let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arccot )) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot )) `| Z) . x = - ((arccot . x) / (1 + (x ^2 ))) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arccot )) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( (1 / 2) (#) ((#Z 2) * arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot )) `| Z) . x = - ((arccot . x) / (1 + (x ^2 ))) ) )
A3: Z c= dom ((#Z 2) * arccot ) by A1, VALUED_1:def 5;
then A4: (#Z 2) * arccot is_differentiable_on Z by A2, Th92;
for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot )) `| Z) . x = - ((arccot . x) / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arccot )) `| Z) . x = - ((arccot . x) / (1 + (x ^2 ))) )
assume A5: x in Z ; :: thesis: (((1 / 2) (#) ((#Z 2) * arccot )) `| Z) . x = - ((arccot . x) / (1 + (x ^2 )))
then (((1 / 2) (#) ((#Z 2) * arccot )) `| Z) . x = (1 / 2) * (diff ((#Z 2) * arccot ),x) by A1, A4, FDIFF_1:28
.= (1 / 2) * ((((#Z 2) * arccot ) `| Z) . x) by A4, A5, FDIFF_1:def 8
.= (1 / 2) * (- ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2 )))) by A2, A3, A5, Th92
.= - ((1 / 2) * ((2 * ((arccot . x) #Z 1)) / (1 + (x ^2 ))))
.= - ((1 / 2) * ((2 * (arccot . x)) / (1 + (x ^2 )))) by PREPOWER:45
.= - ((arccot . x) / (1 + (x ^2 ))) ;
hence (((1 / 2) (#) ((#Z 2) * arccot )) `| Z) . x = - ((arccot . x) / (1 + (x ^2 ))) ; :: thesis: verum

end;

hence ( (1 / 2) (#) ((#Z 2) * arccot ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccot )) `| Z) . x = - ((arccot . x) / (1 + (x ^2 ))) ) ) by A1, A4, FDIFF_1:28; :: thesis: verum