let n be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL holds integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (upper_bound A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; :: thesis: integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (upper_bound A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (lower_bound A))
A1: [#] REAL = dom (((#Z n) * cos) (#) sin) by FUNCT_2:def 1;
A2: for x being Element of REAL st x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) holds
(((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (((#Z n) * cos) (#) sin) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) implies (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (((#Z n) * cos) (#) sin) . x )
assume x in dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) ; :: thesis: (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (((#Z n) * cos) (#) sin) . x
(((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = ((cos . x) #Z n) * (sin . x) by Th4
.= ((#Z n) . (cos . x)) * (sin . x) by TAYLOR_1:def 1
.= (((#Z n) * cos) . x) * (sin . x) by FUNCT_1:13, SIN_COS:24
.= (((#Z n) * cos) (#) sin) . x by A1, VALUED_1:def 4 ;
hence (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) . x = (((#Z n) * cos) (#) sin) . x ; :: thesis: verum

end;

for x0 being Real holds (#Z n) * cos is_differentiable_in x0

proof

let x0 be Real; :: thesis: (#Z n) * cos is_differentiable_in x0
cos is_differentiable_in x0 by SIN_COS:63;
hence (#Z n) * cos is_differentiable_in x0 by TAYLOR_1:3; :: thesis: verum

end;

then ( dom ((#Z n) * cos) = REAL & ( for x0 being Real st x0 in REAL holds
(#Z n) * cos is_differentiable_in x0 ) ) by FUNCT_2:def 1;
then (#Z n) * cos is_differentiable_on REAL by A1, FDIFF_1:9;
then A3: (((#Z n) * cos) (#) sin) | REAL is continuous by A1, FDIFF_1:21, FDIFF_1:25, SIN_COS:68;
then (((#Z n) * cos) (#) sin) | A is continuous by FCONT_1:16;
then A4: ((#Z n) * cos) (#) sin is_integrable_on A by A1, INTEGRA5:11;
(- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos) is_differentiable_on REAL by Th4;
then dom (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL) = dom (((#Z n) * cos) (#) sin) by A1, FDIFF_1:def 7;
then A5: ((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) `| REAL = ((#Z n) * cos) (#) sin by A2, PARTFUN1:5;
(((#Z n) * cos) (#) sin) | A is bounded by A1, A3, INTEGRA5:10;
hence integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (upper_bound A)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (lower_bound A)) by A4, A5, Th4, INTEGRA5:13; :: thesis: verum