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)) * (cos . 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)) * (cos . 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 & ( for x being Real st x in Z holds
((sin + cos ) `| Z) . x = (cos . x) - (sin . x) ) ) by Th38;
A4: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
for x being Real st x in Z holds
((exp_R (#) (sin + cos )) `| Z) . x = (2 * (exp_R . x)) * (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) (sin + cos )) `| Z) . x = (2 * (exp_R . x)) * (cos . x) )
assume A5: x in Z ; :: thesis: ((exp_R (#) (sin + cos )) `| Z) . x = (2 * (exp_R . x)) * (cos . 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_1:29
.= (((sin . x) + (cos . x)) * (diff exp_R ,x)) + ((exp_R . x) * (diff (sin + cos ),x)) by VALUED_1:1
.= (((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 8
.= (((sin . x) + (cos . x)) * (exp_R . x)) + ((exp_R . x) * ((cos . x) - (sin . x))) by A2, A5, Th38
.= (2 * (exp_R . x)) * (cos . x) ;
hence ((exp_R (#) (sin + cos )) `| Z) . x = (2 * (exp_R . x)) * (cos . 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)) * (cos . x) ) ) by A1, A3, A4, FDIFF_1:29; :: thesis: verum