let h, x be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan holds
(cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2))
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = (tan (#) tan) . x ) & x + (h / 2) in dom tan & x - (h / 2) in dom tan implies (cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) )
assume that
A1: for x being Real holds f . x = (tan (#) tan) . x and
A2: ( x + (h / 2) in dom tan & x - (h / 2) in dom tan ) ; :: thesis: (cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2))
A3: ( cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 ) by A2, FDIFF_8:1;
(cD (f,h)) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5
.= ((tan (#) tan) . (x + (h / 2))) - (f . (x - (h / 2))) by A1
.= ((tan (#) tan) . (x + (h / 2))) - ((tan (#) tan) . (x - (h / 2))) by A1
.= ((tan . (x + (h / 2))) * (tan . (x + (h / 2)))) - ((tan (#) tan) . (x - (h / 2))) by VALUED_1:5
.= ((tan . (x + (h / 2))) * (tan . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) by VALUED_1:5
.= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * (tan . (x + (h / 2)))) - ((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) by A2, RFUNCT_1:def 1
.= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * ((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) "))) - ((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) by A2, RFUNCT_1:def 1
.= (((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * ((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) "))) - (((sin . (x - (h / 2))) * ((cos . (x - (h / 2))) ")) * (tan . (x - (h / 2)))) by A2, RFUNCT_1:def 1
.= ((tan (x + (h / 2))) ^2) - ((tan (x - (h / 2))) ^2) by A2, RFUNCT_1:def 1
.= ((tan (x + (h / 2))) - (tan (x - (h / 2)))) * ((tan (x + (h / 2))) + (tan (x - (h / 2))))
.= ((sin ((x + (h / 2)) - (x - (h / 2)))) / ((cos (x + (h / 2))) * (cos (x - (h / 2))))) * ((tan (x + (h / 2))) + (tan (x - (h / 2)))) by A3, SIN_COS4:20
.= ((sin h) / ((cos (x + (h / 2))) * (cos (x - (h / 2))))) * ((sin ((x + (h / 2)) + (x - (h / 2)))) / ((cos (x + (h / 2))) * (cos (x - (h / 2))))) by A3, SIN_COS4:19
.= ((sin h) * (sin (2 * x))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2) by XCMPLX_1:76
.= (- ((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x)))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2) by SIN_COS4:29
.= - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) ;
hence (cD (f,h)) . x = - (((1 / 2) * ((cos (h + (2 * x))) - (cos (h - (2 * x))))) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2)) ; :: thesis: verum