let n be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL st A = [.(- (PI / 2)),(PI / 2).] holds
integral ((((#Z n) * cos) (#) sin),A) = 0
let A be non empty closed_interval Subset of REAL; :: thesis: ( A = [.(- (PI / 2)),(PI / 2).] implies integral ((((#Z n) * cos) (#) sin),A) = 0 )
assume A = [.(- (PI / 2)),(PI / 2).] ; :: thesis: integral ((((#Z n) * cos) (#) sin),A) = 0
then ( upper_bound A = PI / 2 & lower_bound A = - (PI / 2) ) by INTEGRA8:37;
then integral ((((#Z n) * cos) (#) sin),A) = (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (PI / 2)) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (- (PI / 2))) by Th22
.= ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (PI / 2))) - (((- (1 / (n + 1))) (#) ((#Z (n + 1)) * cos)) . (- (PI / 2))) by VALUED_1:6
.= ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (PI / 2))) - ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . (- (PI / 2)))) by VALUED_1:6
.= ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . pd))) - ((- (1 / (n + 1))) * (((#Z (n + 1)) * cos) . mpd)) by FUNCT_1:13, SIN_COS:24
.= ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . pd))) - ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (- (PI / 2))))) by FUNCT_1:13, SIN_COS:24
.= ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (PI / 2)))) - ((- (1 / (n + 1))) * ((#Z (n + 1)) . (cos . (PI / 2)))) by SIN_COS:30
.= 0 ;
hence integral ((((#Z n) * cos) (#) sin),A) = 0 ; :: thesis: verum