:: Several Integrability Formulas of Special Functions
:: by Cuiying Peng , Fuguo Ge and Xiquan Liang
::
:: Received August 28, 2007
:: Copyright (c) 2007-2011 Association of Mizar Users

begin

Lm1: sin (PI / 4) > 0

proof end;

Lm2: sin (- (PI / 4)) < 0

proof end;

theorem Th1: :: INTEGRA8:1

for x being Real
for n being Element of NAT holds sin (x + ((2 * n) * PI)) = sin x

proof end;

theorem Th2: :: INTEGRA8:2

for x being Real
for n being Element of NAT holds sin (x + (((2 * n) + 1) * PI)) = - (sin x)

proof end;

theorem Th3: :: INTEGRA8:3

for x being Real
for n being Element of NAT holds cos (x + ((2 * n) * PI)) = cos x

proof end;

theorem Th4: :: INTEGRA8:4

for x being Real
for n being Element of NAT holds cos (x + (((2 * n) + 1) * PI)) = - (cos x)

proof end;

theorem Th5: :: INTEGRA8:5

for x being Real st sin (x / 2) >= 0 holds
sin (x / 2) = sqrt ((1 - (cos x)) / 2)

proof end;

theorem Th6: :: INTEGRA8:6

for x being Real st sin (x / 2) < 0 holds
sin (x / 2) = - (sqrt ((1 - (cos x)) / 2))

proof end;

theorem Th7: :: INTEGRA8:7

sin (PI / 4) = (sqrt 2) / 2

proof end;

theorem Th8: :: INTEGRA8:8

sin (- (PI / 4)) = - ((sqrt 2) / 2)

proof end;

theorem Th9: :: INTEGRA8:9

[.(- ((sqrt 2) / 2)),((sqrt 2) / 2).] c= ].(- 1),1.[

proof end;

theorem Th10: :: INTEGRA8:10

arcsin ((sqrt 2) / 2) = PI / 4

proof end;

theorem Th11: :: INTEGRA8:11

arcsin (- ((sqrt 2) / 2)) = - (PI / 4)

proof end;

theorem Th12: :: INTEGRA8:12

for x being Real st cos (x / 2) >= 0 holds
cos (x / 2) = sqrt ((1 + (cos x)) / 2)

proof end;

Lm3: cos (PI / 4) > 0

proof end;

theorem Th13: :: INTEGRA8:13

cos (PI / 4) = (sqrt 2) / 2

proof end;

theorem Th14: :: INTEGRA8:14

cos ((3 * PI) / 4) = - ((sqrt 2) / 2)

proof end;

theorem Th15: :: INTEGRA8:15

arccos ((sqrt 2) / 2) = PI / 4

proof end;

theorem Th16: :: INTEGRA8:16

arccos (- ((sqrt 2) / 2)) = (3 * PI) / 4

proof end;

theorem Th17: :: INTEGRA8:17

sinh . 1 = ((number_e ^2) - 1) / (2 * number_e)

proof end;

theorem Th18: :: INTEGRA8:18

cosh . 0 = 1

proof end;

theorem Th19: :: INTEGRA8:19

cosh . 1 = ((number_e ^2) + 1) / (2 * number_e)

proof end;

Lm4: for f1, f2 being PartFunc of REAL,REAL holds (- f1) (#) (- f2) = f1 (#) f2

proof end;

theorem Th20: :: INTEGRA8:20

for L1 being LINEAR holds - L1 is LINEAR

proof end;

theorem Th21: :: INTEGRA8:21

for R1 being REST holds - R1 is REST

proof end;

theorem Th22: :: INTEGRA8:22

for f1 being PartFunc of REAL,REAL
for x0 being Real st f1 is_differentiable_in x0 holds
( - f1 is_differentiable_in x0 & diff ((- f1),x0) = - (diff (f1,x0)) )

proof end;

theorem Th23: :: INTEGRA8:23

for f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st Z c= dom (- f1) & f1 is_differentiable_on Z holds
( - f1 is_differentiable_on Z & ( for x being Real st x in Z holds
((- f1) `| Z) . x = - (diff (f1,x)) ) )

proof end;

