let Z be open Subset of REAL; ( 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
; ( (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 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;
( 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
;
(((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))
;
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; verum