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 = (sin . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (cos * sin)) . (upper_bound A)) - ((- (cos * sin)) . (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 = (sin . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (cos * sin)) . (upper_bound A)) - ((- (cos * sin)) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & ( for x being Real st x in Z holds
f . x = (sin . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous implies integral (f,A) = ((- (cos * sin)) . (upper_bound A)) - ((- (cos * sin)) . (lower_bound A)) )
assume A1:
( A c= Z & ( for x being Real st x in Z holds
f . x = (sin . (sin . x)) * (cos . x) ) & Z = dom f & f | A is continuous )
; integral (f,A) = ((- (cos * sin)) . (upper_bound A)) - ((- (cos * sin)) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
dom sin = REAL
by SIN_COS:24;
( rng sin c= dom sin & dom sin = dom cos )
by SIN_COS:24;
then
dom (cos * sin) = REAL
by A3, RELAT_1:27;
then A4:
dom (- (cos * sin)) = REAL
by VALUED_1:8;
A5:
cos * sin is_differentiable_on Z
by FDIFF_10:9;
then A6:
(- 1) (#) (cos * sin) is_differentiable_on Z
by A4, FDIFF_1:20;
A7:
for x being Real st x in Z holds
((- (cos * sin)) `| Z) . x = (sin . (sin . x)) * (cos . x)
A9:
for x being Element of REAL st x in dom ((- (cos * sin)) `| Z) holds
((- (cos * sin)) `| Z) . x = f . x
dom ((- (cos * sin)) `| Z) = dom f
by A1, A6, FDIFF_1:def 7;
then
(- (cos * sin)) `| Z = f
by A9, PARTFUN1:5;
hence
integral (f,A) = ((- (cos * sin)) . (upper_bound A)) - ((- (cos * sin)) . (lower_bound A))
by A1, A2, A6, INTEGRA5:13; verum