let Z be open Subset of REAL; :: thesis: ( Z c= ].(- 1),1.[ implies ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2))) ) ) )
assume A1: Z c= ].(- 1),1.[ ; :: thesis: ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2))) ) )
then A2: arccot is_differentiable_on Z by Th74, FDIFF_1:26;
for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies (arccot `| Z) . x = - (1 / (1 + (x ^2))) )
assume A3: x in Z ; :: thesis: (arccot `| Z) . x = - (1 / (1 + (x ^2)))
then A4: - 1 <= x by A1, XXREAL_1:4;
A5: x <= 1 by A1, A3, XXREAL_1:4;
thus (arccot `| Z) . x = diff (arccot,x) by A2, A3, FDIFF_1:def 7
.= - (1 / (1 + (x ^2))) by A4, A5, Th76 ; :: thesis: verum

end;

hence ( arccot is_differentiable_on Z & ( for x being Real st x in Z holds
(arccot `| Z) . x = - (1 / (1 + (x ^2))) ) ) by A1, Th74, FDIFF_1:26; :: thesis: verum