let Z be open Subset of REAL; :: thesis: ( Z c= dom (exp_R (#) (sin - cos)) implies ( exp_R (#) (sin - cos) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (sin - cos)) `| Z) . x = (2 * (exp_R . x)) * (sin . x) ) ) )
assume A1: Z c= dom (exp_R (#) (sin - cos)) ; :: thesis: ( exp_R (#) (sin - cos) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (sin - cos)) `| Z) . x = (2 * (exp_R . x)) * (sin . x) ) )
then Z c= (dom (sin - cos)) /\ (dom exp_R) by VALUED_1:def 4;
then A2: Z c= dom (sin - cos) by XBOOLE_1:18;
then A3: sin - cos is_differentiable_on Z by Th39;
A4: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
for x being Real st x in Z holds
((exp_R (#) (sin - cos)) `| Z) . x = (2 * (exp_R . x)) * (sin . x)

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) (sin - cos)) `| Z) . x = (2 * (exp_R . x)) * (sin . x) )
assume A5: x in Z ; :: thesis: ((exp_R (#) (sin - cos)) `| Z) . x = (2 * (exp_R . x)) * (sin . x)
then ((exp_R (#) (sin - cos)) `| Z) . x = (((sin - cos) . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((sin - cos),x))) by A1, A3, A4, FDIFF_1:21
.= (((sin . x) - (cos . x)) * (diff (exp_R,x))) + ((exp_R . x) * (diff ((sin - cos),x))) by A2, A5, VALUED_1:13
.= (((sin . x) - (cos . x)) * (exp_R . x)) + ((exp_R . x) * (diff ((sin - cos),x))) by TAYLOR_1:16
.= (((sin . x) - (cos . x)) * (exp_R . x)) + ((exp_R . x) * (((sin - cos) `| Z) . x)) by A3, A5, FDIFF_1:def 7
.= (((sin . x) - (cos . x)) * (exp_R . x)) + ((exp_R . x) * ((cos . x) + (sin . x))) by A2, A5, Th39
.= (2 * (exp_R . x)) * (sin . x) ;
hence ((exp_R (#) (sin - cos)) `| Z) . x = (2 * (exp_R . x)) * (sin . x) ; :: thesis: verum

end;

hence ( exp_R (#) (sin - cos) is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) (sin - cos)) `| Z) . x = (2 * (exp_R . x)) * (sin . x) ) ) by A1, A3, A4, FDIFF_1:21; :: thesis: verum