let Z be open Subset of REAL; :: thesis: ( Z c= dom (cosec (#) arctan) & Z c= ].(- 1),1.[ implies ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (cosec (#) arctan) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
Z c= (dom cosec) /\ (dom arctan) by A1, VALUED_1:def 4;
then A4: Z c= dom cosec by XBOOLE_1:18;
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 A4, RFUNCT_1:3;
hence cosec is_differentiable_in x by FDIFF_9:2; :: thesis: verum

end;

then A5: cosec is_differentiable_on Z by A4, FDIFF_1:9;
for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2))))

proof

let x be Real; :: thesis: ( x in Z implies ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) )
assume A6: x in Z ; :: thesis: ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2))))
then A7: sin . x <> 0 by A4, RFUNCT_1:3;
((cosec (#) arctan) `| Z) . x = ((arctan . x) * (diff (cosec,x))) + ((cosec . x) * (diff (arctan,x))) by A1, A5, A3, A6, FDIFF_1:21
.= ((arctan . x) * (- ((cos . x) / ((sin . x) ^2)))) + ((cosec . x) * (diff (arctan,x))) by A7, FDIFF_9:2
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def 7
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((cosec . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + ((1 / (sin . x)) * (1 / (1 + (x ^2)))) by A4, A6, RFUNCT_1:def 2
.= (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) by XCMPLX_1:102 ;
hence ((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ; :: thesis: verum

end;

hence ( cosec (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) arctan) `| Z) . x = (- (((cos . x) * (arctan . x)) / ((sin . x) ^2))) + (1 / ((sin . x) * (1 + (x ^2)))) ) ) by A1, A5, A3, FDIFF_1:21; :: thesis: verum