let x0, x1 be Real; ( x0 in dom tan & x1 in dom tan implies [!(tan (#) sin),x0,x1!] = ((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1) )
assume A1:
( x0 in dom tan & x1 in dom tan )
; [!(tan (#) sin),x0,x1!] = ((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1)
[!(tan (#) sin),x0,x1!] =
(((tan . x0) * (sin . x0)) - ((tan (#) sin) . x1)) / (x0 - x1)
by VALUED_1:5
.=
(((tan . x0) * (sin . x0)) - ((tan . x1) * (sin . x1))) / (x0 - x1)
by VALUED_1:5
.=
((((sin . x0) * ((cos . x0) ")) * (sin . x0)) - ((tan . x1) * (sin . x1))) / (x0 - x1)
by A1, RFUNCT_1:def 1
.=
((((sin x0) / (cos x0)) * (sin x0)) - (((sin x1) / (cos x1)) * (sin x1))) / (x0 - x1)
by A1, RFUNCT_1:def 1
.=
(((sin x0) / ((cos x0) / (sin x0))) - (((sin x1) / (cos x1)) * (sin x1))) / (x0 - x1)
by XCMPLX_1:82
.=
(((sin x0) / ((cos x0) / (sin x0))) - ((sin x1) / ((cos x1) / (sin x1)))) / (x0 - x1)
by XCMPLX_1:82
.=
((((sin x0) * (sin x0)) / (cos x0)) - ((sin x1) / ((cos x1) / (sin x1)))) / (x0 - x1)
by XCMPLX_1:77
.=
((((sin x0) * (sin x0)) / (cos x0)) - (((sin x1) * (sin x1)) / (cos x1))) / (x0 - x1)
by XCMPLX_1:77
.=
(((1 - ((cos x0) * (cos x0))) / (cos x0)) - (((sin x1) * (sin x1)) / (cos x1))) / (x0 - x1)
by SIN_COS4:4
.=
(((1 / (cos x0)) - (((cos x0) * (cos x0)) / (cos x0))) - ((1 - ((cos x1) * (cos x1))) / (cos x1))) / (x0 - x1)
by SIN_COS4:4
.=
(((1 / (cos x0)) - (cos x0)) - ((1 / (cos x1)) - (((cos x1) * (cos x1)) / (cos x1)))) / (x0 - x1)
by A1, FDIFF_8:1, XCMPLX_1:89
.=
(((1 / (cos x0)) - (cos x0)) - ((1 / (cos x1)) - (cos x1))) / (x0 - x1)
by A1, FDIFF_8:1, XCMPLX_1:89
.=
((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1)
;
hence
[!(tan (#) sin),x0,x1!] = ((((1 / (cos x0)) - (cos x0)) - (1 / (cos x1))) + (cos x1)) / (x0 - x1)
; verum