let Z be open Subset of REAL; :: thesis: ( Z c= dom (sin (#) ()) implies ( sin (#) () is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) ()) `| Z) . x = (((cos . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) ) ) )

A1: for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
assume A2: Z c= dom (sin (#) ()) ; :: thesis: ( sin (#) () is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) ()) `| Z) . x = (((cos . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) ) )

then A3: Z c= (dom ()) /\ () by VALUED_1:def 4;
then A4: Z c= dom () by XBOOLE_1:18;
then A5: sin + cos is_differentiable_on Z by FDIFF_7:38;
Z c= dom sin by ;
then A6: sin is_differentiable_on Z by ;
for x being Real st x in Z holds
((sin (#) ()) `| Z) . x = (((cos . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2)
proof
let x be Real; :: thesis: ( x in Z implies ((sin (#) ()) `| 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 ; :: thesis: ((sin (#) ()) `| Z) . x = (((cos . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2)
then ((sin (#) ()) `| Z) . x = ((() . x) * (diff (sin,x))) + ((sin . x) * (diff ((),x))) by
.= (((sin . xx) + (cos . xx)) * (diff (sin,x))) + ((sin . x) * (diff ((),x))) by VALUED_1:1
.= (((sin . x) + (cos . x)) * (cos . x)) + ((sin . x) * (diff ((),x))) by SIN_COS:64
.= (((sin . x) + (cos . x)) * (cos . x)) + ((sin . x) * ((() `| Z) . x)) by
.= (((sin . x) + (cos . x)) * (cos . x)) + ((sin . x) * ((cos . x) - (sin . x))) by ;
hence ((sin (#) ()) `| Z) . x = (((cos . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) ; :: thesis: verum
end;
hence ( sin (#) () is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) ()) `| Z) . x = (((cos . x) ^2) + ((2 * (sin . x)) * (cos . x))) - ((sin . x) ^2) ) ) by ; :: thesis: verum