theorem :: INTEGRA8:24

- sin is_differentiable_on REAL

proof end;

theorem Th25: :: INTEGRA8:25

for x being Real holds
( - cos is_differentiable_in x & diff ((- cos),x) = sin . x )

proof end;

theorem Th26: :: INTEGRA8:26

( - cos is_differentiable_on REAL & ( for x being Real st x in REAL holds
diff ((- cos),x) = sin . x ) )

proof end;

theorem Th27: :: INTEGRA8:27

sin `| REAL = cos

proof end;

theorem Th28: :: INTEGRA8:28

cos `| REAL = - sin

proof end;

theorem Th29: :: INTEGRA8:29

(- cos) `| REAL = sin by Th26, FDIFF_1:def 8, SIN_COS:27;

theorem Th30: :: INTEGRA8:30

sinh `| REAL = cosh

proof end;

theorem Th31: :: INTEGRA8:31

cosh `| REAL = sinh

proof end;

theorem Th32: :: INTEGRA8:32

exp_R `| REAL = exp_R

proof end;

theorem Th33: :: INTEGRA8:33

for Z being open Subset of REAL st Z c= dom tan holds
( tan is_differentiable_on Z & ( for x being Real st x in Z holds
(tan `| Z) . x = 1 / ((cos . x) ^2) ) )

proof end;

theorem Th34: :: INTEGRA8:34

for Z being open Subset of REAL st Z c= dom cot holds
( cot is_differentiable_on Z & ( for x being Real st x in Z holds
(cot `| Z) . x = - (1 / ((sin . x) ^2)) ) )

proof end;

theorem :: INTEGRA8:35

for r being Real holds rng (REAL --> r) c= REAL

proof end;

definition

let r be Real;
func Cst r -> Function of REAL,REAL equals :: INTEGRA8:def 1
REAL --> r;
coherence
REAL --> r is Function of REAL,REAL ;

end;

:: deftheorem defines Cst INTEGRA8:def 1 :
for r being Real holds Cst r = REAL --> r;

theorem Th36: :: INTEGRA8:36

for a, b being Real
for A being closed-interval Subset of REAL holds chi (A,A) = (Cst 1) | A

proof end;

theorem Th37: :: INTEGRA8:37

for a, b being Real
for A being closed-interval Subset of REAL st A = [.a,b.] holds
( upper_bound A = b & lower_bound A = a )

proof end;

begin

theorem :: INTEGRA8:38

for a, b being Real st a <= b holds
integral ((Cst 1),a,b) = b - a

proof end;

Lm5: dom sin = REAL
by FUNCT_2:def 1;

Lm6: dom cos = REAL
by FUNCT_2:def 1;

Lm7: dom (- sin) = REAL
by FUNCT_2:def 1;

Lm8: dom exp_R = REAL
by FUNCT_2:def 1;

Lm9: dom sinh = REAL
by FUNCT_2:def 1;

Lm10: dom cosh = REAL
by FUNCT_2:def 1;

Lm11: for A being closed-interval Subset of REAL holds
( cos is_integrable_on A & cos | A is bounded )

proof end;

theorem Th39: :: INTEGRA8:39

for A being closed-interval Subset of REAL holds integral (cos,A) = (sin . (upper_bound A)) - (sin . (lower_bound A))

proof end;

theorem :: INTEGRA8:40

for A being closed-interval Subset of REAL st A = [.0,(PI / 2).] holds
integral (cos,A) = 1

proof end;

theorem :: INTEGRA8:41

for A being closed-interval Subset of REAL st A = [.0,PI.] holds
integral (cos,A) = 0

proof end;

theorem :: INTEGRA8:42

for A being closed-interval Subset of REAL st A = [.0,((PI * 3) / 2).] holds
integral (cos,A) = - 1

proof end;

theorem :: INTEGRA8:43

for A being closed-interval Subset of REAL st A = [.0,(PI * 2).] holds
integral (cos,A) = 0

proof end;

theorem :: INTEGRA8:44

for A being closed-interval Subset of REAL
for n being Element of NAT st A = [.((2 * n) * PI),(((2 * n) + 1) * PI).] holds
integral (cos,A) = 0

proof end;

theorem :: INTEGRA8:45

for A being closed-interval Subset of REAL
for x being Real
for n being Element of NAT st A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] holds
integral (cos,A) = - (2 * (sin x))

proof end;

