let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln * arctan) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x > 0 ) implies ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) ) )
assume that
A1: Z c= dom (ln * arctan) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arctan . x > 0 ; :: thesis: ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) )
A4: for x being Real st x in Z holds
ln * arctan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * arctan is_differentiable_in x )
assume A5: x in Z ; :: thesis: ln * arctan is_differentiable_in x
arctan is_differentiable_on Z by A2, Th81;
then A6: arctan is_differentiable_in x by A5, FDIFF_1:9;
arctan . x > 0 by A3, A5;
hence ln * arctan is_differentiable_in x by A6, TAYLOR_1:20; :: thesis: verum

end;

then A7: ln * arctan is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x))

proof

let x be Real; :: thesis: ( x in Z implies ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) )
assume A8: x in Z ; :: thesis: ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x))
then A9: - 1 < x by A2, XXREAL_1:4;
arctan is_differentiable_on Z by A2, Th81;
then A10: arctan is_differentiable_in x by A8, FDIFF_1:9;
A11: x < 1 by A2, A8, XXREAL_1:4;
arctan . x > 0 by A3, A8;
then diff ((ln * arctan),x) = (diff (arctan,x)) / (arctan . x) by A10, TAYLOR_1:20
.= (1 / (1 + (x ^2))) / (arctan . x) by A9, A11, Th75
.= 1 / ((1 + (x ^2)) * (arctan . x)) by XCMPLX_1:78 ;
hence ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) by A7, A8, FDIFF_1:def 7; :: thesis: verum

end;

hence ( ln * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum