let x, h be Real; :: thesis: for f being Function of REAL ,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2 ) ) & cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 holds
(cD f,h) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2 )
let f be Function of REAL ,REAL ; :: thesis: ( ( for x being Real holds f . x = 1 / ((cos x) ^2 ) ) & cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 implies (cD f,h) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2 ) )
assume that
A1: for x being Real holds f . x = 1 / ((cos x) ^2 ) and
A2: ( cos (x + (h / 2)) <> 0 & cos (x - (h / 2)) <> 0 ) ; :: thesis: (cD f,h) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2 )
(cD f,h) . x = (f . (x + (h / 2))) - (f . (x - (h / 2))) by DIFF_1:5
.= (1 / ((cos (x + (h / 2))) ^2 )) - (f . (x - (h / 2))) by A1
.= (1 / ((cos (x + (h / 2))) ^2 )) - (1 / ((cos (x - (h / 2))) ^2 )) by A1
.= ((1 * ((cos (x - (h / 2))) ^2 )) - (1 * ((cos (x + (h / 2))) ^2 ))) / (((cos (x + (h / 2))) ^2 ) * ((cos (x - (h / 2))) ^2 )) by A2, XCMPLX_1:131
.= (((cos (x - (h / 2))) ^2 ) - ((cos (x + (h / 2))) ^2 )) / (((cos (x + (h / 2))) * (cos (x - (h / 2)))) ^2 )
.= (((cos (x - (h / 2))) ^2 ) - ((cos (x + (h / 2))) ^2 )) / (((1 / 2) * ((cos ((x + (h / 2)) + (x - (h / 2)))) + (cos ((x + (h / 2)) - (x - (h / 2)))))) ^2 ) by SIN_COS4:36
.= (((cos (x - (h / 2))) ^2 ) - ((cos (x + (h / 2))) ^2 )) / ((1 / 4) * (((cos (2 * x)) + (cos h)) ^2 ))
.= ((((cos (x - (h / 2))) ^2 ) - ((cos (x + (h / 2))) ^2 )) / (1 / 4)) / (((cos (2 * x)) + (cos h)) ^2 ) by XCMPLX_1:79
.= 4 * ((((cos (x - (h / 2))) - (cos (x + (h / 2)))) * ((cos (x - (h / 2))) + (cos (x + (h / 2))))) / (((cos (2 * x)) + (cos h)) ^2 ))
.= 4 * (((- (2 * ((sin (((x - (h / 2)) + (x + (h / 2))) / 2)) * (sin (((x - (h / 2)) - (x + (h / 2))) / 2))))) * ((cos (x - (h / 2))) + (cos (x + (h / 2))))) / (((cos (2 * x)) + (cos h)) ^2 )) by SIN_COS4:22
.= 4 * (((- (2 * ((sin ((2 * x) / 2)) * (sin ((- h) / 2))))) * (2 * ((cos ((2 * x) / 2)) * (cos ((- h) / 2))))) / (((cos (2 * x)) + (cos h)) ^2 )) by SIN_COS4:21
.= (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2 ) ;
hence (cD f,h) . x = (((((- 16) * (sin x)) * (sin ((- h) / 2))) * (cos x)) * (cos ((- h) / 2))) / (((cos (2 * x)) + (cos h)) ^2 ) ; :: thesis: verum