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 = (cot . 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 = (cot . 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, FDIFF_8:2;
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 = (cot . x) ^2

proof

let x be Real; :: thesis: ( x in Z implies (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2 )
assume A10: x in Z ; :: thesis: (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2
then A11: sin . x <> 0 by A5, FDIFF_8:2;
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) by XCMPLX_1:121
.= ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2)) / ((sin . x) ^2) by SIN_COS:31
.= ((cos x) / (sin x)) * ((cos . x) / (sin . x)) by XCMPLX_1:77
.= (cot . x) * (cot x) by A5, A10, FDIFF_8:2, SIN_COS9:16
.= (cot . x) ^2 by A5, A10, FDIFF_8:2, SIN_COS9:16 ;
hence (((- cot) - (id Z)) `| Z) . x = (cot . 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 = (cot . x) ^2 ) ) by A2, A9, A7, FDIFF_1:27; :: thesis: verum