let Z be open Subset of REAL ; :: thesis: ( Z c= dom (cot (#) arccot ) & Z c= ].(- 1),1.[ implies ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arccot ) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2 ))) - ((cot . x) / (1 + (x ^2 ))) ) ) )
assume that
A1: Z c= dom (cot (#) arccot ) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arccot ) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2 ))) - ((cot . x) / (1 + (x ^2 ))) ) )
Z c= (dom cot ) /\ (dom arccot ) by A1, VALUED_1:def 4;
then A3: Z c= dom cot by XBOOLE_1:18;
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 A3, FDIFF_8:2;
hence cot is_differentiable_in x by FDIFF_7:47; :: thesis: verum

end;

then A4: cot is_differentiable_on Z by A3, FDIFF_1:16;
A5: arccot is_differentiable_on Z by A2, SIN_COS9:82;
for x being Real st x in Z holds
((cot (#) arccot ) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2 ))) - ((cot . x) / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((cot (#) arccot ) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2 ))) - ((cot . x) / (1 + (x ^2 ))) )
assume A6: x in Z ; :: thesis: ((cot (#) arccot ) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2 ))) - ((cot . x) / (1 + (x ^2 )))
then A7: sin . x <> 0 by A3, FDIFF_8:2;
((cot (#) arccot ) `| Z) . x = ((arccot . x) * (diff cot ,x)) + ((cot . x) * (diff arccot ,x)) by A1, A4, A5, A6, FDIFF_1:29
.= ((arccot . x) * (- (1 / ((sin . x) ^2 )))) + ((cot . x) * (diff arccot ,x)) by A7, FDIFF_7:47
.= (- ((arccot . x) / ((sin . x) ^2 ))) + ((cot . x) * ((arccot `| Z) . x)) by A5, A6, FDIFF_1:def 8
.= (- ((arccot . x) / ((sin . x) ^2 ))) + ((cot . x) * (- (1 / (1 + (x ^2 ))))) by A2, A6, SIN_COS9:82
.= (- ((arccot . x) / ((sin . x) ^2 ))) - ((cot . x) / (1 + (x ^2 ))) ;
hence ((cot (#) arccot ) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2 ))) - ((cot . x) / (1 + (x ^2 ))) ; :: thesis: verum

end;

hence ( cot (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((cot (#) arccot ) `| Z) . x = (- ((arccot . x) / ((sin . x) ^2 ))) - ((cot . x) / (1 + (x ^2 ))) ) ) by A1, A4, A5, FDIFF_1:29; :: thesis: verum