let Z be open Subset of REAL; ( 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
; ( 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
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;
( x in Z implies ((ln * arctan) `| Z) . x = 1 / ((1 + (x ^2)) * (arctan . x)) )
assume A8:
x in Z
;
((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;
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; verum