let A be non empty closed_interval Subset of REAL; :: thesis: ( A = [.(- (2 * PI)),(2 * PI).] implies sin is_orthogonal_with cos ,A )
assume A = [.(- (2 * PI)),(2 * PI).] ; :: thesis: sin is_orthogonal_with cos ,A
then A1: ( upper_bound A = 2 * PI & lower_bound A = - (2 * PI) ) by INTEGRA8:37;
|||(sin,cos,A)||| = (1 / 2) * (((cos . (lower_bound A)) * (cos . (lower_bound A))) - ((cos . (upper_bound A)) * (cos . (upper_bound A)))) by INTEGRA8:90
.= (1 / 2) * (((cos . (2 * PI)) * (cos . (- (2 * PI)))) - ((cos . (2 * PI)) * (cos . (2 * PI)))) by A1, SIN_COS:30
.= (1 / 2) * (((cos . (2 * PI)) * (cos . (2 * PI))) - ((cos . (2 * PI)) * (cos . (2 * PI)))) by SIN_COS:30 ;
hence sin is_orthogonal_with cos ,A ; :: thesis: verum