let x0, x1 be Real; ( x0 in dom cosec & x1 in dom cosec implies [!(cosec (#) cosec),x0,x1!] = (4 * ((sin (x1 + x0)) * (sin (x1 - x0)))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) )
assume A1:
( x0 in dom cosec & x1 in dom cosec )
; [!(cosec (#) cosec),x0,x1!] = (4 * ((sin (x1 + x0)) * (sin (x1 - x0)))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1))
A2:
( sin . x0 <> 0 & sin . x1 <> 0 )
by A1, RFUNCT_1:3;
[!(cosec (#) cosec),x0,x1!] =
(((cosec . x0) * (cosec . x0)) - ((cosec (#) cosec) . x1)) / (x0 - x1)
by VALUED_1:5
.=
(((cosec . x0) * (cosec . x0)) - ((cosec . x1) * (cosec . x1))) / (x0 - x1)
by VALUED_1:5
.=
((((sin . x0) ") * (cosec . x0)) - ((cosec . x1) * (cosec . x1))) / (x0 - x1)
by A1, RFUNCT_1:def 2
.=
((((sin . x0) ") * ((sin . x0) ")) - ((cosec . x1) * (cosec . x1))) / (x0 - x1)
by A1, RFUNCT_1:def 2
.=
((((sin . x0) ") * ((sin . x0) ")) - (((sin . x1) ") * (cosec . x1))) / (x0 - x1)
by A1, RFUNCT_1:def 2
.=
((((sin . x0) ") ^2) - (((sin . x1) ") ^2)) / (x0 - x1)
by A1, RFUNCT_1:def 2
.=
(((1 / (sin . x0)) - (1 / (sin . x1))) * ((1 / (sin . x0)) + (1 / (sin . x1)))) / (x0 - x1)
.=
((((1 * (sin . x1)) - (1 * (sin . x0))) / ((sin . x0) * (sin . x1))) * ((1 / (sin . x0)) + (1 / (sin . x1)))) / (x0 - x1)
by A2, XCMPLX_1:130
.=
((((sin . x1) - (sin . x0)) / ((sin . x0) * (sin . x1))) * (((sin . x1) + (sin . x0)) / ((sin . x0) * (sin . x1)))) / (x0 - x1)
by A2, XCMPLX_1:116
.=
((((sin . x1) - (sin . x0)) * ((sin . x1) + (sin . x0))) / (((sin . x0) * (sin . x1)) * ((sin . x0) * (sin . x1)))) / (x0 - x1)
by XCMPLX_1:76
.=
((((sin x1) * (sin x1)) - ((sin x0) * (sin x0))) / (((sin x0) * (sin x1)) ^2)) / (x0 - x1)
.=
(((sin (x1 + x0)) * (sin (x1 - x0))) / (((sin x0) * (sin x1)) ^2)) / (x0 - x1)
by SIN_COS4:37
.=
(((sin (x1 + x0)) * (sin (x1 - x0))) / ((- ((1 / 2) * ((cos (x0 + x1)) - (cos (x0 - x1))))) ^2)) / (x0 - x1)
by SIN_COS4:29
.=
((1 * ((sin (x1 + x0)) * (sin (x1 - x0)))) / ((1 / 4) * (((cos (x0 + x1)) - (cos (x0 - x1))) ^2))) / (x0 - x1)
.=
((1 / (1 / 4)) * (((sin (x1 + x0)) * (sin (x1 - x0))) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2))) / (x0 - x1)
by XCMPLX_1:76
.=
((4 * ((sin (x1 + x0)) * (sin (x1 - x0)))) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1)
.=
(4 * ((sin (x1 + x0)) * (sin (x1 - x0)))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1))
by XCMPLX_1:78
;
hence
[!(cosec (#) cosec),x0,x1!] = (4 * ((sin (x1 + x0)) * (sin (x1 - x0)))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1))
; verum