let n be Element of NAT ; :: thesis: for A being 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 = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^ )) & Z = dom f & f | A is continuous holds
integral f,A = (((1 / n) (#) ((#Z n) * (cos ^ ))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^ ))) . (lower_bound A))
let A be 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 = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^ )) & Z = dom f & f | A is continuous holds
integral f,A = (((1 / n) (#) ((#Z n) * (cos ^ ))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^ ))) . (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 = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^ )) & Z = dom f & f | A is continuous holds
integral f,A = (((1 / n) (#) ((#Z n) * (cos ^ ))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^ ))) . (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 = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^ )) & Z = dom f & f | A is continuous implies integral f,A = (((1 / n) (#) ((#Z n) * (cos ^ ))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^ ))) . (lower_bound A)) )
assume A1: ( A c= Z & n > 0 & ( for x being Real st x in Z holds
( f . x = (sin . x) / ((cos . x) #Z (n + 1)) & cos . x <> 0 ) ) & Z c= dom ((#Z n) * (cos ^ )) & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = (((1 / n) (#) ((#Z n) * (cos ^ ))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^ ))) . (lower_bound A))
then AA: Z c= dom ((1 / n) (#) ((#Z n) * (cos ^ ))) by VALUED_1:def 5;
A2: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A3: for x being Real st x in Z holds
cos . x <> 0 by A1;
then A4: (1 / n) (#) ((#Z n) * (cos ^ )) is_differentiable_on Z by A1, AA, FDIFF_7:31;
A5: for x being Real st x in dom (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) holds
(((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) implies (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = f . x )
assume x in dom (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) ; :: thesis: (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = f . x
then A6: x in Z by A4, FDIFF_1:def 8;
then (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) by A1, AA, A3, FDIFF_7:31
.= f . x by A1, A6 ;
hence (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) = dom f by A1, A4, FDIFF_1:def 8;
then ((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z = f by A5, PARTFUN1:34;
hence integral f,A = (((1 / n) (#) ((#Z n) * (cos ^ ))) . (upper_bound A)) - (((1 / n) (#) ((#Z n) * (cos ^ ))) . (lower_bound A)) by A1, A2, A4, INTEGRA5:13; :: thesis: verum