let Z be open Subset of REAL; :: thesis: ( Z c= dom cot & Z c= ].(- 1),1.[ implies ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom cot and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) )
A3: arctan - arccot is_differentiable_on Z by A2, Th38;
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 A1, FDIFF_8:2;
hence cot is_differentiable_in x by FDIFF_7:47; :: thesis: verum

end;

then A4: cot is_differentiable_on Z by A1, FDIFF_1:9;
A5: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A6: Z c= dom arccot by A2, XBOOLE_1:1;
].(- 1),1.[ c= dom arctan by A5, SIN_COS9:23, XBOOLE_1:1;
then Z c= dom arctan by A2, XBOOLE_1:1;
then Z c= (dom arctan) /\ (dom arccot) by A6, XBOOLE_1:19;
then A7: Z c= dom (arctan - arccot) by VALUED_1:12;
then Z c= (dom cot) /\ (dom (arctan - arccot)) by A1, XBOOLE_1:19;
then A8: Z c= dom (cot (#) (arctan - arccot)) by VALUED_1:def 4;
for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) )
assume A9: x in Z ; :: thesis: ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2)))
then A10: sin . x <> 0 by A1, FDIFF_8:2;
((cot (#) (arctan - arccot)) `| Z) . x = (((arctan - arccot) . x) * (diff (cot,x))) + ((cot . x) * (diff ((arctan - arccot),x))) by A8, A4, A3, A9, FDIFF_1:21
.= (((arctan . x) - (arccot . x)) * (diff (cot,x))) + ((cot . x) * (diff ((arctan - arccot),x))) by A7, A9, VALUED_1:13
.= (((arctan . x) - (arccot . x)) * (- (1 / ((sin . x) ^2)))) + ((cot . x) * (diff ((arctan - arccot),x))) by A10, FDIFF_7:47
.= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (((arctan - arccot) `| Z) . x)) by A3, A9, FDIFF_1:def 7
.= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((cot . x) * (2 / (1 + (x ^2)))) by A2, A9, Th38
.= (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ;
hence ((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ; :: thesis: verum

end;

hence ( cot (#) (arctan - arccot) is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) (arctan - arccot)) `| Z) . x = (- (((arctan . x) - (arccot . x)) / ((sin . x) ^2))) + ((2 * (cot . x)) / (1 + (x ^2))) ) ) by A8, A4, A3, FDIFF_1:21; :: thesis: verum