let Z be open Subset of REAL ; :: thesis: ( Z c= dom (exp_R * arctan ) & 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 . (arctan . x)) / (1 + (x ^2 )) ) ) )
assume that
A1:
Z c= dom (exp_R * arctan )
and
A2:
Z c= ].(- 1),1.[
; :: thesis: ( exp_R * arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 )) ) )
A3:
for x being Real st x in Z holds
exp_R * arctan is_differentiable_in x
then A6:
exp_R * arctan is_differentiable_on Z
by A1, FDIFF_1:16;
for x being Real st x in Z holds
((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 ))
proof
let x be
Real;
:: thesis: ( x in Z implies ((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 )) )
assume A7:
x in Z
;
:: thesis: ((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 ))
A8:
arctan is_differentiable_on Z
by A2, Th79;
then A9:
arctan is_differentiable_in x
by A7, FDIFF_1:16;
exp_R is_differentiable_in arctan . x
by SIN_COS:70;
then diff (exp_R * arctan ),
x =
(diff exp_R ,(arctan . x)) * (diff arctan ,x)
by A9, FDIFF_2:13
.=
(diff exp_R ,(arctan . x)) * ((arctan `| Z) . x)
by A7, A8, FDIFF_1:def 8
.=
(diff exp_R ,(arctan . x)) * (1 / (1 + (x ^2 )))
by A2, A7, Th79
.=
(exp_R . (arctan . x)) / (1 + (x ^2 ))
by SIN_COS:70
;
hence
((exp_R * arctan ) `| Z) . x = (exp_R . (arctan . x)) / (1 + (x ^2 ))
by A6, A7, FDIFF_1:def 8;
:: thesis: 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 . (arctan . x)) / (1 + (x ^2 )) ) )
by A1, A3, FDIFF_1:16; :: thesis: verum