let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cot * exp_R ) implies ( cot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * exp_R ) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) ) ) )
assume A1: Z c= dom (cot * exp_R ) ; :: thesis: ( cot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * exp_R ) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) ) )
A2: for x being Real st x in Z holds
sin . (exp_R . x) <> 0

proof

let x be Real; :: thesis: ( x in Z implies sin . (exp_R . x) <> 0 )
assume x in Z ; :: thesis: sin . (exp_R . x) <> 0
then exp_R . x in dom (cos / sin ) by A1, FUNCT_1:21;
hence sin . (exp_R . x) <> 0 by Th2; :: thesis: verum

end;

A3: for x being Real st x in Z holds
cot * exp_R is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cot * exp_R is_differentiable_in x )
assume x in Z ; :: thesis: cot * exp_R is_differentiable_in x
then sin . (exp_R . x) <> 0 by A2;
then ( exp_R is_differentiable_in x & cot is_differentiable_in exp_R . x ) by FDIFF_7:47, SIN_COS:70;
hence cot * exp_R is_differentiable_in x by FDIFF_2:13; :: thesis: verum

end;

then A4: cot * exp_R is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((cot * exp_R ) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((cot * exp_R ) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) )
assume A5: x in Z ; :: thesis: ((cot * exp_R ) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 ))
then A6: sin . (exp_R . x) <> 0 by A2;
then ( exp_R is_differentiable_in x & cot is_differentiable_in exp_R . x ) by FDIFF_7:47, SIN_COS:70;
then diff (cot * exp_R ),x = (diff cot ,(exp_R . x)) * (diff exp_R ,x) by FDIFF_2:13
.= (- (1 / ((sin . (exp_R . x)) ^2 ))) * (diff exp_R ,x) by A6, FDIFF_7:47
.= - ((diff exp_R ,x) / ((sin . (exp_R . x)) ^2 ))
.= - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) by SIN_COS:70 ;
hence ((cot * exp_R ) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) by A4, A5, FDIFF_1:def 8; :: thesis: verum

end;

hence ( cot * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * exp_R ) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2 )) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum