let Z be open Subset of REAL ; :: thesis: ( Z c= dom (((1 / 2) (#) ((#Z 2) * sin )) - cos ) & ( for x being Real st x in Z holds
( sin . x > 0 & cos . x < 1 ) ) implies ( ((1 / 2) (#) ((#Z 2) * sin )) - cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / 2) (#) ((#Z 2) * sin )) - cos ) `| Z) . x = ((sin . x) |^ 3) / (1 - (cos . x)) ) ) )
assume that
A1: Z c= dom (((1 / 2) (#) ((#Z 2) * sin )) - cos ) and
A2: for x being Real st x in Z holds
( sin . x > 0 & cos . x < 1 ) ; :: thesis: ( ((1 / 2) (#) ((#Z 2) * sin )) - cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / 2) (#) ((#Z 2) * sin )) - cos ) `| Z) . x = ((sin . x) |^ 3) / (1 - (cos . x)) ) )
Z c= (dom ((1 / 2) (#) ((#Z 2) * sin ))) /\ (dom cos ) by A1, VALUED_1:12;
then A3: Z c= dom ((1 / 2) (#) ((#Z 2) * sin )) by XBOOLE_1:18;
then A4: (1 / 2) (#) ((#Z 2) * sin ) is_differentiable_on Z by Th49;
A5: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;

now

let x be Real; :: thesis: ( x in Z implies ((((1 / 2) (#) ((#Z 2) * sin )) - cos ) `| Z) . x = ((sin . x) |^ 3) / (1 - (cos . x)) )
assume A6: x in Z ; :: thesis: ((((1 / 2) (#) ((#Z 2) * sin )) - cos ) `| Z) . x = ((sin . x) |^ 3) / (1 - (cos . x))
then A7: 1 - (cos . x) > 0 by A2, XREAL_1:52;
((((1 / 2) (#) ((#Z 2) * sin )) - cos ) `| Z) . x = (diff ((1 / 2) (#) ((#Z 2) * sin )),x) - (diff cos ,x) by A1, A4, A5, A6, FDIFF_1:27
.= (diff ((1 / 2) (#) ((#Z 2) * sin )),x) - (- (sin . x)) by SIN_COS:68
.= ((((1 / 2) (#) ((#Z 2) * sin )) `| Z) . x) - (- (sin . x)) by A4, A6, FDIFF_1:def 8
.= ((sin . x) * (cos . x)) - (- (sin . x)) by A3, A6, Th49
.= (((sin . x) * (1 + (cos . x))) * (1 - (cos . x))) / (1 - (cos . x)) by A7, XCMPLX_1:90
.= ((sin . x) * (1 - ((cos . x) ^2 ))) / (1 - (cos . x))
.= ((sin . x) * (1 - ((cos x) ^2 ))) / (1 - (cos . x)) by SIN_COS:def 23
.= ((sin . x) * ((sin x) * (sin x))) / (1 - (cos . x)) by SIN_COS4:6
.= ((sin . x) * ((sin x) |^ 2)) / (1 - (cos . x)) by WSIERP_1:2
.= ((sin . x) * ((sin . x) |^ 2)) / (1 - (cos . x)) by SIN_COS:def 21
.= ((sin . x) |^ (2 + 1)) / (1 - (cos . x)) by NEWTON:11
.= ((sin . x) |^ 3) / (1 - (cos . x)) ;
hence ((((1 / 2) (#) ((#Z 2) * sin )) - cos ) `| Z) . x = ((sin . x) |^ 3) / (1 - (cos . x)) ; :: thesis: verum

end;

hence ( ((1 / 2) (#) ((#Z 2) * sin )) - cos is_differentiable_on Z & ( for x being Real st x in Z holds
((((1 / 2) (#) ((#Z 2) * sin )) - cos ) `| Z) . x = ((sin . x) |^ 3) / (1 - (cos . x)) ) ) by A1, A4, A5, FDIFF_1:27; :: thesis: verum