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 = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ) & Z c= dom (tan * cot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (tan * cot)) . (upper_bound A)) - ((- (tan * cot)) . (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 = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ) & Z c= dom (tan * cot) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (tan * cot)) . (upper_bound A)) - ((- (tan * cot)) . (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 = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ) & Z c= dom (tan * cot) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (tan * cot)) . (upper_bound A)) - ((- (tan * cot)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f . x = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ) & Z c= dom (tan * cot) & Z = dom f & f | A is continuous ) ; :: thesis: integral (f,A) = ((- (tan * cot)) . (upper_bound A)) - ((- (tan * cot)) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: Z c= dom (- (tan * cot)) by A1, VALUED_1:8;
A4: tan * cot is_differentiable_on Z by A1, FDIFF_10:1;
then A5: (- 1) (#) (tan * cot) is_differentiable_on Z by A3, FDIFF_1:20;
A6: for x being Real st x in Z holds
((- (tan * cot)) `| Z) . x = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((- (tan * cot)) `| Z) . x = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) )
assume A7: x in Z ; :: thesis: ((- (tan * cot)) `| Z) . x = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2))
((- (tan * cot)) `| Z) . x = ((- 1) (#) ((tan * cot) `| Z)) . x by A4, FDIFF_2:19
.= (- 1) * (((tan * cot) `| Z) . x) by VALUED_1:6
.= (- 1) * ((1 / ((cos . (cot . x)) ^2)) * (- (1 / ((sin . x) ^2)))) by A1, A7, FDIFF_10:1
.= (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ;
hence ((- (tan * cot)) `| Z) . x = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) ; :: thesis: verum

end;

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

proof

let x be Element of REAL ; :: thesis: ( x in dom ((- (tan * cot)) `| Z) implies ((- (tan * cot)) `| Z) . x = f . x )
assume x in dom ((- (tan * cot)) `| Z) ; :: thesis: ((- (tan * cot)) `| Z) . x = f . x
then A9: x in Z by A5, FDIFF_1:def 7;
then ((- (tan * cot)) `| Z) . x = (1 / ((cos . (cot . x)) ^2)) * (1 / ((sin . x) ^2)) by A6
.= f . x by A1, A9 ;
hence ((- (tan * cot)) `| Z) . x = f . x ; :: thesis: verum

end;

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