let h, x be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / (cos x) ) & cos x <> 0 & cos (x - h) <> 0 holds
(bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h))
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = 1 / (cos x) ) & cos x <> 0 & cos (x - h) <> 0 implies (bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h)) )
assume that
A1: for x being Real holds f . x = 1 / (cos x) and
A2: ( cos x <> 0 & cos (x - h) <> 0 ) ; :: thesis: (bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h))
(bD (f,h)) . x = (f . x) - (f . (x - h)) by DIFF_1:4
.= (1 / (cos x)) - (f . (x - h)) by A1
.= (1 / (cos x)) - (1 / (cos (x - h))) by A1
.= ((1 * (cos (x - h))) - (1 * (cos x))) / ((cos x) * (cos (x - h))) by A2, XCMPLX_1:130
.= ((cos (x - h)) - (cos x)) / ((1 / 2) * ((cos (x + (x - h))) + (cos (x - (x - h))))) by SIN_COS4:32
.= (((cos (x - h)) - (cos x)) / (1 / 2)) / ((cos ((2 * x) - h)) + (cos h)) by XCMPLX_1:78
.= 2 * (((cos (x - h)) - (cos x)) / ((cos ((2 * x) - h)) + (cos h))) ;
hence (bD (f,h)) . x = (2 * ((cos (x - h)) - (cos x))) / ((cos ((2 * x) - h)) + (cos h)) ; :: thesis: verum