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:70;
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_1:16;
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_1:16;
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_1:29
.= ((tan . x) * (exp_R . x)) + ((exp_R . x) * (diff tan ,x)) by SIN_COS:70
.= ((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 4
.= (((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_1:29; :: thesis: verum