let Z be open Subset of REAL; :: thesis: ( Z c= dom ((- arccot) - (id Z)) & Z c= ].(- 1),1.[ implies ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom ((- arccot) - (id Z)) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) )
A3: arccot is_differentiable_on Z by A2, Th82;
A4: Z c= (dom (- arccot)) /\ (dom (id Z)) by A1, VALUED_1:12;
then A5: Z c= dom (id Z) by XBOOLE_1:18;
A6: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
then A7: id Z is_differentiable_on Z by A5, FDIFF_1:23;
A8: Z c= dom ((- 1) (#) arccot) by A4, XBOOLE_1:18;
then A9: - arccot is_differentiable_on Z by A3, FDIFF_1:20;
for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) )
A10: 1 + (x ^2) > 0 by XREAL_1:34, XREAL_1:63;
assume A11: x in Z ; :: thesis: (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2)))
then (((- arccot) - (id Z)) `| Z) . x = (diff ((- arccot),x)) - (diff ((id Z),x)) by A1, A7, A9, FDIFF_1:19
.= (((- arccot) `| Z) . x) - (diff ((id Z),x)) by A9, A11, FDIFF_1:def 7
.= ((- 1) * (diff (arccot,x))) - (diff ((id Z),x)) by A8, A3, A11, FDIFF_1:20
.= ((- 1) * ((arccot `| Z) . x)) - (diff ((id Z),x)) by A3, A11, FDIFF_1:def 7
.= ((- 1) * (- (1 / (1 + (x ^2))))) - (diff ((id Z),x)) by A2, A11, Th82
.= (1 / (1 + (x ^2))) - (((id Z) `| Z) . x) by A7, A11, FDIFF_1:def 7
.= (1 / (1 + (x ^2))) - 1 by A5, A6, A11, FDIFF_1:23
.= (1 / (1 + (x ^2))) - ((1 + (x ^2)) / (1 + (x ^2))) by A10, XCMPLX_1:60
.= - ((x ^2) / (1 + (x ^2))) ;
hence (((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ; :: thesis: verum

end;

hence ( (- arccot) - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- arccot) - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) by A1, A7, A9, FDIFF_1:19; :: thesis: verum