let h, x be Real; :: thesis: ( x in dom sec & x + h in dom sec implies (fD ((sec (#) sec),h)) . x = ((4 * (sin ((2 * x) + h))) * (sin h)) / (((cos ((2 * x) + h)) + (cos h)) ^2) )
set f = sec (#) sec;
assume A1: ( x in dom sec & x + h in dom sec ) ; :: thesis: (fD ((sec (#) sec),h)) . x = ((4 * (sin ((2 * x) + h))) * (sin h)) / (((cos ((2 * x) + h)) + (cos h)) ^2)
A2: ( cos . x <> 0 & cos . (x + h) <> 0 ) by A1, RFUNCT_1:3;
( x in dom (sec (#) sec) & x + h in dom (sec (#) sec) )

proof

( x in (dom sec) /\ (dom sec) & x + h in (dom sec) /\ (dom sec) ) by A1;
hence ( x in dom (sec (#) sec) & x + h in dom (sec (#) sec) ) by VALUED_1:def 4; :: thesis: verum

end;

then (fD ((sec (#) sec),h)) . x = ((sec (#) sec) . (x + h)) - ((sec (#) sec) . x) by DIFF_1:1
.= ((sec . (x + h)) * (sec . (x + h))) - ((sec (#) sec) . x) by VALUED_1:5
.= ((sec . (x + h)) * (sec . (x + h))) - ((sec . x) * (sec . x)) by VALUED_1:5
.= (((cos . (x + h)) ") * (sec . (x + h))) - ((sec . x) * (sec . x)) by A1, RFUNCT_1:def 2
.= (((cos . (x + h)) ") * ((cos . (x + h)) ")) - ((sec . x) * (sec . x)) by A1, RFUNCT_1:def 2
.= (((cos . (x + h)) ") * ((cos . (x + h)) ")) - (((cos . x) ") * (sec . x)) by A1, RFUNCT_1:def 2
.= (((cos . (x + h)) ") ^2) - (((cos . x) ") ^2) by A1, RFUNCT_1:def 2
.= ((1 / (cos . (x + h))) - (1 / (cos . x))) * ((1 / (cos . (x + h))) + (1 / (cos . x)))
.= (((1 * (cos . x)) - (1 * (cos . (x + h)))) / ((cos . (x + h)) * (cos . x))) * ((1 / (cos . (x + h))) + (1 / (cos . x))) by A2, XCMPLX_1:130
.= (((cos . x) - (cos . (x + h))) / ((cos . (x + h)) * (cos . x))) * (((cos . x) + (cos . (x + h))) / ((cos . (x + h)) * (cos . x))) by A2, XCMPLX_1:116
.= (((cos . x) - (cos . (x + h))) * ((cos . x) + (cos . (x + h)))) / (((cos . (x + h)) * (cos . x)) * ((cos . (x + h)) * (cos . x))) by XCMPLX_1:76
.= (((cos x) * (cos x)) - ((cos (x + h)) * (cos (x + h)))) / (((cos (x + h)) * (cos x)) ^2)
.= ((sin ((x + h) + x)) * (sin ((x + h) - x))) / (((cos (x + h)) * (cos x)) ^2) by SIN_COS4:38
.= ((sin ((2 * x) + h)) * (sin h)) / (((1 / 2) * ((cos ((x + h) + x)) + (cos ((x + h) - x)))) ^2) by SIN_COS4:32
.= (1 * ((sin ((2 * x) + h)) * (sin h))) / ((1 / 4) * (((cos ((2 * x) + h)) + (cos h)) ^2))
.= (1 / (1 / 4)) * (((sin ((2 * x) + h)) * (sin h)) / (((cos ((2 * x) + h)) + (cos h)) ^2)) by XCMPLX_1:76
.= ((4 * (sin ((2 * x) + h))) * (sin h)) / (((cos ((2 * x) + h)) + (cos h)) ^2) ;
hence (fD ((sec (#) sec),h)) . x = ((4 * (sin ((2 * x) + h))) * (sin h)) / (((cos ((2 * x) + h)) + (cos h)) ^2) ; :: thesis: verum