let Z be open Subset of REAL ; :: thesis:
for x being Real st x in Z holds
cos is_differentiable_in x by SIN_COS:68;
then A1: cos is_differentiable_on Z by FDIFF_1:16, SIN_COS:27;
assume A2: Z c= ].(- 1),1.[ ; :: thesis:
then A3: arctan + arccot is_differentiable_on Z by Th37;
A4: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A5: Z c= dom arccot by A2, XBOOLE_1:1;
].(- 1),1.[ c= dom arctan by A4, SIN_COS9:23, XBOOLE_1:1;
then Z c= dom arctan by A2, XBOOLE_1:1;
then Z c= (dom arctan ) /\ (dom arccot ) by A5, XBOOLE_1:19;
then A6: Z c= dom (arctan + arccot ) by VALUED_1:def 1;
then Z c= (dom cos ) /\ (dom (arctan + arccot )) by SIN_COS:27, XBOOLE_1:19;
then A7: Z c= dom (cos (#) (arctan + arccot )) by VALUED_1:def 4;
for x being Real st x in Z holds
((cos (#) (arctan + arccot )) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x)))

proof

let x be Real; :: thesis:
assume A8: x in Z ; :: thesis:
then ((cos (#) (arctan + arccot )) `| Z) . x = (((arctan + arccot ) . x) * (diff cos ,x)) + ((cos . x) * (diff (arctan + arccot ),x)) by A7, A1, A3, FDIFF_1:29
.= (((arctan . x) + (arccot . x)) * (diff cos ,x)) + ((cos . x) * (diff (arctan + arccot ),x)) by A6, A8, VALUED_1:def 1
.= (((arctan . x) + (arccot . x)) * (- (sin . x))) + ((cos . x) * (diff (arctan + arccot ),x)) by SIN_COS:68
.= (((arctan . x) + (arccot . x)) * (- (sin . x))) + ((cos . x) * (((arctan + arccot ) `| Z) . x)) by A3, A8, FDIFF_1:def 8
.= (- (((arctan . x) + (arccot . x)) * (sin . x))) + ((cos . x) * 0 ) by A2, A8, Th37
.= - ((sin . x) * ((arctan . x) + (arccot . x))) ;
hence ((cos (#) (arctan + arccot )) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ; :: thesis:

end;

hence ( cos (#) (arctan + arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
((cos (#) (arctan + arccot )) `| Z) . x = - ((sin . x) * ((arctan . x) + (arccot . x))) ) ) by A7, A1, A3, FDIFF_1:29; :: thesis: