let Z be open Subset of REAL; :: thesis: ( Z c= dom ((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) & not 0 in Z implies ( (sin * ((id Z) ^)) (#) (cos * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2)) ) ) )
set f = id Z;
assume that
A1: Z c= dom ((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) and
A2: not 0 in Z ; :: thesis: ( (sin * ((id Z) ^)) (#) (cos * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2)) ) )
A3: sin * ((id Z) ^) is_differentiable_on Z by A2, Th5;
A4: Z c= (dom (sin * ((id Z) ^))) /\ (dom (cos * ((id Z) ^))) by A1, VALUED_1:def 4;
then A5: Z c= dom (cos * ((id Z) ^)) by XBOOLE_1:18;
then A6: cos * ((id Z) ^) is_differentiable_on Z by A2, Th6;
A7: Z c= dom (sin * ((id Z) ^)) by A4, XBOOLE_1:18;
then for y being object st y in Z holds
y in dom ((id Z) ^) by FUNCT_1:11;
then A8: Z c= dom ((id Z) ^) ;

now :: thesis: for x being Real st x in Z holds
(((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2))

let x be Real; :: thesis: ( x in Z implies (((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2)) )
assume A9: x in Z ; :: thesis: (((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2))
then (((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (((cos * ((id Z) ^)) . x) * (diff ((sin * ((id Z) ^)),x))) + (((sin * ((id Z) ^)) . x) * (diff ((cos * ((id Z) ^)),x))) by A1, A6, A3, FDIFF_1:21
.= (((cos * ((id Z) ^)) . x) * (((sin * ((id Z) ^)) `| Z) . x)) + (((sin * ((id Z) ^)) . x) * (diff ((cos * ((id Z) ^)),x))) by A3, A9, FDIFF_1:def 7
.= (((cos * ((id Z) ^)) . x) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + (((sin * ((id Z) ^)) . x) * (diff ((cos * ((id Z) ^)),x))) by A2, A9, Th5
.= (((cos * ((id Z) ^)) . x) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + (((sin * ((id Z) ^)) . x) * (((cos * ((id Z) ^)) `| Z) . x)) by A6, A9, FDIFF_1:def 7
.= (((cos * ((id Z) ^)) . x) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + (((sin * ((id Z) ^)) . x) * ((1 / (x ^2)) * (sin . (1 / x)))) by A2, A5, A9, Th6
.= ((cos . (((id Z) ^) . x)) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + (((sin * ((id Z) ^)) . x) * ((1 / (x ^2)) * (sin . (1 / x)))) by A5, A9, FUNCT_1:12
.= ((cos . (((id Z) . x) ")) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + (((sin * ((id Z) ^)) . x) * ((1 / (x ^2)) * (sin . (1 / x)))) by A8, A9, RFUNCT_1:def 2
.= ((cos . (1 * (x "))) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + (((sin * ((id Z) ^)) . x) * ((1 / (x ^2)) * (sin . (1 / x)))) by A9, FUNCT_1:18
.= ((cos . (1 / x)) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + (((sin * ((id Z) ^)) . x) * ((1 / (x ^2)) * (sin . (1 / x)))) by XCMPLX_0:def 9
.= ((cos . (1 / x)) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + ((sin . (((id Z) ^) . x)) * ((1 / (x ^2)) * (sin . (1 / x)))) by A7, A9, FUNCT_1:12
.= ((cos . (1 / x)) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + ((sin . (((id Z) . x) ")) * ((1 / (x ^2)) * (sin . (1 / x)))) by A8, A9, RFUNCT_1:def 2
.= ((cos . (1 / x)) * (- ((1 / (x ^2)) * (cos . (1 / x))))) + ((sin . (1 * (x "))) * ((1 / (x ^2)) * (sin . (1 / x)))) by A9, FUNCT_1:18
.= (- (((cos . (1 / x)) * (1 / (x ^2))) * (cos . (1 / x)))) + ((sin . (1 / x)) * ((1 / (x ^2)) * (sin . (1 / x)))) by XCMPLX_0:def 9
.= (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2)) ;
hence (((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2)) ; :: thesis: verum

end;

hence ( (sin * ((id Z) ^)) (#) (cos * ((id Z) ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((sin * ((id Z) ^)) (#) (cos * ((id Z) ^))) `| Z) . x = (1 / (x ^2)) * (((sin . (1 / x)) ^2) - ((cos . (1 / x)) ^2)) ) ) by A1, A6, A3, FDIFF_1:21; :: thesis: verum