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