let Z be open Subset of REAL; :: thesis: ( Z c= dom ((- cot) - cosec) & ( for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ) implies ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) ) )
assume that
A1: Z c= dom ((- cot) - cosec) and
A2: for x being Real st x in Z holds
( 1 + (cos . x) <> 0 & 1 - (cos . x) <> 0 ) ; :: thesis: ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) )
Z c= (dom (- cot)) /\ (dom (sin ^)) by A1, 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: for x being Real st x in Z holds
sin . x <> 0 by A4, FDIFF_8:2;
then A8: sin ^ is_differentiable_on Z by FDIFF_4:40;
for x being Real st x in Z holds
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x))

proof

let x be Real; :: thesis: ( x in Z implies (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) )
assume A9: x in Z ; :: thesis: (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x))
then A10: sin . x <> 0 by A4, FDIFF_8:2;
A11: 1 + (cos . x) <> 0 by A2, A9;
(((- cot) - cosec) `| Z) . x = (diff ((- cot),x)) - (diff ((sin ^),x)) by A1, A8, A6, A9, FDIFF_1:19
.= ((((- 1) (#) cot) `| Z) . x) - (diff ((sin ^),x)) by A6, A9, FDIFF_1:def 7
.= ((- 1) * (diff (cot,x))) - (diff ((sin ^),x)) by A3, A5, A9, FDIFF_1:20
.= ((- 1) * (- (1 / ((sin . x) ^2)))) - (diff ((sin ^),x)) by A10, FDIFF_7:47
.= (1 / ((sin . x) ^2)) - (((sin ^) `| Z) . x) by A8, A9, FDIFF_1:def 7
.= (1 / ((sin . x) ^2)) - (- ((cos . x) / ((sin . x) ^2))) by A7, A9, FDIFF_4:40
.= (1 / ((sin . x) ^2)) + ((cos . x) / ((sin . x) ^2))
.= (1 + (cos . x)) / ((((sin . x) ^2) + ((cos . x) ^2)) - ((cos . x) ^2)) by XCMPLX_1:62
.= (1 + (cos . x)) / (1 - ((cos . x) ^2)) by SIN_COS:28
.= (1 + (cos . x)) / ((1 + (cos . x)) * (1 - (cos . x)))
.= ((1 + (cos . x)) / (1 + (cos . x))) / (1 - (cos . x)) by XCMPLX_1:78
.= 1 / (1 - (cos . x)) by A11, XCMPLX_1:60 ;
hence (((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ; :: thesis: verum

end;

hence ( (- cot) - cosec is_differentiable_on Z & ( for x being Real st x in Z holds
(((- cot) - cosec) `| Z) . x = 1 / (1 - (cos . x)) ) ) by A1, A8, A6, FDIFF_1:19; :: thesis: verum