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

proof

let x be Element of REAL ; :: thesis: ( x in dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) implies (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = (((#Z n) * sin) (#) cos) . x )
assume x in dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) ; :: thesis: (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = (((#Z n) * sin) (#) cos) . x
(((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = ((sin . x) #Z n) * (cos . x) by Th3
.= ((#Z n) . (sin . x)) * (cos . x) by TAYLOR_1:def 1
.= (((#Z n) * sin) . x) * (cos . x) by FUNCT_1:13, SIN_COS:24
.= (((#Z n) * sin) (#) cos) . x by A1, VALUED_1:def 4 ;
hence (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) . x = (((#Z n) * sin) (#) cos) . x ; :: thesis: verum

end;

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

proof

let x0 be Real; :: thesis: (#Z n) * sin is_differentiable_in x0
sin is_differentiable_in x0 by SIN_COS:64;
hence (#Z n) * sin is_differentiable_in x0 by TAYLOR_1:3; :: thesis: verum

end;

then ( dom ((#Z n) * sin) = REAL & ( for x0 being Real st x0 in REAL holds
(#Z n) * sin is_differentiable_in x0 ) ) by FUNCT_2:def 1;
then (#Z n) * sin is_differentiable_on REAL by A1, FDIFF_1:9;
then A3: (((#Z n) * sin) (#) cos) | REAL is continuous by A1, FDIFF_1:21, FDIFF_1:25, SIN_COS:67;
then (((#Z n) * sin) (#) cos) | A is continuous by FCONT_1:16;
then A4: ((#Z n) * sin) (#) cos is_integrable_on A by A1, INTEGRA5:11;
(1 / (n + 1)) (#) ((#Z (n + 1)) * sin) is_differentiable_on REAL by Th3;
then dom (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL) = dom (((#Z n) * sin) (#) cos) by A1, FDIFF_1:def 7;
then A5: ((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) `| REAL = ((#Z n) * sin) (#) cos by A2, PARTFUN1:5;
(((#Z n) * sin) (#) cos) | A is bounded by A1, A3, INTEGRA5:10;
hence integral ((((#Z n) * sin) (#) cos),A) = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (upper_bound A)) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin)) . (lower_bound A)) by A4, A5, Th3, INTEGRA5:13; :: thesis: verum