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_1:18;
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_1:3;
hence cosec is_differentiable_in x by Th2; :: thesis: verum

end;

then A6: cosec is_differentiable_on Z by A5, FDIFF_1:9;
then A7: (- 1) (#) cosec is_differentiable_on Z by A4, FDIFF_1:20;
A8: Z c= dom (id Z) by A3, XBOOLE_1:18;
then A9: id Z is_differentiable_on Z by A1, FDIFF_1:23;
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_1:3;
then A12: (sin . x) ^2 > 0 by SQUARE_1:12;
(((- cosec) - (id Z)) `| Z) . x = (diff ((- cosec),x)) - (diff ((id Z),x)) by A2, A9, A7, A10, FDIFF_1:19
.= ((((- 1) (#) cosec) `| Z) . x) - (diff ((id Z),x)) by A7, A10, FDIFF_1:def 7
.= ((- 1) * (diff (cosec,x))) - (diff ((id Z),x)) by A4, A6, A10, FDIFF_1:20
.= ((- 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 7
.= ((cos . x) / ((sin . x) ^2)) - 1 by A8, A1, A10, FDIFF_1:23
.= ((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_1:19; :: thesis: verum