let x, h be Real; :: thesis: for f being Function of REAL ,REAL st ( for x being Real holds f . x = (cot (#) cos ) . x ) & x in dom cot & x - h in dom cot holds
(bD f,h) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h))
let f be Function of REAL ,REAL ; :: thesis: ( ( for x being Real holds f . x = (cot (#) cos ) . x ) & x in dom cot & x - h in dom cot implies (bD f,h) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) )
assume that
A1: for x being Real holds f . x = (cot (#) cos ) . x and
A2: ( x in dom cot & x - h in dom cot ) ; :: thesis: (bD f,h) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h))
(bD f,h) . x = (f . x) - (f . (x - h)) by DIFF_1:4
.= ((cot (#) cos ) . x) - (f . (x - h)) by A1
.= ((cot (#) cos ) . x) - ((cot (#) cos ) . (x - h)) by A1
.= ((cot . x) * (cos . x)) - ((cot (#) cos ) . (x - h)) by VALUED_1:5
.= ((cot . x) * (cos . x)) - ((cot . (x - h)) * (cos . (x - h))) by VALUED_1:5
.= (((cos . x) * ((sin . x) " )) * (cos . x)) - ((cot . (x - h)) * (cos . (x - h))) by A2, RFUNCT_1:def 4
.= (((cos x) / (sin x)) * (cos x)) - (((cos (x - h)) / (sin (x - h))) * (cos (x - h))) by A2, RFUNCT_1:def 4
.= ((cos x) / ((sin x) / (cos x))) - (((cos (x - h)) / (sin (x - h))) * (cos (x - h))) by XCMPLX_1:83
.= ((cos x) / ((sin x) / (cos x))) - ((cos (x - h)) / ((sin (x - h)) / (cos (x - h)))) by XCMPLX_1:83
.= (((cos x) * (cos x)) / (sin x)) - ((cos (x - h)) / ((sin (x - h)) / (cos (x - h)))) by XCMPLX_1:78
.= (((cos x) * (cos x)) / (sin x)) - (((cos (x - h)) * (cos (x - h))) / (sin (x - h))) by XCMPLX_1:78
.= ((1 - ((sin x) * (sin x))) / (sin x)) - (((cos (x - h)) * (cos (x - h))) / (sin (x - h))) by SIN_COS4:7
.= ((1 / (sin x)) - (((sin x) * (sin x)) / (sin x))) - ((1 - ((sin (x - h)) * (sin (x - h)))) / (sin (x - h))) by SIN_COS4:7
.= ((1 / (sin x)) - (sin x)) - ((1 / (sin (x - h))) - (((sin (x - h)) * (sin (x - h))) / (sin (x - h)))) by A2, FDIFF_8:2, XCMPLX_1:90
.= ((1 / (sin x)) - (sin x)) - ((1 / (sin (x - h))) - (sin (x - h))) by A2, FDIFF_8:2, XCMPLX_1:90
.= (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) ;
hence (bD f,h) . x = (((1 / (sin x)) - (sin x)) - (1 / (sin (x - h)))) + (sin (x - h)) ; :: thesis: verum