Lm12: for A being closed-interval Subset of REAL holds
( - sin is_integrable_on A & (- sin) | A is bounded )

proof end;

theorem Th46: :: INTEGRA8:46

for A being closed-interval Subset of REAL holds integral ((- sin),A) = (cos . (upper_bound A)) - (cos . (lower_bound A))

proof end;

theorem :: INTEGRA8:47

for A being closed-interval Subset of REAL st A = [.0,(PI / 2).] holds
integral ((- sin),A) = - 1

proof end;

theorem :: INTEGRA8:48

for A being closed-interval Subset of REAL st A = [.0,PI.] holds
integral ((- sin),A) = - 2

proof end;

theorem :: INTEGRA8:49

for A being closed-interval Subset of REAL st A = [.0,((PI * 3) / 2).] holds
integral ((- sin),A) = - 1

proof end;

theorem :: INTEGRA8:50

for A being closed-interval Subset of REAL st A = [.0,(PI * 2).] holds
integral ((- sin),A) = 0

proof end;

theorem :: INTEGRA8:51

for A being closed-interval Subset of REAL
for n being Element of NAT st A = [.((2 * n) * PI),(((2 * n) + 1) * PI).] holds
integral ((- sin),A) = - 2

proof end;

theorem :: INTEGRA8:52

for A being closed-interval Subset of REAL
for x being Real
for n being Element of NAT st A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] holds
integral ((- sin),A) = - (2 * (cos x))

proof end;

Lm13: for A being closed-interval Subset of REAL holds
( exp_R is_integrable_on A & exp_R | A is bounded )

proof end;

theorem Th53: :: INTEGRA8:53

for A being closed-interval Subset of REAL holds integral (exp_R,A) = (exp_R . (upper_bound A)) - (exp_R . (lower_bound A))

proof end;

theorem :: INTEGRA8:54

for A being closed-interval Subset of REAL st A = [.0,1.] holds
integral (exp_R,A) = number_e - 1

proof end;

Lm14: for A being closed-interval Subset of REAL holds sinh | A is continuous

proof end;

Lm15: for A being closed-interval Subset of REAL holds
( sinh is_integrable_on A & sinh | A is bounded )

proof end;

theorem Th55: :: INTEGRA8:55

for A being closed-interval Subset of REAL holds integral (sinh,A) = (cosh . (upper_bound A)) - (cosh . (lower_bound A))

proof end;

theorem :: INTEGRA8:56

for A being closed-interval Subset of REAL st A = [.0,1.] holds
integral (sinh,A) = ((number_e - 1) ^2) / (2 * number_e)

proof end;

Lm16: for A being closed-interval Subset of REAL holds cosh | A is continuous

proof end;

Lm17: for A being closed-interval Subset of REAL holds
( cosh is_integrable_on A & cosh | A is bounded )

proof end;

theorem Th57: :: INTEGRA8:57

for A being closed-interval Subset of REAL holds integral (cosh,A) = (sinh . (upper_bound A)) - (sinh . (lower_bound A))

proof end;

theorem :: INTEGRA8:58

for A being closed-interval Subset of REAL st A = [.0,1.] holds
integral (cosh,A) = ((number_e ^2) - 1) / (2 * number_e)

proof end;

theorem :: INTEGRA8:59

for f2 being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL
for Z being open Subset of REAL st A c= Z & dom tan = Z & dom tan = dom f2 & ( for x being Real st x in Z holds
( f2 . x = 1 / ((cos . x) ^2) & cos . x <> 0 ) ) & f2 | A is continuous holds
integral (f2,A) = (tan . (upper_bound A)) - (tan . (lower_bound A))

proof end;

theorem :: INTEGRA8:60

for f2 being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL
for Z being open Subset of REAL st A c= Z & dom cot = Z & dom cot = dom f2 & ( for x being Real st x in Z holds
( f2 . x = - (1 / ((sin . x) ^2)) & sin . x <> 0 ) ) & f2 | A is continuous holds
integral (f2,A) = (cot . (upper_bound A)) - (cot . (lower_bound A))

proof end;

theorem :: INTEGRA8:61

for f2 being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL st dom tanh = dom f2 & ( for x being Real st x in REAL holds
f2 . x = 1 / ((cosh . x) ^2) ) & f2 | A is continuous holds
integral (f2,A) = (tanh . (upper_bound A)) - (tanh . (lower_bound A))

proof end;

Lm18: dom (arcsin `| ].(- 1),1.[) = ].(- 1),1.[
by FDIFF_1:def 8, SIN_COS6:84;

theorem Th62: :: INTEGRA8:62

for f2 being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL st A c= ].(- 1),1.[ & dom (arcsin `| ].(- 1),1.[) = dom f2 & ( for x being Real holds
( x in ].(- 1),1.[ & f2 . x = 1 / (sqrt (1 - (x ^2))) ) ) & f2 | A is continuous holds
integral (f2,A) = (arcsin . (upper_bound A)) - (arcsin . (lower_bound A))

proof end;

theorem Th63: :: INTEGRA8:63

for f2 being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL st A c= ].(- 1),1.[ & dom (arccos `| ].(- 1),1.[) = dom f2 & ( for x being Real holds
( x in ].(- 1),1.[ & f2 . x = - (1 / (sqrt (1 - (x ^2)))) ) ) & f2 | A is continuous holds
integral (f2,A) = (arccos . (upper_bound A)) - (arccos . (lower_bound A))

proof end;

theorem :: INTEGRA8:64

for f2 being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL st A = [.(- ((sqrt 2) / 2)),((sqrt 2) / 2).] & dom (arcsin `| ].(- 1),1.[) = dom f2 & ( for x being Real holds
( x in ].(- 1),1.[ & f2 . x = 1 / (sqrt (1 - (x ^2))) ) ) & f2 | A is continuous holds
integral (f2,A) = PI / 2

proof end;

theorem :: INTEGRA8:65

for f2 being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL st A = [.(- ((sqrt 2) / 2)),((sqrt 2) / 2).] & dom (arccos `| ].(- 1),1.[) = dom f2 & ( for x being Real holds
( x in ].(- 1),1.[ & f2 . x = - (1 / (sqrt (1 - (x ^2)))) ) ) & f2 | A is continuous holds
integral (f2,A) = - (PI / 2)

proof end;

theorem Th66: :: INTEGRA8:66

for f, g being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL
for Z being open Subset of REAL st f is_differentiable_on Z & g is_differentiable_on Z & A c= Z & f `| Z is_integrable_on A & (f `| Z) | A is bounded & g `| Z is_integrable_on A & (g `| Z) | A is bounded holds
integral (((f `| Z) + (g `| Z)),A) = (((f . (upper_bound A)) - (f . (lower_bound A))) + (g . (upper_bound A))) - (g . (lower_bound A))

proof end;

theorem Th67: :: INTEGRA8:67

for f, g being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL
for Z being open Subset of REAL st f is_differentiable_on Z & g is_differentiable_on Z & A c= Z & f `| Z is_integrable_on A & (f `| Z) | A is bounded & g `| Z is_integrable_on A & (g `| Z) | A is bounded holds
integral (((f `| Z) - (g `| Z)),A) = ((f . (upper_bound A)) - (f . (lower_bound A))) - ((g . (upper_bound A)) - (g . (lower_bound A)))

proof end;

theorem Th68: :: INTEGRA8:68

for f being PartFunc of REAL,REAL
for A being closed-interval Subset of REAL
for r being Real
for Z being open Subset of REAL st f is_differentiable_on Z & A c= Z & f `| Z is_integrable_on A & (f `| Z) | A is bounded holds
integral ((r (#) (f `| Z)),A) = (r * (f . (upper_bound A))) - (r * (f . (lower_bound A)))

proof end;

