let A be 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 & ( for x being Real st x in Z holds
f . x = ((1 / ((sin . x) ^2)) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2)) ) & Z c= dom (cot (#) sec) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (cot (#) sec)) . (upper_bound A)) - ((- (cot (#) sec)) . (lower_bound A))
let f be 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 / ((sin . x) ^2)) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2)) ) & Z c= dom (cot (#) sec) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (cot (#) sec)) . (upper_bound A)) - ((- (cot (#) sec)) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & ( for x being Real st x in Z holds
f . x = ((1 / ((sin . x) ^2)) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2)) ) & Z c= dom (cot (#) sec) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (cot (#) sec)) . (upper_bound A)) - ((- (cot (#) sec)) . (lower_bound A)) )
assume A1:
( A c= Z & ( for x being Real st x in Z holds
f . x = ((1 / ((sin . x) ^2)) / (cos . x)) - (((cot . x) * (sin . x)) / ((cos . x) ^2)) ) & Z c= dom (cot (#) sec) & Z = dom f & f | A is continuous )
; integral (f,A) = ((- (cot (#) sec)) . (upper_bound A)) - ((- (cot (#) sec)) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
- (cot (#) sec) is_differentiable_on Z
by A1, Th12;
A4:
for x being Element of REAL st x in dom ((- (cot (#) sec)) `| Z) holds
((- (cot (#) sec)) `| Z) . x = f . x
dom ((- (cot (#) sec)) `| Z) = dom f
by A1, A3, FDIFF_1:def 7;
then
(- (cot (#) sec)) `| Z = f
by A4, PARTFUN1:5;
hence
integral (f,A) = ((- (cot (#) sec)) . (upper_bound A)) - ((- (cot (#) sec)) . (lower_bound A))
by A1, A2, A3, INTEGRA5:13; verum