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_1:18;
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 Z c= (dom (- cot)) /\ (dom (id Z)) by VALUED_1:12;
then A3: Z c= dom (- cot) by XBOOLE_1:18;
then A4: 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 A4, FDIFF_8:2;
hence cot is_differentiable_in x by FDIFF_7:47; :: thesis: verum

end;

then A5: cot is_differentiable_on Z by A4, FDIFF_1:9;
then A6: (- 1) (#) cot is_differentiable_on Z by A3, FDIFF_1:20;
A7: Z c= dom (id Z) ;
then A8: id Z is_differentiable_on Z by A1, FDIFF_1:23;
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 A9: x in Z ; :: thesis: (((- cot) - (id Z)) `| Z) . x = (cot . x) ^2
then A10: sin . x <> 0 by A4, FDIFF_8:2;
then A11: (sin . x) ^2 > 0 by SQUARE_1:12;
(((- cot) - (id Z)) `| Z) . x = (diff ((- cot),x)) - (diff ((id Z),x)) by A2, A8, A6, A9, FDIFF_1:19
.= ((((- 1) (#) cot) `| Z) . x) - (diff ((id Z),x)) by A6, A9, FDIFF_1:def 7
.= ((- 1) * (diff (cot,x))) - (diff ((id Z),x)) by A3, A5, A9, FDIFF_1:20
.= ((- 1) * (- (1 / ((sin . x) ^2)))) - (diff ((id Z),x)) by A10, FDIFF_7:47
.= (1 / ((sin . x) ^2)) - (((id Z) `| Z) . x) by A8, A9, FDIFF_1:def 7
.= (1 / ((sin . x) ^2)) - 1 by A7, A1, A9, FDIFF_1:23
.= (1 / ((sin . x) ^2)) - (((sin . x) ^2) / ((sin . x) ^2)) by A11, XCMPLX_1:60
.= (1 - ((sin . x) ^2)) / ((sin . x) ^2) by XCMPLX_1:120
.= ((((cos . x) ^2) + ((sin . x) ^2)) - ((sin . x) ^2)) / ((sin . x) ^2) by SIN_COS:28
.= ((cos x) / (sin x)) * ((cos . x) / (sin . x)) by XCMPLX_1:76
.= (cot . x) * (cot x) by A4, A9, FDIFF_8:2, SIN_COS9:16
.= (cot . x) ^2 by A4, A9, 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, A8, A6, FDIFF_1:19; :: thesis: verum