let Z be open Subset of REAL; ( Z c= dom (cos (#) (sin + cos)) implies ( cos (#) (sin + cos) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) ) ) )
A1:
for x being Real st x in Z holds
cos is_differentiable_in x
by SIN_COS:63;
assume A2:
Z c= dom (cos (#) (sin + cos))
; ( cos (#) (sin + cos) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) ) )
then A3:
Z c= (dom (sin + cos)) /\ (dom cos)
by VALUED_1:def 4;
then A4:
Z c= dom (sin + cos)
by XBOOLE_1:18;
then A5:
sin + cos is_differentiable_on Z
by FDIFF_7:38;
Z c= dom cos
by A3, XBOOLE_1:18;
then A6:
cos is_differentiable_on Z
by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2)
proof
let x be
Real;
( x in Z implies ((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) )
reconsider xx =
x as
Element of
REAL by XREAL_0:def 1;
assume A7:
x in Z
;
((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2)
then ((cos (#) (sin + cos)) `| Z) . x =
(((sin + cos) . x) * (diff (cos,x))) + ((cos . x) * (diff ((sin + cos),x)))
by A2, A5, A6, FDIFF_1:21
.=
(((sin . xx) + (cos . xx)) * (diff (cos,x))) + ((cos . x) * (diff ((sin + cos),x)))
by VALUED_1:1
.=
(((sin . x) + (cos . x)) * (- (sin . x))) + ((cos . x) * (diff ((sin + cos),x)))
by SIN_COS:63
.=
(((sin . x) + (cos . x)) * (- (sin . x))) + ((cos . x) * (((sin + cos) `| Z) . x))
by A5, A7, FDIFF_1:def 7
.=
(((sin . x) + (cos . x)) * (- (sin . x))) + ((cos . x) * ((cos . x) - (sin . x)))
by A4, A7, FDIFF_7:38
;
hence
((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2)
;
verum
end;
hence
( cos (#) (sin + cos) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (sin + cos)) `| Z) . x = (((cos . x) ^2) - ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) ) )
by A2, A5, A6, FDIFF_1:21; verum