let Z be open Subset of REAL; :: thesis:
for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
then A1: sin is_differentiable_on Z by FDIFF_1:9, SIN_COS:24;
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 sin) /\ (dom (arctan + arccot)) by SIN_COS:24, XBOOLE_1:19;
then A7: Z c= dom (sin (#) (arctan + arccot)) by VALUED_1:def 4;
for x being Real st x in Z holds
((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x))

proof

let x be Real; :: thesis:
assume A8: x in Z ; :: thesis:
then ((sin (#) (arctan + arccot)) `| Z) . x = (((arctan + arccot) . x) * (diff (sin,x))) + ((sin . x) * (diff ((arctan + arccot),x))) by A7, A1, A3, FDIFF_1:21
.= (((arctan . x) + (arccot . x)) * (diff (sin,x))) + ((sin . x) * (diff ((arctan + arccot),x))) by A6, A8, VALUED_1:def 1
.= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * (diff ((arctan + arccot),x))) by SIN_COS:64
.= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * (((arctan + arccot) `| Z) . x)) by A3, A8, FDIFF_1:def 7
.= (((arctan . x) + (arccot . x)) * (cos . x)) + ((sin . x) * 0) by A2, A8, Th37
.= (cos . x) * ((arctan . x) + (arccot . x)) ;
hence ((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ; :: thesis:

end;

hence ( sin (#) (arctan + arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) (arctan + arccot)) `| Z) . x = (cos . x) * ((arctan . x) + (arccot . x)) ) ) by A7, A1, A3, FDIFF_1:21; :: thesis: