let Z be open Subset of REAL ; :: thesis: ( Z c= dom cosec & Z c= ].(- 1),1.[ implies ( cosec (#) (arctan + arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot )) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 )) ) ) )
assume that
A1: Z c= dom cosec and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cosec (#) (arctan + arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot )) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 )) ) )
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arctan by SIN_COS9:23, XBOOLE_1:1;
then A4: Z c= dom arctan by A2, XBOOLE_1:1;
].(- 1),1.[ c= dom arccot by A3, SIN_COS9:24, XBOOLE_1:1;
then Z c= dom arccot by A2, XBOOLE_1:1;
then Z c= (dom arctan ) /\ (dom arccot ) by A4, XBOOLE_1:19;
then A5: Z c= dom (arctan + arccot ) by VALUED_1:def 1;
then Z c= (dom cosec ) /\ (dom (arctan + arccot )) by A1, XBOOLE_1:19;
then A6: Z c= dom (cosec (#) (arctan + arccot )) by VALUED_1:def 4;
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 A1, RFUNCT_1:13;
hence cosec is_differentiable_in x by FDIFF_9:2; :: thesis: verum

end;

then A7: cosec is_differentiable_on Z by A1, FDIFF_1:16;
A8: ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan + arccot ) `| Z) . x = 0 ) ) by A2, Th37;
for x being Real st x in Z holds
((cosec (#) (arctan + arccot )) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((cosec (#) (arctan + arccot )) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 )) )
assume A9: x in Z ; :: thesis: ((cosec (#) (arctan + arccot )) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 ))
then A10: sin . x <> 0 by A1, RFUNCT_1:13;
((cosec (#) (arctan + arccot )) `| Z) . x = (((arctan + arccot ) . x) * (diff cosec ,x)) + ((cosec . x) * (diff (arctan + arccot ),x)) by A6, A7, A8, A9, FDIFF_1:29
.= (((arctan . x) + (arccot . x)) * (diff cosec ,x)) + ((cosec . x) * (diff (arctan + arccot ),x)) by A5, A9, VALUED_1:def 1
.= (((arctan . x) + (arccot . x)) * (- ((cos . x) / ((sin . x) ^2 )))) + ((cosec . x) * (diff (arctan + arccot ),x)) by A10, FDIFF_9:2
.= (- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 ))) + ((cosec . x) * (((arctan + arccot ) `| Z) . x)) by A8, A9, FDIFF_1:def 8
.= (- ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 ))) + ((cosec . x) * 0 ) by A2, A9, Th37
.= - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 )) ;
hence ((cosec (#) (arctan + arccot )) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 )) ; :: thesis: verum

end;

hence ( cosec (#) (arctan + arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cosec (#) (arctan + arccot )) `| Z) . x = - ((((arctan . x) + (arccot . x)) * (cos . x)) / ((sin . x) ^2 )) ) ) by A6, A7, A8, FDIFF_1:29; :: thesis: verum