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

proof

( x + (h / 2) in (dom cosec) /\ (dom cosec) & x - (h / 2) in (dom cosec) /\ (dom cosec) ) by A1;
hence ( x + (h / 2) in dom (cosec (#) cosec) & x - (h / 2) in dom (cosec (#) cosec) ) by VALUED_1:def 4; :: thesis: verum

end;

then (cD ((cosec (#) cosec),h)) . x = ((cosec (#) cosec) . (x + (h / 2))) - ((cosec (#) cosec) . (x - (h / 2))) by DIFF_1:39
.= ((cosec . (x + (h / 2))) * (cosec . (x + (h / 2)))) - ((cosec (#) cosec) . (x - (h / 2))) by VALUED_1:5
.= ((cosec . (x + (h / 2))) * (cosec . (x + (h / 2)))) - ((cosec . (x - (h / 2))) * (cosec . (x - (h / 2)))) by VALUED_1:5
.= (((sin . (x + (h / 2))) ") * (cosec . (x + (h / 2)))) - ((cosec . (x - (h / 2))) * (cosec . (x - (h / 2)))) by A1, RFUNCT_1:def 2
.= (((sin . (x + (h / 2))) ") * ((sin . (x + (h / 2))) ")) - ((cosec . (x - (h / 2))) * (cosec . (x - (h / 2)))) by A1, RFUNCT_1:def 2
.= (((sin . (x + (h / 2))) ") * ((sin . (x + (h / 2))) ")) - (((sin . (x - (h / 2))) ") * (cosec . (x - (h / 2)))) by A1, RFUNCT_1:def 2
.= (((sin . (x + (h / 2))) ") ^2) - (((sin . (x - (h / 2))) ") ^2) by A1, RFUNCT_1:def 2
.= ((1 / (sin . (x + (h / 2)))) - (1 / (sin . (x - (h / 2))))) * ((1 / (sin . (x + (h / 2)))) + (1 / (sin . (x - (h / 2)))))
.= (((1 * (sin . (x - (h / 2)))) - (1 * (sin . (x + (h / 2))))) / ((sin . (x + (h / 2))) * (sin . (x - (h / 2))))) * ((1 / (sin . (x + (h / 2)))) + (1 / (sin . (x - (h / 2))))) by A2, XCMPLX_1:130
.= (((sin . (x - (h / 2))) - (sin . (x + (h / 2)))) / ((sin . (x + (h / 2))) * (sin . (x - (h / 2))))) * (((sin . (x - (h / 2))) + (sin . (x + (h / 2)))) / ((sin . (x + (h / 2))) * (sin . (x - (h / 2))))) by A2, XCMPLX_1:116
.= (((sin . (x - (h / 2))) - (sin . (x + (h / 2)))) * ((sin . (x - (h / 2))) + (sin . (x + (h / 2))))) / (((sin . (x + (h / 2))) * (sin . (x - (h / 2)))) * ((sin . (x + (h / 2))) * (sin . (x - (h / 2))))) by XCMPLX_1:76
.= (((sin (x - (h / 2))) * (sin (x - (h / 2)))) - ((sin (x + (h / 2))) * (sin (x + (h / 2))))) / (((sin (x + (h / 2))) * (sin (x - (h / 2)))) ^2)
.= ((sin ((x - (h / 2)) + (x + (h / 2)))) * (sin ((x - (h / 2)) - (x + (h / 2))))) / (((sin (x + (h / 2))) * (sin (x - (h / 2)))) ^2) by SIN_COS4:37
.= ((sin (2 * x)) * (sin (- h))) / ((- ((1 / 2) * ((cos ((x + (h / 2)) + (x - (h / 2)))) - (cos ((x + (h / 2)) - (x - (h / 2))))))) ^2) by SIN_COS4:29
.= ((sin (2 * x)) * (- (sin h))) / ((1 / 4) * (((cos (2 * x)) - (cos h)) ^2)) by SIN_COS:31
.= - ((1 * ((sin (2 * x)) * (sin h))) / ((1 / 4) * (((cos (2 * x)) - (cos h)) ^2)))
.= - ((1 / (1 / 4)) * (((sin (2 * x)) * (sin h)) / (((cos (2 * x)) - (cos h)) ^2))) by XCMPLX_1:76
.= - (((4 * (sin (2 * x))) * (sin h)) / (((cos (2 * x)) - (cos h)) ^2)) ;
hence (cD ((cosec (#) cosec),h)) . x = - (((4 * (sin (2 * x))) * (sin h)) / (((cos (2 * x)) - (cos h)) ^2)) ; :: thesis: verum