let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((- cosec ) - (id Z)) implies ( (- cosec ) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cosec ) - (id Z)) `| Z) . x = ((cos . x) - ((sin . x) ^2 )) / ((sin . x) ^2 ) ) ) )
set f = - cosec ;
A1: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
assume A2: Z c= dom ((- cosec ) - (id Z)) ; :: thesis: ( (- cosec ) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cosec ) - (id Z)) `| Z) . x = ((cos . x) - ((sin . x) ^2 )) / ((sin . x) ^2 ) ) )
then A3: Z c= (dom (- cosec )) /\ (dom (id Z)) by VALUED_1:12;
then A4: Z c= dom (- cosec ) by XBOOLE_1:18;
then A5: Z c= dom cosec by VALUED_1:8;
for x being Real st x in Z holds
cosec is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cosec is_differentiable_in x )
assume x in Z ; :: thesis: cosec is_differentiable_in x
then sin . x <> 0 by A5, RFUNCT_1:13;
hence cosec is_differentiable_in x by Th2; :: thesis: verum

end;

then A6: cosec is_differentiable_on Z by A5, FDIFF_1:16;
then A7: (- 1) (#) cosec is_differentiable_on Z by A4, FDIFF_1:28;
A8: Z c= dom (id Z) by A3, XBOOLE_1:18;
then A9: id Z is_differentiable_on Z by A1, FDIFF_1:31;
for x being Real st x in Z holds
(((- cosec ) - (id Z)) `| Z) . x = ((cos . x) - ((sin . x) ^2 )) / ((sin . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies (((- cosec ) - (id Z)) `| Z) . x = ((cos . x) - ((sin . x) ^2 )) / ((sin . x) ^2 ) )
assume A10: x in Z ; :: thesis: (((- cosec ) - (id Z)) `| Z) . x = ((cos . x) - ((sin . x) ^2 )) / ((sin . x) ^2 )
then A11: sin . x <> 0 by A5, RFUNCT_1:13;
then A12: (sin . x) ^2 > 0 by SQUARE_1:74;
(((- cosec ) - (id Z)) `| Z) . x = (diff (- cosec ),x) - (diff (id Z),x) by A2, A9, A7, A10, FDIFF_1:27
.= ((((- 1) (#) cosec ) `| Z) . x) - (diff (id Z),x) by A7, A10, FDIFF_1:def 8
.= ((- 1) * (diff cosec ,x)) - (diff (id Z),x) by A4, A6, A10, FDIFF_1:28
.= ((- 1) * (- ((cos . x) / ((sin . x) ^2 )))) - (diff (id Z),x) by A11, Th2
.= ((cos . x) / ((sin . x) ^2 )) - (((id Z) `| Z) . x) by A9, A10, FDIFF_1:def 8
.= ((cos . x) / ((sin . x) ^2 )) - 1 by A8, A1, A10, FDIFF_1:31
.= ((cos . x) / ((sin . x) ^2 )) - (((sin . x) ^2 ) / ((sin . x) ^2 )) by A12, XCMPLX_1:60
.= ((cos . x) - ((sin . x) ^2 )) / ((sin . x) ^2 ) ;
hence (((- cosec ) - (id Z)) `| Z) . x = ((cos . x) - ((sin . x) ^2 )) / ((sin . x) ^2 ) ; :: thesis: verum

end;

hence ( (- cosec ) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cosec ) - (id Z)) `| Z) . x = ((cos . x) - ((sin . x) ^2 )) / ((sin . x) ^2 ) ) ) by A2, A9, A7, FDIFF_1:27; :: thesis: verum