let Z be open Subset of REAL ; :: thesis: ( Z c= dom (exp_R (#) cosec ) implies ( exp_R (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) cosec ) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . 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 (#) cosec ) ; :: thesis: ( exp_R (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) cosec ) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2 )) ) )
then A3: Z c= (dom exp_R ) /\ (dom cosec ) by VALUED_1:def 4;
then A4: Z c= dom cosec 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
( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) )

proof

let x be Real; :: thesis: ( x in Z implies ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) ) )
assume x in Z ; :: thesis: ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) )
then sin . x <> 0 by A4, RFUNCT_1:13;
hence ( cosec is_differentiable_in x & diff cosec ,x = - ((cos . x) / ((sin . x) ^2 )) ) by Th2; :: thesis: verum

end;

then for x being Real st x in Z holds
cosec is_differentiable_in x ;
then A7: cosec is_differentiable_on Z by A4, FDIFF_1:16;
for x being Real st x in Z holds
((exp_R (#) cosec ) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) cosec ) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2 )) )
assume A8: x in Z ; :: thesis: ((exp_R (#) cosec ) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2 ))
then ((exp_R (#) cosec ) `| Z) . x = ((cosec . x) * (diff exp_R ,x)) + ((exp_R . x) * (diff cosec ,x)) by A2, A5, A7, FDIFF_1:29
.= ((cosec . x) * (exp_R . x)) + ((exp_R . x) * (diff cosec ,x)) by SIN_COS:70
.= ((cosec . x) * (exp_R . x)) + ((exp_R . x) * (- ((cos . x) / ((sin . x) ^2 )))) by A6, A8
.= ((exp_R . x) / (sin . x)) + (((exp_R . x) * (- (cos . x))) / ((sin . x) ^2 )) by A4, A8, RFUNCT_1:def 8 ;
hence ((exp_R (#) cosec ) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2 )) ; :: thesis: verum

end;

hence ( exp_R (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) cosec ) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2 )) ) ) by A2, A5, A7, FDIFF_1:29; :: thesis: verum