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

proof

let x be Real; :: thesis: ( x in Z implies cot is_differentiable_in x )
assume x in Z ; :: thesis: cot is_differentiable_in x
then sin . x <> 0 by A5, Th2;
hence cot is_differentiable_in x by FDIFF_7:47; :: thesis: verum

end;

then A6: cot is_differentiable_on Z by A5, FDIFF_1:16;
then A7: (- 1) (#) cot 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
(((- cot ) - (id Z)) `| Z) . x = ((cos . x) ^2 ) / ((sin . x) ^2 )

proof

let x be Real; :: thesis: ( x in Z implies (((- cot ) - (id Z)) `| Z) . x = ((cos . x) ^2 ) / ((sin . x) ^2 ) )
assume A10: x in Z ; :: thesis: (((- cot ) - (id Z)) `| Z) . x = ((cos . x) ^2 ) / ((sin . x) ^2 )
then A11: sin . x <> 0 by A5, Th2;
then A12: (sin . x) ^2 > 0 by SQUARE_1:74;
(((- cot ) - (id Z)) `| Z) . x = (diff (- cot ),x) - (diff (id Z),x) by A2, A9, A7, A10, FDIFF_1:27
.= ((((- 1) (#) cot ) `| Z) . x) - (diff (id Z),x) by A7, A10, FDIFF_1:def 8
.= ((- 1) * (diff cot ,x)) - (diff (id Z),x) by A4, A6, A10, FDIFF_1:28
.= ((- 1) * (- (1 / ((sin . x) ^2 )))) - (diff (id Z),x) by A11, FDIFF_7:47
.= (1 / ((sin . x) ^2 )) - (((id Z) `| Z) . x) by A9, A10, FDIFF_1:def 8
.= (1 / ((sin . x) ^2 )) - 1 by A8, A1, A10, FDIFF_1:31
.= (1 / ((sin . x) ^2 )) - (((sin . x) ^2 ) / ((sin . x) ^2 )) by A12, XCMPLX_1:60
.= (1 - ((sin . x) ^2 )) / ((sin . x) ^2 )
.= ((((cos . x) ^2 ) + ((sin . x) ^2 )) - ((sin . x) ^2 )) / ((sin . x) ^2 ) by SIN_COS:31
.= ((cos . x) ^2 ) / ((sin . x) ^2 ) ;
hence (((- cot ) - (id Z)) `| Z) . x = ((cos . x) ^2 ) / ((sin . x) ^2 ) ; :: thesis: verum

end;

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