let n be Element of NAT ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: 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

proof

let x be Element of REAL ; :: thesis: ( x in dom (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) implies (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = f . x )
assume x in dom (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) ; :: thesis: (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = f . x
then A7: x in Z by A5, FDIFF_1:def 7;
then (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1)) by A1, A2, A4, FDIFF_7:30
.= f . x by A1, A7 ;
hence (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = f . x ; :: thesis: verum

end;

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; :: thesis: verum