let h, x be Real; for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) sin) . x ) & x in dom tan & x - h in dom tan holds
(bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h))
let f be Function of REAL,REAL; ( ( for x being Real holds f . x = (tan (#) sin) . x ) & x in dom tan & x - h in dom tan implies (bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h)) )
assume that
A1:
for x being Real holds f . x = (tan (#) sin) . x
and
A2:
( x in dom tan & x - h in dom tan )
; (bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h))
(bD (f,h)) . x =
(f . x) - (f . (x - h))
by DIFF_1:4
.=
((tan (#) sin) . x) - (f . (x - h))
by A1
.=
((tan (#) sin) . x) - ((tan (#) sin) . (x - h))
by A1
.=
((tan . x) * (sin . x)) - ((tan (#) sin) . (x - h))
by VALUED_1:5
.=
((tan . x) * (sin . x)) - ((tan . (x - h)) * (sin . (x - h)))
by VALUED_1:5
.=
(((sin . x) * ((cos . x) ")) * (sin . x)) - ((tan . (x - h)) * (sin . (x - h)))
by A2, RFUNCT_1:def 1
.=
(((sin x) / (cos x)) * (sin x)) - (((sin (x - h)) / (cos (x - h))) * (sin (x - h)))
by A2, RFUNCT_1:def 1
.=
((sin x) / ((cos x) / (sin x))) - (((sin (x - h)) / (cos (x - h))) * (sin (x - h)))
by XCMPLX_1:82
.=
((sin x) / ((cos x) / (sin x))) - ((sin (x - h)) / ((cos (x - h)) / (sin (x - h))))
by XCMPLX_1:82
.=
(((sin x) * (sin x)) / (cos x)) - ((sin (x - h)) / ((cos (x - h)) / (sin (x - h))))
by XCMPLX_1:77
.=
(((sin x) * (sin x)) / (cos x)) - (((sin (x - h)) * (sin (x - h))) / (cos (x - h)))
by XCMPLX_1:77
.=
((1 - ((cos x) * (cos x))) / (cos x)) - (((sin (x - h)) * (sin (x - h))) / (cos (x - h)))
by SIN_COS4:4
.=
((1 / (cos x)) - (((cos x) * (cos x)) / (cos x))) - ((1 - ((cos (x - h)) * (cos (x - h)))) / (cos (x - h)))
by SIN_COS4:4
.=
((1 / (cos x)) - (cos x)) - ((1 / (cos (x - h))) - (((cos (x - h)) * (cos (x - h))) / (cos (x - h))))
by A2, FDIFF_8:1, XCMPLX_1:89
.=
((1 / (cos x)) - (cos x)) - ((1 / (cos (x - h))) - (cos (x - h)))
by A2, FDIFF_8:1, XCMPLX_1:89
.=
(((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h))
;
hence
(bD (f,h)) . x = (((1 / (cos x)) - (cos x)) - (1 / (cos (x - h)))) + (cos (x - h))
; verum