let Z be open Subset of REAL ; :: thesis: ( Z c= dom (sin - ((1 / 2) (#) ((#Z 2) * sin ))) & ( for x being Real st x in Z holds
( sin . x > 0 & sin . x > - 1 ) ) implies ( sin - ((1 / 2) (#) ((#Z 2) * sin )) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin - ((1 / 2) (#) ((#Z 2) * sin ))) `| Z) . x = ((cos . x) |^ 3) / (1 + (sin . x)) ) ) )
assume that
A1: Z c= dom (sin - ((1 / 2) (#) ((#Z 2) * sin ))) and
A2: for x being Real st x in Z holds
( sin . x > 0 & sin . x > - 1 ) ; :: thesis: ( sin - ((1 / 2) (#) ((#Z 2) * sin )) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin - ((1 / 2) (#) ((#Z 2) * sin ))) `| Z) . x = ((cos . x) |^ 3) / (1 + (sin . x)) ) )
Z c= (dom ((1 / 2) (#) ((#Z 2) * sin ))) /\ (dom sin ) 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: sin is_differentiable_on Z by FDIFF_1:34, SIN_COS:73;

now

let x be Real; :: thesis: ( x in Z implies ((sin - ((1 / 2) (#) ((#Z 2) * sin ))) `| Z) . x = ((cos . x) |^ 3) / (1 + (sin . x)) )
assume A6: x in Z ; :: thesis: ((sin - ((1 / 2) (#) ((#Z 2) * sin ))) `| Z) . x = ((cos . x) |^ 3) / (1 + (sin . x))
then sin . x > - 1 by A2;
then A7: (sin . x) - (- 1) > 0 by XREAL_1:52;
((sin - ((1 / 2) (#) ((#Z 2) * sin ))) `| Z) . x = (diff sin ,x) - (diff ((1 / 2) (#) ((#Z 2) * sin )),x) by A1, A4, A5, A6, FDIFF_1:27
.= (cos . x) - (diff ((1 / 2) (#) ((#Z 2) * sin )),x) by SIN_COS:69
.= (cos . x) - ((((1 / 2) (#) ((#Z 2) * sin )) `| Z) . x) by A4, A6, FDIFF_1:def 8
.= (cos . x) - ((sin . x) * (cos . x)) by A3, A6, Th49
.= (((cos . x) * (1 - (sin . x))) * (1 + (sin . x))) / (1 + (sin . x)) by A7, XCMPLX_1:90
.= ((cos . x) * (1 - ((sin . x) ^2 ))) / (1 + (sin . x))
.= ((cos . x) * (1 - ((sin x) ^2 ))) / (1 + (sin . x)) by SIN_COS:def 21
.= ((cos . x) * ((cos x) * (cos x))) / (1 + (sin . x)) by SIN_COS4:7
.= ((cos . x) * ((cos x) |^ 2)) / (1 + (sin . x)) by WSIERP_1:2
.= ((cos . x) * ((cos . x) |^ 2)) / (1 + (sin . x)) by SIN_COS:def 23
.= ((cos . x) |^ (2 + 1)) / (1 + (sin . x)) by NEWTON:11
.= ((cos . x) |^ 3) / (1 + (sin . x)) ;
hence ((sin - ((1 / 2) (#) ((#Z 2) * sin ))) `| Z) . x = ((cos . x) |^ 3) / (1 + (sin . x)) ; :: thesis: verum

end;

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