real ext-real set
((Z * x) ^2) * ((1 + ((Z * x) ^2)) ") is V11() real ext-real set
- (((Z * x) ^2) / (1 + ((Z * x) ^2))) is V11() real ext-real Element of REAL
ln (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (ln (#) ()) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(ln (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
{ b<sub>1</sub> where b<sub>1</sub> is V11() real ext-real Element of REAL : not b<sub>1</sub> <= 0 } is set
(dom ln) /\ (dom ()) is V46() V47() V48() set
f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
diff (ln,f) is V11() real ext-real Element of REAL
1 / f is V11() real ext-real Element of REAL
f " is V11() real ext-real set
1 * (f ") is V11() real ext-real set
f is V11() real ext-real Element of REAL
((ln (#) ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(() . f) / f is V11() real ext-real Element of REAL
f " is V11() real ext-real set
(() . f) * (f ") is V11() real ext-real set
ln . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(ln . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(ln . f) * ((1 + (f ^2)) ") is V11() real ext-real set
((() . f) / f) + ((ln . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
diff (ln,f) is V11() real ext-real Element of REAL
(() . f) * (diff (ln,f)) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
(ln . f) * (diff ((),f)) is V11() real ext-real Element of REAL
((() . f) * (diff (ln,f))) + ((ln . f) * (diff ((),f))) is V11() real ext-real Element of REAL
1 / f is V11() real ext-real Element of REAL
1 * (f ") is V11() real ext-real set
(() . f) * (1 / f) is V11() real ext-real Element of REAL
((() . f) * (1 / f)) + ((ln . f) * (diff ((),f))) is V11() real ext-real Element of REAL
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . f is V11() real ext-real Element of REAL
(ln . f) * ((() `| Z) . f) is V11() real ext-real Element of REAL
((() . f) * (1 / f)) + ((ln . f) * ((() `| Z) . f)) is V11() real ext-real Element of REAL
(() . f) * 1 is V11() real ext-real Element of REAL
((() . f) * 1) / f is V11() real ext-real Element of REAL
((() . f) * 1) * (f ") is V11() real ext-real set
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
(ln . f) * (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
(((() . f) * 1) / f) + ((ln . f) * (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((ln (#) ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(() . f) / f is V11() real ext-real Element of REAL
f " is V11() real ext-real set
(() . f) * (f ") is V11() real ext-real set
ln . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(ln . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(ln . f) * ((1 + (f ^2)) ") is V11() real ext-real set
((() . f) / f) + ((ln . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
ln (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (ln (#) ()) is V46() V47() V48() Element of K6(REAL)
Z is open V46() V47() V48() Element of K6(REAL)
(ln (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
{ b₁ where b₁ is V11() real ext-real Element of REAL : not b₁ <= 0 } is set
(dom ln) /\ (dom ()) is V46() V47() V48() set
f is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
diff (ln,f) is V11() real ext-real Element of REAL
1 / f is V11() real ext-real Element of REAL
f " is V11() real ext-real set
1 * (f ") is V11() real ext-real set
f is V11() real ext-real Element of REAL
((ln (#) ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(() . f) / f is V11() real ext-real Element of REAL
f " is V11() real ext-real set
(() . f) * (f ") is V11() real ext-real set
ln . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(ln . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(ln . f) * ((1 + (f ^2)) ") is V11() real ext-real set
((() . f) / f) - ((ln . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((ln . f) / (1 + (f ^2))) is V11() real ext-real set
((() . f) / f) + (- ((ln . f) / (1 + (f ^2)))) is V11() real ext-real set
diff (ln,f) is V11() real ext-real Element of REAL
(() . f) * (diff (ln,f)) is V11() real ext-real Element of REAL
diff ((),f) is V11() real ext-real Element of REAL
