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 & ( for x being Real st x in Z holds
f . x = ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) ) & Z c= dom (sec * cos) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (sec * cos)) . (upper_bound A)) - ((- (sec * cos)) . (lower_bound A))
let f 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 . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) ) & Z c= dom (sec * cos) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (sec * cos)) . (upper_bound A)) - ((- (sec * cos)) . (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 . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) ) & Z c= dom (sec * cos) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (sec * cos)) . (upper_bound A)) - ((- (sec * cos)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f . x = ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) ) & Z c= dom (sec * cos) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((- (sec * cos)) . (upper_bound A)) - ((- (sec * cos)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: Z c= dom (- (sec * cos)) by A1, VALUED_1:8;
A4: sec * cos is_differentiable_on Z by A1, FDIFF_9:35;
then A5: (- 1) (#) (sec * cos) is_differentiable_on Z by A3, FDIFF_1:20;
A6: for x being Real st x in Z holds
((- (sec * cos)) `| Z) . x = ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2)

proof

let x be Real; :: thesis: ( x in Z implies ((- (sec * cos)) `| Z) . x = ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) )
assume A7: x in Z ; :: thesis: ((- (sec * cos)) `| Z) . x = ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2)
((- (sec * cos)) `| Z) . x = ((- 1) (#) ((sec * cos) `| Z)) . x by A4, FDIFF_2:19
.= (- 1) * (((sec * cos) `| Z) . x) by VALUED_1:6
.= (- 1) * (- (((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2))) by A1, A7, FDIFF_9:35
.= ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) ;
hence ((- (sec * cos)) `| Z) . x = ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) ; :: thesis: verum

end;

A8: for x being Element of REAL st x in dom ((- (sec * cos)) `| Z) holds
((- (sec * cos)) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((- (sec * cos)) `| Z) implies ((- (sec * cos)) `| Z) . x = f . x )
assume x in dom ((- (sec * cos)) `| Z) ; :: thesis: ((- (sec * cos)) `| Z) . x = f . x
then A9: x in Z by A5, FDIFF_1:def 7;
then ((- (sec * cos)) `| Z) . x = ((sin . x) * (sin . (cos . x))) / ((cos . (cos . x)) ^2) by A6
.= f . x by A1, A9 ;
hence ((- (sec * cos)) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((- (sec * cos)) `| Z) = dom f by A1, A5, FDIFF_1:def 7;
then (- (sec * cos)) `| Z = f by A8, PARTFUN1:5;
hence integral (f,A) = ((- (sec * cos)) . (upper_bound A)) - ((- (sec * cos)) . (lower_bound A)) by A1, A2, A5, INTEGRA5:13; :: thesis: verum