let Z be open Subset of REAL; :: thesis: ( Z c= dom (exp_R (#) cot) implies ( exp_R (#) cot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) cot) `| Z) . x = (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin . 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 (#) cot) ; :: thesis: ( exp_R (#) cot is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) cot) `| Z) . x = (((exp_R . x) * (cos . x)) / (sin . x)) - ((exp_R . x) / ((sin . x) ^2)) ) )
then A3: Z c= (dom exp_R) /\ (dom cot) by VALUED_1:def 4;
then A4: Z c= dom cot 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
( cot is_differentiable_in x & diff (cot,x) = - (1 / ((sin . x) ^2)) )

proof

let x be Real; :: thesis: ( x in Z implies ( cot is_differentiable_in x & diff (cot,x) = - (1 / ((sin . x) ^2)) ) )
assume x in Z ; :: thesis: ( cot is_differentiable_in x & diff (cot,x) = - (1 / ((sin . x) ^2)) )
then sin . x <> 0 by A4, Th2;
hence ( cot is_differentiable_in x & diff (cot,x) = - (1 / ((sin . x) ^2)) ) by FDIFF_7:47; :: thesis: verum

end;

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

proof

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

end;

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