let Z be open Subset of REAL; :: thesis: ( Z c= dom (cosec * exp_R) implies ( cosec * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * exp_R) `| Z) . x = - (((exp_R . x) * (cos . (exp_R . x))) / ((sin . (exp_R . x)) ^2)) ) ) )
assume A1: Z c= dom (cosec * exp_R) ; :: thesis: ( cosec * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * exp_R) `| Z) . x = - (((exp_R . x) * (cos . (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 cosec by A1, FUNCT_1:11;
hence sin . (exp_R . x) <> 0 by RFUNCT_1:3; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies cosec * exp_R is_differentiable_in x )
assume x in Z ; :: thesis: cosec * exp_R is_differentiable_in x
then sin . (exp_R . x) <> 0 by A2;
then ( exp_R is_differentiable_in x & cosec is_differentiable_in exp_R . x ) by Th2, SIN_COS:65;
hence cosec * exp_R is_differentiable_in x by FDIFF_2:13; :: thesis: verum

end;

then A4: cosec * exp_R is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cosec * exp_R) `| Z) . x = - (((exp_R . x) * (cos . (exp_R . x))) / ((sin . (exp_R . x)) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((cosec * exp_R) `| Z) . x = - (((exp_R . x) * (cos . (exp_R . x))) / ((sin . (exp_R . x)) ^2)) )
assume A5: x in Z ; :: thesis: ((cosec * exp_R) `| Z) . x = - (((exp_R . x) * (cos . (exp_R . x))) / ((sin . (exp_R . x)) ^2))
then A6: sin . (exp_R . x) <> 0 by A2;
then ( exp_R is_differentiable_in x & cosec is_differentiable_in exp_R . x ) by Th2, SIN_COS:65;
then diff ((cosec * exp_R),x) = (diff (cosec,(exp_R . x))) * (diff (exp_R,x)) by FDIFF_2:13
.= (- ((cos . (exp_R . x)) / ((sin . (exp_R . x)) ^2))) * (diff (exp_R,x)) by A6, Th2
.= (exp_R . x) * (- ((cos . (exp_R . x)) / ((sin . (exp_R . x)) ^2))) by SIN_COS:65 ;
hence ((cosec * exp_R) `| Z) . x = - (((exp_R . x) * (cos . (exp_R . x))) / ((sin . (exp_R . x)) ^2)) by A4, A5, FDIFF_1:def 7; :: thesis: verum

end;

hence ( cosec * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec * exp_R) `| Z) . x = - (((exp_R . x) * (cos . (exp_R . x))) / ((sin . (exp_R . x)) ^2)) ) ) by A1, A3, FDIFF_1:9; :: thesis: verum