let n be Element of NAT ; for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; for f being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
let f be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous implies integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A)) )
assume A1:
( A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (cos . x) / ((sin . x) #Z (n + 1)) & sin . x <> 0 ) ) & Z c= dom ((#Z n) * (sin ^)) & Z = dom f & f | A is continuous )
; integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
then A2:
Z c= dom ((- (1 / n)) (#) ((#Z n) * (sin ^)))
by VALUED_1:def 5;
A3:
( f is_integrable_on A & f | A is bounded )
by A1, INTEGRA5:10, INTEGRA5:11;
A4:
for x being Real st x in Z holds
sin . x <> 0
by A1;
then A5:
(- (1 / n)) (#) ((#Z n) * (sin ^)) is_differentiable_on Z
by A1, A2, FDIFF_7:30;
A6:
for x being Element of REAL st x in dom (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) holds
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = f . x
dom (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) = dom f
by A1, A5, FDIFF_1:def 7;
then
((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z = f
by A6, PARTFUN1:5;
hence
integral (f,A) = (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (upper_bound A)) - (((- (1 / n)) (#) ((#Z n) * (sin ^))) . (lower_bound A))
by A1, A3, A5, INTEGRA5:13; verum