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 in dom tan & x - h in dom tan holds
(bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2))
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = (tan (#) tan) . x ) & x in dom tan & x - h in dom tan implies (bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) )
assume that
A1: for x being Real holds f . x = (tan (#) tan) . x and
A2: ( x in dom tan & x - h in dom tan ) ; :: thesis: (bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2))
A3: ( cos x <> 0 & cos (x - h) <> 0 ) by A2, FDIFF_8:1;
(bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4
.= ((tan (#) tan) . x) - (f . (x - h)) by A1
.= ((tan (#) tan) . x) - ((tan (#) tan) . (x - h)) by A1
.= ((tan . x) * (tan . x)) - ((tan (#) tan) . (x - h)) by VALUED_1:5
.= ((tan . x) * (tan . x)) - ((tan . (x - h)) * (tan . (x - h))) by VALUED_1:5
.= (((sin . x) * ((cos . x) ")) * (tan . x)) - ((tan . (x - h)) * (tan . (x - h))) by A2, RFUNCT_1:def 1
.= (((sin . x) * ((cos . x) ")) * ((sin . x) * ((cos . x) "))) - ((tan . (x - h)) * (tan . (x - h))) by A2, RFUNCT_1:def 1
.= (((sin . x) * ((cos . x) ")) * ((sin . x) * ((cos . x) "))) - (((sin . (x - h)) * ((cos . (x - h)) ")) * (tan . (x - h))) by A2, RFUNCT_1:def 1
.= ((tan x) ^2) - ((tan (x - h)) ^2) by A2, RFUNCT_1:def 1
.= ((tan x) - (tan (x - h))) * ((tan x) + (tan (x - h)))
.= ((sin (x - (x - h))) / ((cos x) * (cos (x - h)))) * ((tan x) + (tan (x - h))) by A3, SIN_COS4:20
.= ((sin h) / ((cos x) * (cos (x - h)))) * ((sin (x + (x - h))) / ((cos x) * (cos (x - h)))) by A3, SIN_COS4:19
.= ((sin h) * (sin ((2 * x) - h))) / (((cos x) * (cos (x - h))) ^2) by XCMPLX_1:76
.= (- ((1 / 2) * ((cos (h + ((2 * x) - h))) - (cos (h - ((2 * x) - h)))))) / (((cos x) * (cos (x - h))) ^2) by SIN_COS4:29
.= - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) ;
hence (bD (f,h)) . x = - (((1 / 2) * ((cos (2 * x)) - (cos (2 * (h - x))))) / (((cos x) * (cos (x - h))) ^2)) ; :: thesis: verum