let x be Real; :: thesis: for n being Element of NAT
for A being closed-interval Subset of REAL st A = [.(x - ((2 * n) * PI )),(x + ((2 * n) * PI )).] holds
sin is_orthogonal_with cos ,A
let n be Element of NAT ; :: thesis: for A being closed-interval Subset of REAL st A = [.(x - ((2 * n) * PI )),(x + ((2 * n) * PI )).] holds
sin is_orthogonal_with cos ,A
let A be closed-interval Subset of REAL ; :: thesis: ( A = [.(x - ((2 * n) * PI )),(x + ((2 * n) * PI )).] implies sin is_orthogonal_with cos ,A )
assume A = [.(x - ((2 * n) * PI )),(x + ((2 * n) * PI )).] ; :: thesis: sin is_orthogonal_with cos ,A
then A1: ( upper_bound A = x + ((2 * n) * PI ) & lower_bound A = x - ((2 * n) * 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 * n) * PI ) - x)) * (cos . (- (((2 * n) * PI ) - x)))) - ((cos . (x + ((2 * n) * PI ))) * (cos . (x + ((2 * n) * PI ))))) by A1, SIN_COS:33
.= (1 / 2) * (((cos . ((- x) + ((2 * n) * PI ))) * (cos . ((- x) + ((2 * n) * PI )))) - ((cos . (x + ((2 * n) * PI ))) * (cos . (x + ((2 * n) * PI ))))) by SIN_COS:33
.= (1 / 2) * (((cos (- x)) * (cos ((- x) + ((2 * n) * PI )))) - ((cos . (x + ((2 * n) * PI ))) * (cos . (x + ((2 * n) * PI ))))) by INTEGRA8:3
.= (1 / 2) * (((cos (- x)) * (cos (- x))) - ((cos . (x + ((2 * n) * PI ))) * (cos . (x + ((2 * n) * PI ))))) by INTEGRA8:3
.= (1 / 2) * (((cos x) * (cos (- x))) - ((cos . (x + ((2 * n) * PI ))) * (cos . (x + ((2 * n) * PI ))))) by SIN_COS:34
.= (1 / 2) * (((cos x) * (cos x)) - ((cos (x + ((2 * n) * PI ))) * (cos . (x + ((2 * n) * PI ))))) by SIN_COS:34
.= (1 / 2) * (((cos x) * (cos x)) - ((cos x) * (cos (x + ((2 * n) * PI ))))) by INTEGRA8:3
.= (1 / 2) * (((cos x) * (cos x)) - ((cos x) * (cos x))) by INTEGRA8:3 ;
hence sin is_orthogonal_with cos ,A by Def2; :: thesis: verum