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 . (cot . 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 . (cot . 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 (cos / sin ) by A1, FUNCT_1:21;
hence sin . x <> 0 by Th2; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies exp_R * cot is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * cot is_differentiable_in x
then sin . x <> 0 by A2;
then A4: cot is_differentiable_in x by FDIFF_7:47;
exp_R is_differentiable_in cot . x by SIN_COS:70;
hence exp_R * cot is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

then A5: exp_R * cot is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((exp_R * cot ) `| Z) . x = - ((exp_R . (cot . x)) / ((sin . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * cot ) `| Z) . x = - ((exp_R . (cot . x)) / ((sin . x) ^2 )) )
assume A6: x in Z ; :: thesis: ((exp_R * cot ) `| Z) . x = - ((exp_R . (cot . x)) / ((sin . x) ^2 ))
then A7: sin . x <> 0 by A2;
then A8: cot is_differentiable_in x by FDIFF_7:47;
exp_R is_differentiable_in cot . x by SIN_COS:70;
then diff (exp_R * cot ),x = (diff exp_R ,(cot . x)) * (diff cot ,x) by A8, FDIFF_2:13
.= (diff exp_R ,(cot . x)) * (- (1 / ((sin . x) ^2 ))) by A7, FDIFF_7:47
.= (exp_R . (cot . x)) * (- (1 / ((sin . x) ^2 ))) by SIN_COS:70
.= - ((exp_R . (cot . x)) / ((sin . x) ^2 )) ;
hence ((exp_R * cot ) `| Z) . x = - ((exp_R . (cot . x)) / ((sin . x) ^2 )) by A5, A6, FDIFF_1:def 8; :: 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 . (cot . x)) / ((sin . x) ^2 )) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum