let Z be open Subset of REAL; :: thesis: ( Z c= dom (exp_R (#) tan) implies ( exp_R (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) tan) `| Z) . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) ) ) )
A1: for x being Real st x in Z holds
exp_R is_differentiable_in x by SIN_COS:65;
assume A2: Z c= dom (exp_R (#) tan) ; :: thesis: ( exp_R (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) tan) `| Z) . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) ) )
then A3: Z c= (dom exp_R) /\ (dom tan) by VALUED_1:def 4;
then A4: Z c= dom tan by XBOOLE_1:18;
Z c= dom exp_R by A3, XBOOLE_1:18;
then A5: exp_R is_differentiable_on Z by A1, FDIFF_1:9;
A6: for x being Real st x in Z holds
( tan is_differentiable_in x & diff (tan,x) = 1 / ((cos . x) ^2) )

proof

let x be Real; :: thesis: ( x in Z implies ( tan is_differentiable_in x & diff (tan,x) = 1 / ((cos . x) ^2) ) )
assume x in Z ; :: thesis: ( tan is_differentiable_in x & diff (tan,x) = 1 / ((cos . x) ^2) )
then cos . x <> 0 by A4, Th1;
hence ( tan is_differentiable_in x & diff (tan,x) = 1 / ((cos . x) ^2) ) by FDIFF_7:46; :: thesis: verum

end;

then for x being Real st x in Z holds
tan is_differentiable_in x ;
then A7: tan is_differentiable_on Z by A4, FDIFF_1:9;
for x being Real st x in Z holds
((exp_R (#) tan) `| Z) . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) tan) `| Z) . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) )
assume A8: x in Z ; :: thesis: ((exp_R (#) tan) `| Z) . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2))
then ((exp_R (#) tan) `| Z) . x = ((tan . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff (tan,x))) by A2, A5, A7, FDIFF_1:21
.= ((tan . x) * (exp_R . x)) + ((exp_R . x) * (diff (tan,x))) by SIN_COS:65
.= ((tan . x) * (exp_R . x)) + ((exp_R . x) * (1 / ((cos . x) ^2))) by A6, A8
.= (((sin . x) / (cos . x)) * ((exp_R . x) / 1)) + ((exp_R . x) / ((cos . x) ^2)) by A4, A8, RFUNCT_1:def 1
.= (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) ;
hence ((exp_R (#) tan) `| Z) . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) ; :: thesis: verum

end;

hence ( exp_R (#) tan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) tan) `| Z) . x = (((exp_R . x) * (sin . x)) / (cos . x)) + ((exp_R . x) / ((cos . x) ^2)) ) ) by A2, A5, A7, FDIFF_1:21; :: thesis: verum