let h, x 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
(fD (f,h)) . x = (sin (x + h)) - (sin x)
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 (fD (f,h)) . x = (sin (x + h)) - (sin x) )
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: (fD (f,h)) . x = (sin (x + h)) - (sin x)
(fD (f,h)) . x = (f . (x + h)) - (f . x) by DIFF_1:3
.= ((tan (#) cos) . (x + h)) - (f . x) by A1
.= ((tan (#) cos) . (x + h)) - ((tan (#) cos) . x) by A1
.= ((tan . (x + h)) * (cos . (x + h))) - ((tan (#) cos) . x) by VALUED_1:5
.= ((tan . (x + h)) * (cos . (x + h))) - ((tan . x) * (cos . x)) by VALUED_1:5
.= (((sin . (x + h)) * ((cos . (x + h)) ")) * (cos . (x + h))) - ((tan . x) * (cos . x)) by A2, RFUNCT_1:def 1
.= (((sin (x + h)) / (cos (x + h))) * (cos (x + h))) - (((sin x) / (cos x)) * (cos x)) by A2, RFUNCT_1:def 1
.= ((sin (x + h)) / ((cos (x + h)) / (cos (x + h)))) - (((sin x) / (cos x)) * (cos x)) by XCMPLX_1:82
.= ((sin (x + h)) / ((cos (x + h)) * (1 / (cos (x + h))))) - ((sin x) / ((cos x) / (cos x))) by XCMPLX_1:82
.= ((sin (x + h)) / 1) - ((sin x) / ((cos x) * (1 / (cos x)))) by A2, FDIFF_8:1, XCMPLX_1:106
.= ((sin (x + h)) / 1) - ((sin x) / 1) by A2, FDIFF_8:1, XCMPLX_1:106
.= (sin (x + h)) - (sin x) ;
hence (fD (f,h)) . x = (sin (x + h)) - (sin x) ; :: thesis: verum