let Z be open Subset of REAL; :: thesis: ( Z c= dom ((#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 A1: ( Z c= dom ((#Z 2) * arccot) & 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)) ) )
then A2: Z c= dom ((1 / 2) (#) ((#Z 2) * arccot)) by VALUED_1:def 5;
then A3: Z c= dom (- ((1 / 2) (#) ((#Z 2) * arccot))) by VALUED_1:8;
A4: (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_on Z by A2, A1, SIN_COS9:94;
then A5: (- 1) (#) ((1 / 2) (#) ((#Z 2) * arccot)) is_differentiable_on Z by A3, FDIFF_1:20;
A6: ( (#Z 2) * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * arccot) `| Z) . x = - ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2))) ) ) by A1, SIN_COS9:92;
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 A7: x in Z ; :: thesis: ((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = (arccot . x) / (1 + (x ^2))
then A8: (1 / 2) (#) ((#Z 2) * arccot) is_differentiable_in x by A4, FDIFF_1:9;
A9: (#Z 2) * arccot is_differentiable_in x by A6, A7, FDIFF_1:9;
((- ((1 / 2) (#) ((#Z 2) * arccot))) `| Z) . x = diff ((- ((1 / 2) (#) ((#Z 2) * arccot))),x) by A5, A7, FDIFF_1:def 7
.= (- 1) * (diff (((1 / 2) (#) ((#Z 2) * arccot)),x)) by A8, FDIFF_1:15
.= (- 1) * ((1 / 2) * (diff (((#Z 2) * arccot),x))) by A9, FDIFF_1:15
.= (- 1) * ((1 / 2) * ((((#Z 2) * arccot) `| Z) . x)) by A6, A7, FDIFF_1:def 7
.= (- 1) * ((1 / 2) * (- ((2 * ((arccot . x) #Z (2 - 1))) / (1 + (x ^2))))) by A1, A7, SIN_COS9:92
.= (- 1) * (- ((1 / 2) * ((2 * ((arccot . x) #Z 1)) / (1 + (x ^2)))))
.= (- 1) * (- ((1 / 2) * ((2 * (arccot . x)) / (1 + (x ^2))))) by PREPOWER:35
.= (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 A3, A4, FDIFF_1:20; :: thesis: verum