let x, h be Real; :: thesis: for f being Function of REAL ,REAL st ( for x being Real holds f . x = (tan (#) sin ) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan holds
(cD f,h) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2)))
let f be Function of REAL ,REAL ; :: thesis: ( ( for x being Real holds f . x = (tan (#) sin ) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan implies (cD f,h) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) )
assume that
A1: for x being Real holds f . x = (tan (#) sin ) . x and
A2: ( x + (h / 2) in dom tan & x - (h / 2) in dom tan ) ; :: thesis: (cD f,h) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2)))
(cD f,h) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5
.= ((tan (#) sin ) . (x + (h / 2))) - (f . (x - (h / 2))) by A1
.= ((tan (#) sin ) . (x + (h / 2))) - ((tan (#) sin ) . (x - (h / 2))) by A1
.= ((tan . (x + (h / 2))) * (sin . (x + (h / 2)))) - ((tan (#) sin ) . (x - (h / 2))) by VALUED_1:5
.= ((tan . (x + (h / 2))) * (sin . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (sin . (x - (h / 2)))) by VALUED_1:5
.= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) " )) * (sin . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (sin . (x - (h / 2)))) by A2, RFUNCT_1:def 4
.= (((sin (x + (h / 2))) / (cos (x + (h / 2)))) * (sin (x + (h / 2)))) - (((sin (x - (h / 2))) / (cos (x - (h / 2)))) * (sin (x - (h / 2)))) by A2, RFUNCT_1:def 4
.= ((sin (x + (h / 2))) / ((cos (x + (h / 2))) / (sin (x + (h / 2))))) - (((sin (x - (h / 2))) / (cos (x - (h / 2)))) * (sin (x - (h / 2)))) by XCMPLX_1:83
.= ((sin (x + (h / 2))) / ((cos (x + (h / 2))) / (sin (x + (h / 2))))) - ((sin (x - (h / 2))) / ((cos (x - (h / 2))) / (sin (x - (h / 2))))) by XCMPLX_1:83
.= (((sin (x + (h / 2))) * (sin (x + (h / 2)))) / (cos (x + (h / 2)))) - ((sin (x - (h / 2))) / ((cos (x - (h / 2))) / (sin (x - (h / 2))))) by XCMPLX_1:78
.= (((sin (x + (h / 2))) * (sin (x + (h / 2)))) / (cos (x + (h / 2)))) - (((sin (x - (h / 2))) * (sin (x - (h / 2)))) / (cos (x - (h / 2)))) by XCMPLX_1:78
.= ((1 - ((cos (x + (h / 2))) * (cos (x + (h / 2))))) / (cos (x + (h / 2)))) - (((sin (x - (h / 2))) * (sin (x - (h / 2)))) / (cos (x - (h / 2)))) by SIN_COS4:6
.= ((1 / (cos (x + (h / 2)))) - (((cos (x + (h / 2))) * (cos (x + (h / 2)))) / (cos (x + (h / 2))))) - ((1 - ((cos (x - (h / 2))) * (cos (x - (h / 2))))) / (cos (x - (h / 2)))) by SIN_COS4:6
.= ((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - ((1 / (cos (x - (h / 2)))) - (((cos (x - (h / 2))) * (cos (x - (h / 2)))) / (cos (x - (h / 2))))) by A2, FDIFF_8:1, XCMPLX_1:90
.= ((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - ((1 / (cos (x - (h / 2)))) - (cos (x - (h / 2)))) by A2, FDIFF_8:1, XCMPLX_1:90
.= (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) ;
hence (cD f,h) . x = (((1 / (cos (x + (h / 2)))) - (cos (x + (h / 2)))) - (1 / (cos (x - (h / 2))))) + (cos (x - (h / 2))) ; :: thesis: verum