Lm19: for A being closed-interval Subset of REAL holds
( sin is_integrable_on A & sin | A is bounded )

proof end;

theorem Th69: :: INTEGRA8:69

for A being closed-interval Subset of REAL holds integral ((sin + cos),A) = ((((- cos) . (upper_bound A)) - ((- cos) . (lower_bound A))) + (sin . (upper_bound A))) - (sin . (lower_bound A))

proof end;

theorem :: INTEGRA8:70

for A being closed-interval Subset of REAL st A = [.0,(PI / 2).] holds
integral ((sin + cos),A) = 2

proof end;

theorem :: INTEGRA8:71

for A being closed-interval Subset of REAL st A = [.0,PI.] holds
integral ((sin + cos),A) = 2

proof end;

theorem :: INTEGRA8:72

for A being closed-interval Subset of REAL st A = [.0,((PI * 3) / 2).] holds
integral ((sin + cos),A) = 0

proof end;

theorem :: INTEGRA8:73

for A being closed-interval Subset of REAL st A = [.0,(PI * 2).] holds
integral ((sin + cos),A) = 0

proof end;

theorem :: INTEGRA8:74

for A being closed-interval Subset of REAL
for n being Element of NAT st A = [.((2 * n) * PI),(((2 * n) + 1) * PI).] holds
integral ((sin + cos),A) = 2

proof end;

theorem :: INTEGRA8:75

for A being closed-interval Subset of REAL
for x being Real
for n being Element of NAT st A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] holds
integral ((sin + cos),A) = (2 * (cos x)) - (2 * (sin x))

proof end;

theorem Th76: :: INTEGRA8:76

for A being closed-interval Subset of REAL holds integral ((sinh + cosh),A) = (((cosh . (upper_bound A)) - (cosh . (lower_bound A))) + (sinh . (upper_bound A))) - (sinh . (lower_bound A))

proof end;

theorem :: INTEGRA8:77

for A being closed-interval Subset of REAL st A = [.0,1.] holds
integral ((sinh + cosh),A) = number_e - 1

proof end;

theorem Th78: :: INTEGRA8:78

for A being closed-interval Subset of REAL holds integral ((sin - cos),A) = (((- cos) . (upper_bound A)) - ((- cos) . (lower_bound A))) - ((sin . (upper_bound A)) - (sin . (lower_bound A)))

proof end;

theorem :: INTEGRA8:79

for A being closed-interval Subset of REAL st A = [.0,(PI / 2).] holds
integral ((sin - cos),A) = 0

proof end;

theorem :: INTEGRA8:80

for A being closed-interval Subset of REAL st A = [.0,PI.] holds
integral ((sin - cos),A) = 2

proof end;

theorem :: INTEGRA8:81

for A being closed-interval Subset of REAL st A = [.0,((PI * 3) / 2).] holds
integral ((sin - cos),A) = 2

proof end;

theorem :: INTEGRA8:82

for A being closed-interval Subset of REAL st A = [.0,(PI * 2).] holds
integral ((sin - cos),A) = 0

proof end;

theorem :: INTEGRA8:83

for A being closed-interval Subset of REAL
for n being Element of NAT st A = [.((2 * n) * PI),(((2 * n) + 1) * PI).] holds
integral ((sin - cos),A) = 2

proof end;

theorem :: INTEGRA8:84

for A being closed-interval Subset of REAL
for x being Real
for n being Element of NAT st A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] holds
integral ((sin - cos),A) = (2 * (cos x)) + (2 * (sin x))

proof end;

theorem :: INTEGRA8:85

for A being closed-interval Subset of REAL
for r being Real holds integral ((r (#) sin),A) = (r * ((- cos) . (upper_bound A))) - (r * ((- cos) . (lower_bound A)))

proof end;

theorem :: INTEGRA8:86

for A being closed-interval Subset of REAL
for r being Real holds integral ((r (#) cos),A) = (r * (sin . (upper_bound A))) - (r * (sin . (lower_bound A)))

proof end;

theorem :: INTEGRA8:87

for A being closed-interval Subset of REAL
for r being Real holds integral ((r (#) sinh),A) = (r * (cosh . (upper_bound A))) - (r * (cosh . (lower_bound A)))

proof end;

theorem :: INTEGRA8:88

for A being closed-interval Subset of REAL
for r being Real holds integral ((r (#) cosh),A) = (r * (sinh . (upper_bound A))) - (r * (sinh . (lower_bound A)))

