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

proof

let x be Real; :: thesis: ( x in Z implies sin . x <> 0 )
assume x in Z ; :: thesis: sin . x <> 0
then x in dom cosec by A1, FUNCT_1:11;
hence sin . x <> 0 by RFUNCT_1:3; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies exp_R * cosec is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * cosec is_differentiable_in x
then sin . x <> 0 by A2;
then A4: cosec is_differentiable_in x by Th2;
exp_R is_differentiable_in cosec . x by SIN_COS:65;
hence exp_R * cosec is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: exp_R * cosec is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((exp_R * cosec) `| Z) . x = - (((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * cosec) `| Z) . x = - (((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2)) )
A6: exp_R is_differentiable_in cosec . x by SIN_COS:65;
assume A7: x in Z ; :: thesis: ((exp_R * cosec) `| Z) . x = - (((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2))
then A8: sin . x <> 0 by A2;
then cosec is_differentiable_in x by Th2;
then diff ((exp_R * cosec),x) = (diff (exp_R,(cosec . x))) * (diff (cosec,x)) by A6, FDIFF_2:13
.= (diff (exp_R,(cosec . x))) * (- ((cos . x) / ((sin . x) ^2))) by A8, Th2
.= (exp_R . (cosec . x)) * (- ((cos . x) / ((sin . x) ^2))) by SIN_COS:65 ;
hence ((exp_R * cosec) `| Z) . x = - (((exp_R . (cosec . x)) * (cos . x)) / ((sin . x) ^2)) by A5, A7, FDIFF_1:def 7; :: thesis: verum

end;

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