let Z be open Subset of REAL ; :: thesis: ( Z c= dom (- cosec ) implies ( - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((- cosec ) `| Z) . x = (cos . x) / ((sin . x) ^2 ) ) ) )
A1: sin is_differentiable_on Z by FDIFF_1:34, SIN_COS:73;
assume A2: Z c= dom (- cosec ) ; :: thesis: ( - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((- cosec ) `| Z) . x = (cos . x) / ((sin . x) ^2 ) ) )
then A3: Z c= dom cosec by VALUED_1:8;
then for x being Real st x in Z holds
sin . x <> 0 by RFUNCT_1:13;
then A4: cosec is_differentiable_on Z by A1, FDIFF_2:22;
A5: for x being Real st x in Z holds
((- cosec ) `| Z) . x = (cos . x) / ((sin . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies ((- cosec ) `| Z) . x = (cos . x) / ((sin . x) ^2 ) )
assume A6: x in Z ; :: thesis: ((- cosec ) `| Z) . x = (cos . x) / ((sin . x) ^2 )
then A7: ( sin . x <> 0 & sin is_differentiable_in x ) by A3, A1, FDIFF_1:16, RFUNCT_1:13;
((- cosec ) `| Z) . x = (- 1) * (diff (sin ^ ),x) by A2, A4, A6, FDIFF_1:28
.= (- 1) * (- ((diff sin ,x) / ((sin . x) ^2 ))) by A7, FDIFF_2:15
.= (cos . x) / ((sin . x) ^2 ) by SIN_COS:69 ;
hence ((- cosec ) `| Z) . x = (cos . x) / ((sin . x) ^2 ) ; :: thesis: verum

end;

(- 1) (#) cosec is_differentiable_on Z by A2, A4, FDIFF_1:28;
hence ( - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((- cosec ) `| Z) . x = (cos . x) / ((sin . x) ^2 ) ) ) by A5; :: thesis: verum