let a be Real; :: thesis: for A being non empty closed_interval Subset of REAL
for f, f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . (a * x)) - ((cos . (a * x)) ^2)) / ((cos . (a * x)) ^2) ) & Z c= dom (((1 / a) (#) (sec * f1)) - (id Z)) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((1 / a) (#) (sec * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (sec * f1)) - (id Z)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; :: thesis: for f, f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . (a * x)) - ((cos . (a * x)) ^2)) / ((cos . (a * x)) ^2) ) & Z c= dom (((1 / a) (#) (sec * f1)) - (id Z)) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((1 / a) (#) (sec * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (sec * f1)) - (id Z)) . (lower_bound A))
let f, f1 be PartFunc of REAL,REAL; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . (a * x)) - ((cos . (a * x)) ^2)) / ((cos . (a * x)) ^2) ) & Z c= dom (((1 / a) (#) (sec * f1)) - (id Z)) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((((1 / a) (#) (sec * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (sec * f1)) - (id Z)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . (a * x)) - ((cos . (a * x)) ^2)) / ((cos . (a * x)) ^2) ) & Z c= dom (((1 / a) (#) (sec * f1)) - (id Z)) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f & f | A is continuous implies integral (f,A) = ((((1 / a) (#) (sec * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (sec * f1)) - (id Z)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . (a * x)) - ((cos . (a * x)) ^2)) / ((cos . (a * x)) ^2) ) & Z c= dom (((1 / a) (#) (sec * f1)) - (id Z)) & ( for x being Real st x in Z holds
( f1 . x = a * x & a <> 0 ) ) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((((1 / a) (#) (sec * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (sec * f1)) - (id Z)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: ((1 / a) (#) (sec * f1)) - (id Z) is_differentiable_on Z by A1, FDIFF_9:26;
A4: for x being Element of REAL st x in dom ((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) holds
((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) implies ((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) . x = f . x )
assume x in dom ((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) ; :: thesis: ((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) . x = f . x
then A5: x in Z by A3, FDIFF_1:def 7;
then ((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) . x = ((sin . (a * x)) - ((cos . (a * x)) ^2)) / ((cos . (a * x)) ^2) by A1, FDIFF_9:26
.= f . x by A1, A5 ;
hence ((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((((1 / a) (#) (sec * f1)) - (id Z)) `| Z) = dom f by A1, A3, FDIFF_1:def 7;
then (((1 / a) (#) (sec * f1)) - (id Z)) `| Z = f by A4, PARTFUN1:5;
hence integral (f,A) = ((((1 / a) (#) (sec * f1)) - (id Z)) . (upper_bound A)) - ((((1 / a) (#) (sec * f1)) - (id Z)) . (lower_bound A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum