let Z be open Subset of REAL; ( Z c= ].(- 1),1.[ implies ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) ) )
assume A1:
Z c= ].(- 1),1.[
; ( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) )
then A2:
arctan is_differentiable_on Z
by Th81;
].(- 1),1.[ c= [.(- 1),1.]
by XXREAL_1:25;
then
].(- 1),1.[ c= dom arctan
by Th23;
then
Z c= dom arctan
by A1;
then
Z c= (dom exp_R) /\ (dom arctan)
by SIN_COS:47, XBOOLE_1:19;
then A3:
Z c= dom (exp_R (#) arctan)
by VALUED_1:def 4;
for x being Real st x in Z holds
exp_R is_differentiable_in x
by SIN_COS:65;
then A4:
exp_R is_differentiable_on Z
by FDIFF_1:9, SIN_COS:47;
for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
proof
let x be
Real;
( x in Z implies ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) )
assume A5:
x in Z
;
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
then ((exp_R (#) arctan) `| Z) . x =
((arctan . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff (arctan,x)))
by A3, A4, A2, FDIFF_1:21
.=
((arctan . x) * (exp_R . x)) + ((exp_R . x) * (diff (arctan,x)))
by SIN_COS:65
.=
((exp_R . x) * (arctan . x)) + ((exp_R . x) * ((arctan `| Z) . x))
by A2, A5, FDIFF_1:def 7
.=
((exp_R . x) * (arctan . x)) + ((exp_R . x) * (1 / (1 + (x ^2))))
by A1, A5, Th81
.=
((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
;
hence
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
;
verum
end;
hence
( exp_R (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) ) )
by A3, A4, A2, FDIFF_1:21; verum