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 )) ) ) )
assume A1: 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 A2: Z c= (dom exp_R ) /\ (dom cot ) by VALUED_1:def 4;
then A3: Z c= dom exp_R by XBOOLE_1:18;
A4: Z c= dom cot by A2, XBOOLE_1:18;
for x being Real st x in Z holds
exp_R is_differentiable_in x by SIN_COS:70;
then A5: exp_R is_differentiable_on Z by A3, FDIFF_1:16;
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_1:16;
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 A1, A5, A7, FDIFF_1:29
.= ((cot . x) * (exp_R . x)) + ((exp_R . x) * (diff cot ,x)) by SIN_COS:70
.= ((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 4
.= (((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 A1, A5, A7, FDIFF_1:29; :: thesis: verum