(ln . f) * (diff ((),f)) is V11() real ext-real Element of REAL
((() . f) * (diff (ln,f))) + ((ln . f) * (diff ((),f))) is V11() real ext-real Element of REAL
1 / f is V11() real ext-real Element of REAL
1 * (f ") is V11() real ext-real set
(() . f) * (1 / f) is V11() real ext-real Element of REAL
((() . f) * (1 / f)) + ((ln . f) * (diff ((),f))) is V11() real ext-real Element of REAL
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . f is V11() real ext-real Element of REAL
(ln . f) * ((() `| Z) . f) is V11() real ext-real Element of REAL
((() . f) * (1 / f)) + ((ln . f) * ((() `| Z) . f)) is V11() real ext-real Element of REAL
1 / (1 + (f ^2)) is V11() real ext-real Element of REAL
1 * ((1 + (f ^2)) ") is V11() real ext-real set
- (1 / (1 + (f ^2))) is V11() real ext-real Element of REAL
(ln . f) * (- (1 / (1 + (f ^2)))) is V11() real ext-real Element of REAL
((() . f) * (1 / f)) + ((ln . f) * (- (1 / (1 + (f ^2))))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
((ln (#) ()) `| Z) . f is V11() real ext-real Element of REAL
() . f is V11() real ext-real Element of REAL
(() . f) / f is V11() real ext-real Element of REAL
f " is V11() real ext-real set
(() . f) * (f ") is V11() real ext-real set
ln . f is V11() real ext-real Element of REAL
f ^2 is V11() real ext-real Element of REAL
f * f is V11() real ext-real set
1 + (f ^2) is V11() real ext-real Element of REAL
(ln . f) / (1 + (f ^2)) is V11() real ext-real Element of REAL
(1 + (f ^2)) " is V11() real ext-real set
(ln . f) * ((1 + (f ^2)) ") is V11() real ext-real set
((() . f) / f) - ((ln . f) / (1 + (f ^2))) is V11() real ext-real Element of REAL
- ((ln . f) / (1 + (f ^2))) is V11() real ext-real set
((() . f) / f) + (- ((ln . f) / (1 + (f ^2)))) is V11() real ext-real set
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) ^) (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (((id Z) ^) (#) ()) is V46() V47() V48() Element of K6(REAL)
(((id Z) ^) (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
(dom ((id Z) ^)) /\ (dom ()) is V46() V47() V48() set
x is V11() real ext-real Element of REAL
((((id Z) ^) (#) ()) `| Z) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
(() . x) / (x ^2) is V11() real ext-real Element of REAL
(x ^2) " is V11() real ext-real set
(() . x) * ((x ^2) ") is V11() real ext-real set
- ((() . x) / (x ^2)) is V11() real ext-real Element of REAL
1 + (x ^2) is V11() real ext-real Element of REAL
x * (1 + (x ^2)) is V11() real ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V11() real ext-real Element of REAL
(x * (1 + (x ^2))) " is V11() real ext-real set
1 * ((x * (1 + (x ^2))) ") is V11() real ext-real set
(- ((() . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) is V11() real ext-real Element of REAL
diff (((id Z) ^),x) is V11() real ext-real Element of REAL
(() . x) * (diff (((id Z) ^),x)) is V11() real ext-real Element of REAL
((id Z) ^) . x is V11() real ext-real Element of REAL
diff ((),x) is V11() real ext-real Element of REAL
(((id Z) ^) . x) * (diff ((),x)) is V11() real ext-real Element of REAL
((() . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff ((),x))) is V11() real ext-real Element of REAL
((id Z) ^) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((id Z) ^) `| Z) . x is V11() real ext-real Element of REAL
(() . x) * ((((id Z) ^) `| Z) . x) is V11() real ext-real Element of REAL
((() . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff ((),x))) is V11() real ext-real Element of REAL
1 / (x ^2) is V11() real ext-real Element of REAL
1 * ((x ^2) ") is V11() real ext-real set
- (1 / (x ^2)) is V11() real ext-real Element of REAL
(() . x) * (- (1 / (x ^2))) is V11() real ext-real Element of REAL
((() . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff ((),x))) is V11() real ext-real Element of REAL
(() . x) * (1 / (x ^2)) is V11() real ext-real Element of REAL
- ((() . x) * (1 / (x ^2))) is V11() real ext-real Element of REAL
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . x is V11() real ext-real Element of REAL
(((id Z) ^) . x) * ((() `| Z) . x) is V11() real ext-real Element of REAL
