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_1:11;
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:65;
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_1:9;
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:65;
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:65 ;
hence ((cot * exp_R) `| Z) . x = - ((exp_R . x) / ((sin . (exp_R . x)) ^2)) by A4, A5, FDIFF_1:def 7; :: 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_1:9; :: thesis: verum