let x, h be Real; :: thesis: for f being Function of REAL ,REAL st ( for x being Real holds f . x = (tan (#) cos ) . x ) & x in dom tan & x - h in dom tan holds
(bD f,h) . x = (sin x) - (sin (x - h))
let f be Function of REAL ,REAL ; :: thesis: ( ( for x being Real holds f . x = (tan (#) cos ) . x ) & x in dom tan & x - h in dom tan implies (bD f,h) . x = (sin x) - (sin (x - h)) )
assume that
A1: for x being Real holds f . x = (tan (#) cos ) . x and
A2: ( x in dom tan & x - h in dom tan ) ; :: thesis: (bD f,h) . x = (sin x) - (sin (x - h))
(bD f,h) . x = (f . x) - (f . (x - h)) by DIFF_1:4
.= ((tan (#) cos ) . x) - (f . (x - h)) by A1
.= ((tan (#) cos ) . x) - ((tan (#) cos ) . (x - h)) by A1
.= ((tan . x) * (cos . x)) - ((tan (#) cos ) . (x - h)) by VALUED_1:5
.= ((tan . x) * (cos . x)) - ((tan . (x - h)) * (cos . (x - h))) by VALUED_1:5
.= (((sin . x) * ((cos . x) " )) * (cos . x)) - ((tan . (x - h)) * (cos . (x - h))) by A2, RFUNCT_1:def 4
.= (((sin x) / (cos x)) * (cos x)) - (((sin (x - h)) / (cos (x - h))) * (cos (x - h))) by A2, RFUNCT_1:def 4
.= ((sin x) / ((cos x) / (cos x))) - (((sin (x - h)) / (cos (x - h))) * (cos (x - h))) by XCMPLX_1:83
.= ((sin x) / ((cos x) * (1 / (cos x)))) - ((sin (x - h)) / ((cos (x - h)) / (cos (x - h)))) by XCMPLX_1:83
.= ((sin x) / 1) - ((sin (x - h)) / ((cos (x - h)) * (1 / (cos (x - h))))) by A2, FDIFF_8:1, XCMPLX_1:107
.= ((sin x) / 1) - ((sin (x - h)) / 1) by A2, FDIFF_8:1, XCMPLX_1:107
.= (sin x) - (sin (x - h)) ;
hence (bD f,h) . x = (sin x) - (sin (x - h)) ; :: thesis: verum