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_7:39;
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) )
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 . x) - (cos . x)) * (diff (cos,x))) + ((cos . x) * (diff ((sin - cos),x)))
by A4, A7, VALUED_1:13
.=
(((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_7:39
;
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