proof end;

theorem :: INTEGRA8:89

for A being closed-interval Subset of REAL
for r being Real holds integral ((r (#) exp_R),A) = (r * (exp_R . (upper_bound A))) - (r * (exp_R . (lower_bound A)))

proof end;

theorem Th90: :: INTEGRA8:90

for A being closed-interval Subset of REAL holds integral ((sin (#) cos),A) = (1 / 2) * (((cos . (lower_bound A)) * (cos . (lower_bound A))) - ((cos . (upper_bound A)) * (cos . (upper_bound A))))

proof end;

theorem :: INTEGRA8:91

for A being closed-interval Subset of REAL st A = [.0,(PI / 2).] holds
integral ((sin (#) cos),A) = 1 / 2

proof end;

theorem :: INTEGRA8:92

for A being closed-interval Subset of REAL st A = [.0,PI.] holds
integral ((sin (#) cos),A) = 0

proof end;

theorem :: INTEGRA8:93

for A being closed-interval Subset of REAL st A = [.0,(PI * (3 / 2)).] holds
integral ((sin (#) cos),A) = 1 / 2

proof end;

theorem :: INTEGRA8:94

for A being closed-interval Subset of REAL st A = [.0,(PI * 2).] holds
integral ((sin (#) cos),A) = 0

proof end;

theorem :: INTEGRA8:95

for A being closed-interval Subset of REAL
for n being Element of NAT st A = [.((2 * n) * PI),(((2 * n) + 1) * PI).] holds
integral ((sin (#) cos),A) = 0

proof end;

theorem :: INTEGRA8:96

for A being closed-interval Subset of REAL
for x being Real
for n being Element of NAT st A = [.(x + ((2 * n) * PI)),(x + (((2 * n) + 1) * PI)).] holds
integral ((sin (#) cos),A) = 0

proof end;

Lm20: for A being closed-interval Subset of REAL holds
( cos (#) cos is_integrable_on A & (cos (#) cos) | A is bounded )

proof end;

theorem :: INTEGRA8:97

for A being closed-interval Subset of REAL holds integral ((sin (#) sin),A) = (((cos . (lower_bound A)) * (sin . (lower_bound A))) - ((cos . (upper_bound A)) * (sin . (upper_bound A)))) + (integral ((cos (#) cos),A))

proof end;

theorem :: INTEGRA8:98

for A being closed-interval Subset of REAL holds integral ((sinh (#) sinh),A) = (((cosh . (upper_bound A)) * (sinh . (upper_bound A))) - ((cosh . (lower_bound A)) * (sinh . (lower_bound A)))) - (integral ((cosh (#) cosh),A))

proof end;

theorem :: INTEGRA8:99

for A being closed-interval Subset of REAL holds integral ((sinh (#) cosh),A) = (1 / 2) * (((cosh . (upper_bound A)) * (cosh . (upper_bound A))) - ((cosh . (lower_bound A)) * (cosh . (lower_bound A))))

proof end;

theorem :: INTEGRA8:100

for A being closed-interval Subset of REAL holds integral ((exp_R (#) exp_R),A) = (1 / 2) * (((exp_R . (upper_bound A)) ^2) - ((exp_R . (lower_bound A)) ^2))

proof end;

Lm21: for A being closed-interval Subset of REAL holds
( exp_R (#) (sin + cos) is_integrable_on A & (exp_R (#) (sin + cos)) | A is bounded )

proof end;

Lm22: for A being closed-interval Subset of REAL holds
( exp_R (#) (cos - sin) is_integrable_on A & (exp_R (#) (cos - sin)) | A is bounded )

proof end;

theorem :: INTEGRA8:101

for A being closed-interval Subset of REAL holds integral ((exp_R (#) (sin + cos)),A) = ((exp_R (#) sin) . (upper_bound A)) - ((exp_R (#) sin) . (lower_bound A))

proof end;

theorem :: INTEGRA8:102

for A being closed-interval Subset of REAL holds integral ((exp_R (#) (cos - sin)),A) = ((exp_R (#) cos) . (upper_bound A)) - ((exp_R (#) cos) . (lower_bound A))

proof end;