let n be Element of NAT ; :: thesis: for A being closed-interval Subset of REAL st A = [.0 ,PI .] holds
integral (((#Z n) * sin ) (#) cos ),A = 0
let A be closed-interval Subset of REAL ; :: thesis: ( A = [.0 ,PI .] implies integral (((#Z n) * sin ) (#) cos ),A = 0 )
assume A = [.0 ,PI .] ; :: thesis: integral (((#Z n) * sin ) (#) cos ),A = 0
then ( upper_bound A = PI & lower_bound A = 0 ) by INTEGRA8:37;
then integral (((#Z n) * sin ) (#) cos ),A = (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . PI ) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . 0 ) by Th19
.= ((1 / (n + 1)) * (((#Z (n + 1)) * sin ) . PI )) - (((1 / (n + 1)) (#) ((#Z (n + 1)) * sin )) . 0 ) by VALUED_1:6
.= ((1 / (n + 1)) * (((#Z (n + 1)) * sin ) . PI )) - ((1 / (n + 1)) * (((#Z (n + 1)) * sin ) . 0 )) by VALUED_1:6
.= ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . PI ))) - ((1 / (n + 1)) * (((#Z (n + 1)) * sin ) . 0 )) by FUNCT_1:23, SIN_COS:27
.= ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . PI ))) - ((1 / (n + 1)) * ((#Z (n + 1)) . (sin . 0 ))) by FUNCT_1:23, SIN_COS:27
.= 0 by SIN_COS:33, SIN_COS:81 ;
hence integral (((#Z n) * sin ) (#) cos ),A = 0 ; :: thesis: verum