let A be closed-interval Subset of REAL ; :: thesis: for n being Element of NAT st A = [.((2 * n) * PI ),(((2 * n) + 1) * PI ).] holds
integral (- sin ),A = - 2
let n be Element of NAT ; :: thesis: ( A = [.((2 * n) * PI ),(((2 * n) + 1) * PI ).] implies integral (- sin ),A = - 2 )
assume A = [.((2 * n) * PI ),(((2 * n) + 1) * PI ).] ; :: thesis: integral (- sin ),A = - 2
then ( upper_bound A = ((2 * n) + 1) * PI & lower_bound A = (2 * n) * PI ) by Th37;
then integral (- sin ),A = (cos (0 + (((2 * n) + 1) * PI ))) - (cos (0 + ((2 * n) * PI ))) by Th46
.= (- (cos 0 )) - (cos (0 + ((2 * n) * PI ))) by Th4
.= (- (cos 0 )) - (cos 0 ) by Th3
.= (- (cos (0 + (2 * PI )))) - (cos 0 ) by SIN_COS:84
.= (- 1) - 1 by SIN_COS:82, SIN_COS:84 ;
hence integral (- sin ),A = - 2 ; :: thesis: verum