let Z be open Subset of REAL; :: thesis: ( Z c= dom (cot * arctan) & Z c= ].(- 1),1.[ implies ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) )
assume that
A1: Z c= dom (cot * arctan) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) )
A3: for x being Real st x in Z holds
cot * arctan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cot * arctan is_differentiable_in x )
assume A4: x in Z ; :: thesis: cot * arctan is_differentiable_in x
then arctan . x in dom cot by A1, FUNCT_1:11;
then sin . (arctan . x) <> 0 by FDIFF_8:2;
then A5: cot is_differentiable_in arctan . x by FDIFF_7:47;
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then arctan is_differentiable_in x by A4, FDIFF_1:9;
hence cot * arctan is_differentiable_in x by A5, FDIFF_2:13; :: thesis: verum

end;

then A6: cot * arctan is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))

proof

let x be Real; :: thesis: ( x in Z implies ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) )
assume A7: x in Z ; :: thesis: ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2))))
then arctan . x in dom cot by A1, FUNCT_1:11;
then A8: sin . (arctan . x) <> 0 by FDIFF_8:2;
then A9: cot is_differentiable_in arctan . x by FDIFF_7:47;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A7, FDIFF_1:9;
((cot * arctan) `| Z) . x = diff ((cot * arctan),x) by A6, A7, FDIFF_1:def 7
.= (diff (cot,(arctan . x))) * (diff (arctan,x)) by A11, A9, FDIFF_2:13
.= (- (1 / ((sin . (arctan . x)) ^2))) * (diff (arctan,x)) by A8, FDIFF_7:47
.= - ((1 / ((sin . (arctan . x)) ^2)) * (diff (arctan,x)))
.= - ((1 / ((sin . (arctan . x)) ^2)) * ((arctan `| Z) . x)) by A7, A10, FDIFF_1:def 7
.= - ((1 / ((sin . (arctan . x)) ^2)) * (1 / (1 + (x ^2)))) by A2, A7, SIN_COS9:81
.= - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) by XCMPLX_1:102 ;
hence ((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ; :: thesis: verum

end;

hence ( cot * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((cot * arctan) `| Z) . x = - (1 / (((sin . (arctan . x)) ^2) * (1 + (x ^2)))) ) ) by A1, A3, FDIFF_1:9; :: thesis: verum