(- ((() . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((() `| Z) . x)) is V11() real ext-real Element of REAL
(() . x) * 1 is V11() real ext-real Element of REAL
((() . x) * 1) / (x ^2) is V11() real ext-real Element of REAL
((() . x) * 1) * ((x ^2) ") is V11() real ext-real set
- (((() . x) * 1) / (x ^2)) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
1 * ((1 + (x ^2)) ") is V11() real ext-real set
(((id Z) ^) . x) * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(- (((() . x) * 1) / (x ^2))) + ((((id Z) ^) . x) * (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
(id Z) . x is V11() real ext-real Element of REAL
((id Z) . x) " is V11() real ext-real Element of REAL
(((id Z) . x) ") * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(- ((() . x) / (x ^2))) + ((((id Z) . x) ") * (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
1 / x is V11() real ext-real Element of REAL
x " is V11() real ext-real set
1 * (x ") is V11() real ext-real set
(1 / x) * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(- ((() . x) / (x ^2))) + ((1 / x) * (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
1 * 1 is non empty V11() real ext-real positive non negative V63() Element of REAL
(1 * 1) / (x * (1 + (x ^2))) is V11() real ext-real Element of REAL
(1 * 1) * ((x * (1 + (x ^2))) ") is V11() real ext-real set
(- ((() . x) / (x ^2))) + ((1 * 1) / (x * (1 + (x ^2)))) is V11() real ext-real Element of REAL
x is V11() real ext-real Element of REAL
((((id Z) ^) (#) ()) `| Z) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
(() . x) / (x ^2) is V11() real ext-real Element of REAL
(x ^2) " is V11() real ext-real set
(() . x) * ((x ^2) ") is V11() real ext-real set
- ((() . x) / (x ^2)) is V11() real ext-real Element of REAL
1 + (x ^2) is V11() real ext-real Element of REAL
x * (1 + (x ^2)) is V11() real ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V11() real ext-real Element of REAL
(x * (1 + (x ^2))) " is V11() real ext-real set
1 * ((x * (1 + (x ^2))) ") is V11() real ext-real set
(- ((() . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) is V11() real ext-real Element of REAL
Z is open V46() V47() V48() Element of K6(REAL)
id Z is V19() V22( REAL ) V23( REAL ) Function-like one-to-one V36() V37() V38() Element of K6(K7(REAL,REAL))
(id Z) ^ is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
((id Z) ^) (#) () is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom (((id Z) ^) (#) ()) is V46() V47() V48() Element of K6(REAL)
(((id Z) ^) (#) ()) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
dom ((id Z) ^) is V46() V47() V48() Element of K6(REAL)
(dom ((id Z) ^)) /\ (dom ()) is V46() V47() V48() set
x is V11() real ext-real Element of REAL
((((id Z) ^) (#) ()) `| Z) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
(() . x) / (x ^2) is V11() real ext-real Element of REAL
(x ^2) " is V11() real ext-real set
(() . x) * ((x ^2) ") is V11() real ext-real set
- ((() . x) / (x ^2)) is V11() real ext-real Element of REAL
1 + (x ^2) is V11() real ext-real Element of REAL
x * (1 + (x ^2)) is V11() real ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V11() real ext-real Element of REAL
(x * (1 + (x ^2))) " is V11() real ext-real set
1 * ((x * (1 + (x ^2))) ") is V11() real ext-real set
(- ((() . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) is V11() real ext-real Element of REAL
- (1 / (x * (1 + (x ^2)))) is V11() real ext-real set
(- ((() . x) / (x ^2))) + (- (1 / (x * (1 + (x ^2))))) is V11() real ext-real set
diff (((id Z) ^),x) is V11() real ext-real Element of REAL
(() . x) * (diff (((id Z) ^),x)) is V11() real ext-real Element of REAL
((id Z) ^) . x is V11() real ext-real Element of REAL
diff ((),x) is V11() real ext-real Element of REAL
(((id Z) ^) . x) * (diff ((),x)) is V11() real ext-real Element of REAL
((() . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff ((),x))) is V11() real ext-real Element of REAL
((id Z) ^) `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(((id Z) ^) `| Z) . x is V11() real ext-real Element of REAL
(() . x) * ((((id Z) ^) `| Z) . x) is V11() real ext-real Element of REAL
((() . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff ((),x))) is V11() real ext-real Element of REAL
1 / (x ^2) is V11() real ext-real Element of REAL
1 * ((x ^2) ") is V11() real ext-real set
- (1 / (x ^2)) is V11() real ext-real Element of REAL
(() . x) * (- (1 / (x ^2))) is V11() real ext-real Element of REAL
((() . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff ((),x))) is V11() real ext-real Element of REAL
(() . x) * (1 / (x ^2)) is V11() real ext-real Element of REAL
- ((() . x) * (1 / (x ^2))) is V11() real ext-real Element of REAL
() `| Z is V19() V22( REAL ) V23( REAL ) Function-like V36() V37() V38() Element of K6(K7(REAL,REAL))
(() `| Z) . x is V11() real ext-real Element of REAL
(((id Z) ^) . x) * ((() `| Z) . x) is V11() real ext-real Element of REAL
(- ((() . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((() `| Z) . x)) is V11() real ext-real Element of REAL
1 / (1 + (x ^2)) is V11() real ext-real Element of REAL
(1 + (x ^2)) " is V11() real ext-real set
1 * ((1 + (x ^2)) ") is V11() real ext-real set
- (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(((id Z) ^) . x) * (- (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
(- ((() . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * (- (1 / (1 + (x ^2))))) is V11() real ext-real Element of REAL
(() . x) * 1 is V11() real ext-real Element of REAL
((() . x) * 1) / (x ^2) is V11() real ext-real Element of REAL
((() . x) * 1) * ((x ^2) ") is V11() real ext-real set
- (((() . x) * 1) / (x ^2)) is V11() real ext-real Element of REAL
(((id Z) ^) . x) * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(- (((() . x) * 1) / (x ^2))) - ((((id Z) ^) . x) * (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
- ((((id Z) ^) . x) * (1 / (1 + (x ^2)))) is V11() real ext-real set
(- (((() . x) * 1) / (x ^2))) + (- ((((id Z) ^) . x) * (1 / (1 + (x ^2))))) is V11() real ext-real set
(id Z) . x is V11() real ext-real Element of REAL
((id Z) . x) " is V11() real ext-real Element of REAL
(((id Z) . x) ") * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(- ((() . x) / (x ^2))) - ((((id Z) . x) ") * (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
- ((((id Z) . x) ") * (1 / (1 + (x ^2)))) is V11() real ext-real set
(- ((() . x) / (x ^2))) + (- ((((id Z) . x) ") * (1 / (1 + (x ^2))))) is V11() real ext-real set
1 / x is V11() real ext-real Element of REAL
x " is V11() real ext-real set
1 * (x ") is V11() real ext-real set
(1 / x) * (1 / (1 + (x ^2))) is V11() real ext-real Element of REAL
(- ((() . x) / (x ^2))) - ((1 / x) * (1 / (1 + (x ^2)))) is V11() real ext-real Element of REAL
- ((1 / x) * (1 / (1 + (x ^2)))) is V11() real ext-real set
(- ((() . x) / (x ^2))) + (- ((1 / x) * (1 / (1 + (x ^2))))) is V11() real ext-real set
1 * 1 is non empty V11() real ext-real positive non negative V63() Element of REAL
(1 * 1) / (x * (1 + (x ^2))) is V11() real ext-real Element of REAL
(1 * 1) * ((x * (1 + (x ^2))) ") is V11() real ext-real set
(- ((() . x) / (x ^2))) - ((1 * 1) / (x * (1 + (x ^2)))) is V11() real ext-real Element of REAL
- ((1 * 1) / (x * (1 + (x ^2)))) is V11() real ext-real set
(- ((() . x) / (x ^2))) + (- ((1 * 1) / (x * (1 + (x ^2))))) is V11() real ext-real set
x is V11() real ext-real Element of REAL
((((id Z) ^) (#) ()) `| Z) . x is V11() real ext-real Element of REAL
() . x is V11() real ext-real Element of REAL
x ^2 is V11() real ext-real Element of REAL
x * x is V11() real ext-real set
(() . x) / (x ^2) is V11() real ext-real Element of REAL
(x ^2) " is V11() real ext-real set
(() . x) * ((x ^2) ") is V11() real ext-real set
- ((() . x) / (x ^2)) is V11() real ext-real Element of REAL
1 + (x ^2) is V11() real ext-real Element of REAL
x * (1 + (x ^2)) is V11() real ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V11() real ext-real Element of REAL
(x * (1 + (x ^2))) " is V11() real ext-real set
1 * ((x * (1 + (x ^2))) ") is V11() real ext-real set
(- ((() . x) / (x ^2))) - (1 / (x * (1 + (x ^2)))) is V11() real ext-real Element of REAL
- (1 / (x * (1 + (x ^2)))) is V11() real ext-real set
(- ((() . x) / (x ^2))) + (- (1 / (x * (1 + (x ^2))))) is V11() real ext-real set