let x0, x1 be Real; ( x0 in dom sec & x1 in dom sec implies [!(sec (#) sec),x0,x1!] = (4 * ((sin (x0 + x1)) * (sin (x0 - x1)))) / ((((cos (x0 + x1)) + (cos (x0 - x1))) ^2) * (x0 - x1)) )
assume A1:
( x0 in dom sec & x1 in dom sec )
; [!(sec (#) sec),x0,x1!] = (4 * ((sin (x0 + x1)) * (sin (x0 - x1)))) / ((((cos (x0 + x1)) + (cos (x0 - x1))) ^2) * (x0 - x1))
A2:
( cos . x0 <> 0 & cos . x1 <> 0 )
by A1, RFUNCT_1:3;
[!(sec (#) sec),x0,x1!] =
(((sec . x0) * (sec . x0)) - ((sec (#) sec) . x1)) / (x0 - x1)
by VALUED_1:5
.=
(((sec . x0) * (sec . x0)) - ((sec . x1) * (sec . x1))) / (x0 - x1)
by VALUED_1:5
.=
((((cos . x0) ") * (sec . x0)) - ((sec . x1) * (sec . x1))) / (x0 - x1)
by A1, RFUNCT_1:def 2
.=
((((cos . x0) ") * ((cos . x0) ")) - ((sec . x1) * (sec . x1))) / (x0 - x1)
by A1, RFUNCT_1:def 2
.=
((((cos . x0) ") * ((cos . x0) ")) - (((cos . x1) ") * (sec . x1))) / (x0 - x1)
by A1, RFUNCT_1:def 2
.=
((((cos . x0) ") ^2) - (((cos . x1) ") ^2)) / (x0 - x1)
by A1, RFUNCT_1:def 2
.=
(((1 / (cos . x0)) - (1 / (cos . x1))) * ((1 / (cos . x0)) + (1 / (cos . x1)))) / (x0 - x1)
.=
((((1 * (cos . x1)) - (1 * (cos . x0))) / ((cos . x0) * (cos . x1))) * ((1 / (cos . x0)) + (1 / (cos . x1)))) / (x0 - x1)
by A2, XCMPLX_1:130
.=
((((cos . x1) - (cos . x0)) / ((cos . x0) * (cos . x1))) * (((cos . x1) + (cos . x0)) / ((cos . x0) * (cos . x1)))) / (x0 - x1)
by A2, XCMPLX_1:116
.=
((((cos . x1) - (cos . x0)) * ((cos . x1) + (cos . x0))) / (((cos . x0) * (cos . x1)) * ((cos . x0) * (cos . x1)))) / (x0 - x1)
by XCMPLX_1:76
.=
((((cos x1) * (cos x1)) - ((cos x0) * (cos x0))) / (((cos x0) * (cos x1)) ^2)) / (x0 - x1)
.=
(((sin (x0 + x1)) * (sin (x0 - x1))) / (((cos x0) * (cos x1)) ^2)) / (x0 - x1)
by SIN_COS4:38
.=
(((sin (x0 + x1)) * (sin (x0 - x1))) / (((1 / 2) * ((cos (x0 + x1)) + (cos (x0 - x1)))) ^2)) / (x0 - x1)
by SIN_COS4:32
.=
((1 * ((sin (x0 + x1)) * (sin (x0 - x1)))) / ((1 / 4) * (((cos (x0 + x1)) + (cos (x0 - x1))) ^2))) / (x0 - x1)
.=
((1 / (1 / 4)) * (((sin (x0 + x1)) * (sin (x0 - x1))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2))) / (x0 - x1)
by XCMPLX_1:76
.=
((4 * ((sin (x0 + x1)) * (sin (x0 - x1)))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2)) / (x0 - x1)
.=
(4 * ((sin (x0 + x1)) * (sin (x0 - x1)))) / ((((cos (x0 + x1)) + (cos (x0 - x1))) ^2) * (x0 - x1))
by XCMPLX_1:78
;
hence
[!(sec (#) sec),x0,x1!] = (4 * ((sin (x0 + x1)) * (sin (x0 - x1)))) / ((((cos (x0 + x1)) + (cos (x0 - x1))) ^2) * (x0 - x1))
; verum