let Z be open Subset of REAL ; :: thesis:
assume that
A1: Z c= dom (ln * arccot ) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arccot . x > 0 ; :: thesis:
A4: for x being Real st x in Z holds
ln * arccot is_differentiable_in x

let x be Real; :: thesis:
assume A5: x in Z ; :: thesis:
arccot is_differentiable_on Z by A2, Th80;
then A6: arccot is_differentiable_in x by A5, FDIFF_1:16;
arccot . x > 0 by A3, A5;
hence ln * arccot is_differentiable_in x by A6, TAYLOR_1:20; :: thesis:

end;

then A7: ln * arccot is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((ln * arccot ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (arccot . x)))

let x be Real; :: thesis:
assume A8: x in Z ; :: thesis:
then A9: ( - 1 < x & x < 1 ) by A2, XXREAL_1:4;
arccot is_differentiable_on Z by A2, Th80;
then A10: arccot is_differentiable_in x by A8, FDIFF_1:16;
arccot . x > 0 by A3, A8;
then diff (ln * arccot ),x = (diff arccot ,x) / (arccot . x) by A10, TAYLOR_1:20
.= (- (1 / (1 + (x ^2 )))) / (arccot . x) by A9, Th74
.= - ((1 / (1 + (x ^2 ))) / (arccot . x))
.= - (1 / ((1 + (x ^2 )) * (arccot . x))) by XCMPLX_1:79 ;
hence ((ln * arccot ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (arccot . x))) by A7, A8, FDIFF_1:def 8; :: thesis:

end;

hence ( ln * arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccot ) `| Z) . x = - (1 / ((1 + (x ^2 )) * (arccot . x))) ) ) by A1, A4, FDIFF_1:16; :: thesis: